1 . Thread limits are configured at module load time to prevent over-subscription. For cross-platform compatibility, the main script should be wrapped with:"},{"type":"heading","level":2,"text":"4.8 Model persistence"},{"type":"para","text":"Trained models can be saved and loaded for reproducibility:"},{"type":"heading","level":2,"text":"4.9 Tuning the number of iterations"},{"type":"para","text":"As anticipated in Section 3.1 , the number of boosting iterations T T is one of the most influential hyperparameters of the model. The package offers two complementary ways of choosing it: early stopping on a held-out evaluation set, and cross-validation over a grid of candidate values. This section describes each mechanism in turn and explains how they should (and should not) be combined."},{"type":"heading","level":4,"text":"Early stopping and the patience parameter."},{"type":"para","text":"Early stopping requires an evaluation set, passed to fit() through the eval_set argument. After every boosting iteration, the model predicts on this set, records the evaluation MSE, and checks whether the active stopping criterion has been triggered. The patience parameter is an integer that controls how tolerant the procedure is to short-term fluctuations in the evaluation curve: training is halted once the evaluation MSE has failed to improve on its best value for patience consecutive iterations. This tolerance is needed because the evaluation MSE does not always decrease monotonically. A small patience (e.g. 5) reacts quickly to real overfitting, but it can also stop on one of these temporary upticks and treat it as overfitting when it is not, halting in a local minimum of the evaluation curve before training reaches a deeper minimum a few iterations later. A larger value (e.g. 20–50) ignores these small fluctuations and is more likely to find the best model, at the cost of more iterations. Stopping too early is partly offset by restore_best_model (discussed below), which rolls the model back to the lowest-MSE checkpoint; patience still controls how far past the current best the search keeps going. The alternative criterion, target_error , halts training as soon as the evaluation MSE drops to or below a user-specified threshold; it is useful when a problem-specific error target is known a priori. Without an eval_set both criteria are silently ignored and training always runs for the full n_iter iterations. When restore_best_model=True (the default), after stopping the model is rolled back to the checkpoint with the lowest evaluation MSE, so the returned estimator is not the (possibly overfitted) one from the final iteration. To be able to use restore_best_model an evaluation dataset (typically a holdout of the training data) must be passed. Choosing the best checkpoint on it is therefore a model-selection step on validation data, just as in ordinary early stopping for gradient boosting or neural networks. Because this set decides when to stop, its MSE is an optimistic measure of accuracy and should not be reported as the final result: that must come from a separate test set that took no part in training or early stopping, as done in the tutorial (Section 5 ) and benchmark (Appendix C )."},{"type":"heading","level":4,"text":"Cross-validation."},{"type":"para","text":"Cross-validation is supported through the standard sklearn interface: PathBoost can be passed directly to utilities such as GridSearchCV together with a grid over the hyperparameters of interest (typically learning_rate , max_path_length , and, when desired, n_iter ). For each candidate configuration the model is refit on every fold and the validation score is averaged across folds, yielding a fair estimate of generalisation error."},{"type":"heading","level":4,"text":"Combining the two strategies."},{"type":"para","text":"Early stopping and cross-validation should not be combined in a single GridSearchCV call: forwarding an eval_set through fit_params reuses the same validation set across every fold, and different folds stop at different iterations, so the cross-validated score no longer corresponds to a single, well-defined model. We recommend keeping n_iter fixed during cross-validation and applying early stopping only in the final refit."},{"type":"heading","level":1,"text":"5 Tutorial"},{"type":"heading","level":2,"text":"5.1 Installation"},{"type":"para","text":"The package can be installed from the Python Package Index via:"},{"type":"para","text":"Dependencies can be installed with:"},{"type":"heading","level":2,"text":"5.2 Molecular property prediction on the tmQMg dataset"},{"type":"para","text":"We demonstrate the package on the tmQMg dataset [ D2DD00129B ] , a collection of transition metal compounds represented as molecular graphs. In this dataset, atoms are nodes and bonds are edges. Each node carries numerical attributes derived from the periodic table (e.g., atomic number, atomic mass), and edges carry bond-related attributes (e.g., bond strength). The target variable is the HOMO–LUMO gap, a continuous molecular property. The atomic number serves as the anchor attribute."},{"type":"para","text":"We initialize PathBoost with the parameters that will remain constant during cross-validation, for example the fixed number of iterations n_iter , together with the variable-importance settings. The grid search will then vary only learning_rate , max_path_length , and kwargs_for_base_learner . A full description of each parameter is available in Appendix A ."},{"type":"para","text":"Note that error_used controls only how variable importance scores are computed after fitting (MSE vs. MAE when measuring a path’s contribution to error reduction); it has no effect on how the base learners are trained."},{"type":"para","text":"To perform cross validation we use the standard sklearn procedure."},{"type":"para","text":"A compact, self-contained version of the same workflow, without the surrounding commentary, is provided in Appendix B for quick reference."},{"type":"para","text":"As discussed in Section 4.9 , we recommend leaving patience and target_error unset during GridSearchCV , so that each fold is trained for the same fixed number of iterations and the cross-validated scores remain directly comparable. Early stopping is reintroduced in the final training run below."},{"type":"heading","level":2,"text":"5.3 Training"},{"type":"para","text":"For the final training run, the model should be re-initialized with the best hyperparameters and a patience value for early stopping. Here n_iter serves as the maximum number of iterations for each sub-model. This is the strategy we adopt here, but it is not the only one: a common alternative is to fix the other hyperparameters in advance and instead cross-validate the number of iterations directly (Section 4.9 )."},{"type":"para","text":"The best hyperparameters from cross-validation should be included in the initialization. After training, the model’s performance and feature importance can be inspected:"},{"type":"para","text":"The result is shown in Figure 3"},{"type":"heading","level":2,"text":"5.4 Prediction"},{"type":"para","text":"To perform a prediction, input a list of nx.Graph objects:"},{"type":"para","text":"When working directly with SequentialPathBoost , the EBM can be pre-computed once and reused across multiple calls to avoid recomputation:"},{"type":"para","text":"Note that PathBoost.predict() does not expose this shortcut; it always recomputes the EBM internally."},{"type":"heading","level":1,"text":"6 Benchmark comparison"},{"type":"para","text":"We compare PathBoost against two established baselines across six molecular regression tasks. The baselines are the Graph Isomorphism Network with Edge features (GINE) [ hu2020strategies ] and the Weisfeiler–Leman graph kernel paired with support vector regression (WL + SVR) [ Kriege2020 ] . All methods are evaluated using 5-fold cross-validation repeated three times, yielding 15 train/test evaluations per method and dataset. Within each fold the training set is further split 90/10 into a training and a validation subset used for early stopping and, where applicable, hyperparameter selection; the validation subset is never used for final evaluation. Reported metrics (MAE, RMSE, R 2 ) are the mean and standard deviation across the 15 folds, together with the average training time per fold. All experiments were run on CPU."},{"type":"para","text":"PathBoost was applied with fixed hyperparameters, with anchor labels automatically extracted from each training fold as the set of unique atomic numbers present in that fold. The two baselines were instead tuned by grid search on the validation subset of each fold (minimising validation MAE) and then refit on the combined train and validation set using the best configuration. The full hyperparameter grids and fixed settings for all three methods are reported in Appendix C . This benchmark focuses on regression tasks; a comparative study that also covers classification on a broader range of graph datasets is presented in meggio_2nd ."},{"type":"heading","level":2,"text":"6.1 Datasets"},{"type":"heading","level":4,"text":"ESOL."},{"type":"para","text":"ESOL [ wu2018moleculenet ] is a small dataset of 1,128 organic compounds with experimentally measured water solubility expressed as log(mol/L). It is part of the MoleculeNet benchmark collection [ wu2018moleculenet ] ."},{"type":"heading","level":4,"text":"FreeSolv."},{"type":"para","text":"FreeSolv [ wu2018moleculenet ] provides experimental hydration free energies (kcal/mol) for 643 small molecules in water, derived from alchemical free energy calculations."},{"type":"heading","level":4,"text":"QM9."},{"type":"para","text":"QM9 [ ramakrishnan2014quantum ] contains quantum-mechanical properties of approximately 134k stable small organic molecules composed of C, H, O, N, and F atoms with up to 9 heavy atoms, computed at the B3LYP/6-31G(2df,p) level of theory. The target property is the HOMO–LUMO gap. Due to its size, we use a random subsample of 10,000 graphs."},{"type":"heading","level":4,"text":"tmQMg."},{"type":"para","text":"The tmQMg dataset [ D2DD00129B ] is a collection of mononuclear transition metal complexes extracted from the Cambridge Structural Database, represented as molecular graphs where atoms are nodes and bonds are edges.From the u-NatQG variant we consider three regression targets, each treated as a separate task:"},{"type":"para","text":"dipole moment;"},{"type":"para","text":"HOMO energy;"},{"type":"para","text":"polarizability."},{"type":"para","text":"Due to the size of the dataset, each target uses a random subsample of 5,000 graphs."},{"type":"heading","level":2,"text":"6.2 Results"},{"type":"para","text":"Table 2 reports the benchmark results. PathBoost achieves the best performance on five of the six tasks. On the two small organic datasets, ESOL and FreeSolv, PathBoost outperforms both baselines by a clear margin across all metrics, suggesting that path-based features are particularly effective when training data is limited. On QM9, GINE performs best, which is consistent with the known strength of graph neural networks on large, homogeneous organic datasets where message-passing can exploit rich local chemical environments. On the three tmQMg targets, PathBoost performs best on all the considered tasks, with especially strong results on polarizability ( R 2 = 0.928 R^{2}=0.928 ) and HOMO energy ( R 2 = 0.584 R^{2}=0.584 ), while all methods struggle on the dipole moment target, reflecting the intrinsic difficulty of predicting this property from graph topology alone."},{"type":"para","text":"Regarding computational cost, GINE is substantially slower than both PathBoost and WL + SVR across all datasets, with training times up to an order of magnitude larger on the tmQMg tasks. Note that all methods were run on CPU; GPU acceleration would be expected to reduce GINE’s training time considerably. PathBoost occupies a middle ground, being faster than GINE while achieving better predictive performance on most tasks."},{"type":"heading","level":1,"text":"7 Summary and discussion"},{"type":"para","text":"PathBoost, introduced in meggio2024pathboost and meggio_2nd , is a gradient boosting framework for learning from graph-structured data. Its distinguishing feature is interpretability: predictions are decomposed into contributions from labelled paths through the input graphs, and the importance of each path can be quantified after training. While previous works introduced the method and released code for their own experiments, they did not provide a general-purpose easy-to-use implementation."},{"type":"para","text":"The path_boost package presented in this paper fills this gap. It provides a tested, open-source, scikit-learn -compatible reference implementation supporting both regression and binary classification, multiple anchor labels, parallel training across sub-models, and early stopping. Two variable importance criteria, absolute and relative with an optional correlation adjustment, let users identify which labelled paths drive the model’s predictions; this matters in scientific applications such as computational chemistry, where understanding why a prediction was made is often as important as the prediction itself. By following scikit-learn conventions and using networkx for graph handling, the package also integrates naturally with existing data science workflows, enabling cross-validation via GridSearchCV and custom scores, without requiring any custom plumbing on the user’s side."},{"type":"heading","level":1,"text":"Computational details"},{"type":"para","text":"The results in this paper were obtained using Python 3.12 with the following key packages: path_boost 2.1.0, scikit-learn 1.6.1 [ pedregosa2011scikit ] , networkx 3.4.2 [ hagberg2008exploring ] , pandas 2.2.3 [ mckinney2010data ] , numpy 2.1.3 [ harris2020array ] , and matplotlib 3.9.3 [ hunter2007matplotlib ] ."},{"type":"heading","level":1,"text":"Statements and Declarations"},{"type":"heading","level":4,"text":"Ethics approval and consent to participate."},{"type":"para","text":"Not applicable."},{"type":"heading","level":4,"text":"Code availability."},{"type":"para","text":"The path_boost package is open-source and freely available. The source code is hosted on GitHub at https://github.com/Claudio-Me/extended_path_boost , and releases are distributed on the Python Package Index (PyPI) at https://pypi.org/project/path-boost/ . The version used to produce the results in this paper is reported in the Computational details section."},{"type":"para","text":"This appendix provides a summary of the main classes and their parameters."},{"type":"heading","level":2,"text":"A.1 PathBoost Parameters"},{"type":"para","text":"n_iter : Maximum number of boosting iterations (default: 100)"},{"type":"para","text":"max_path_length : Maximum length of discovered paths (default: 5)"},{"type":"para","text":"learning_rate : Shrinkage parameter for boosting updates (default: 0.1)"},{"type":"para","text":"n_of_cores : Number of CPU cores for parallel training (default: 1)"},{"type":"para","text":"patience : Early stopping patience; halts training if validation loss does not improve for this many consecutive iterations (default: None)"},{"type":"para","text":"restore_best_model : If True , the model is restored to the checkpoint with the best validation performance after early stopping, rather than retaining the weights from the final iteration (default: True)"},{"type":"para","text":"verbose : Whether to print progress information (default: False)"},{"type":"heading","level":2,"text":"A.2 SequentialPathBoost Parameters"},{"type":"para","text":"The SequentialPathBoost class accepts the same parameters as PathBoost , except for n_of_cores which is not applicable to single-anchor training."},{"type":"para","text":"A complete workflow for using PathBoost on molecular data:"},{"type":"heading","level":2,"text":"C.1 PathBoost hyperparameters"},{"type":"para","text":"PathBoost was run with fixed hyperparameters without grid search. Anchor labels are automatically extracted from each training fold as the set of unique atomic numbers present. The parameters used are listed in Table 3 ."},{"type":"heading","level":2,"text":"C.2 WL Kernel + SVR hyperparameters"},{"type":"para","text":"The number of WL iterations and the SVR regularisation parameter C C were selected by grid search minimising validation MAE, and then the model was refit on the combined train and validation set using the best configuration. The target y y was standardised with StandardScaler . The search grid and fixed settings are listed in Table 4 ."},{"type":"heading","level":2,"text":"C.3 GINE hyperparameters"},{"type":"para","text":"Six GINE configurations (combinations of hidden_dim and num_layers ) were evaluated by grid search minimising validation MAE; the best configuration was then refit on the combined train and validation set with early stopping. Node features are one-hot encoded atomic numbers (11 classes: H, C, N, O, F, P, S, Cl, Br, I, and other); edge features are bond type encoded as a scalar. The architecture consists of a linear embedding layer followed by num_layers GINEConv blocks (with batch normalisation and ReLU activations) and a global mean pooling layer feeding an MLP prediction head. The hyperparameter grid and fixed settings are listed in Table 5 ."}],"string":"[\n {\n \"type\": \"para\",\n \"text\": \"[1,] \\\\fnm Claudio \\\\sur Meggio\"\n },\n {\n \"type\": \"para\",\n \"text\": \"1] \\\\orgdiv Department of Mathematics, \\\\orgname University of Oslo, \\\\orgaddress \\\\city Oslo, \\\\postcode 0371, \\\\state Oslo, \\\\country Norway\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Graph-structured data are increasingly prevalent across scientific and industrial domains, including molecular chemistry, materials science, and social network analysis. Unlike traditional tabular data, graphs capture relationships and interactions between entities, making them a powerful yet challenging data structure for machine learning. In particular, learning predictive models from graphs requires methods that can extract meaningful features from their structure and, ideally, do so in an interpretable way.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Among the existing approaches for this prediction task, the most common are Graph Neural Network (GNN) and graph kernel methods. GNNs have become the dominant paradigm for learning from graph data, achieving strong empirical results on many benchmarks [ gilmer2017neural , hu2020ogb ] . However, GNNs present several practical limitations: they are generally considered difficult to interpret [ Wu2021 ] and often require a lot of training data to generalize effectively [ Ding2024 ] . Classical graph kernel methods paired with support vector machines remain competitive on small-scale datasets [ Kriege2020 ] , but rely on fixed feature representations that do not adapt during learning, and the resulting models offer limited interpretability in practice.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Previous work by meggio2024pathboost introduced a path-based boosting framework for regression on molecular graphs of transition metal compounds, where a designated metal-centre node, called the anchor node, is used as the starting node for path exploration. meggio_2nd extended this framework in several directions: incorporating multiple node and edge attributes into the path feature space via a two-step base learner that separates path selection from path fitting; introducing automatic anchor node selection based on categorical attribute diversity; and adapting the algorithm to binary classification through the logistic loss. These works also formalized absolute and relative variants of path-level variable importance, and demonstrated competitive performance against graph neural networks and kernel methods on twelve TUDataset benchmarks.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In this paper, we present path_boost , an open source Python package that implements the PathBoost algorithm developed in meggio2024pathboost and meggio_2nd . In addition, path_boost introduces two methodological extensions: a multi-anchor parallel architecture that trains independent sub-models for each anchor label and aggregates their predictions, and an enhanced variable importance framework that includes a correlation adjustment to account for structural dependencies between nested paths. The implementation follows scikit-learn conventions, implementing fit() , predict() , get_params() , and set_params() methods, enabling compatibility with standard tools such as GridSearchCV for hyperparameter tuning. Graphs are represented using the networkx library [ hagberg2008exploring ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The remainder of this paper is organized as follows. Section 2 introduces key terminology and defines the problem setting. Section 3 describes the algorithm, including the boosting procedure with a step-by-step illustration, and variable importance measures. Section 4 covers software design and implementation details. Section 5 provides a tutorial demonstrating the package on a molecular property prediction task using the tmQMg dataset. Section 6 reports a benchmark comparison against GINE and WL + SVR across six molecular datasets. Section 7 concludes with some remarks on the performance of the package.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Preliminaries\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.1 Terminology\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We introduce the key terms used throughout this paper:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Labeled path : an ordered sequence of anchor-attribute values ( l 1 , l 2 , … , l m ) (l_{1},l_{2},\\\\ldots,l_{m}) corresponding to a walk through the graph where consecutive nodes are connected by edges.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Boosting Matrix (BM) : a frequency matrix where each column corresponds to a labeled path and each row to a graph. Entry ( i , u ) (i,u) records how many times labeled path u u occurs in graph G i G_{i} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Extended Boosting Matrix (EBM) : an extended version of the BM that includes, for each labeled path, the averaged numerical node and edge attributes along that path and all its prefixes.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prefix : a labeled path q = ( q 1 , … , q k ) q=(q_{1},\\\\ldots,q_{k}) is a prefix of path p = ( p 1 , … , p m ) p=(p_{1},\\\\ldots,p_{m}) if k < m k<m and q i = p i q_{i}=p_{i} for all i ∈ { 1 , … , k } i\\\\in\\\\{1,\\\\ldots,k\\\\} ; that is, q q is an initial sub-sequence of p p .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Selector model : a simple model (by default, a decision tree stump) fitted on the BM to identify the most informative labeled path at each boosting iteration.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Base learner : the model fitted on the EBM columns relative to the selected path to produce the boosting update. By default, a decision tree regressor.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Anchor attribute ( anchor_nodes_label_name ): the categorical node attribute that determines where paths can start. Each of its distinct values picks out the nodes that carry that value, and paths are allowed to begin from such nodes. For example, in molecular graphs the anchor attribute might be \\\"feature_atomic_number\\\" : this means we use the atomic number to distinguish the different anchor nodes.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Anchor label ( list_anchor_nodes_labels ): a specific value of the anchor attribute that identifies a class of nodes as starting points for path exploration. Only nodes whose anchor attribute equals a selected anchor label are used as path origins. For example, in a molecular graph where the anchor attribute is the atomic number, anchor labels [25, 47, 78] restrict path exploration to start from manganese, silver, and platinum atoms, respectively.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.2 Problem setting\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We are given a dataset of N N graphs 𝒢 = { G 1 , G 2 , … , G N } \\\\mathcal{G}=\\\\{G_{1},G_{2},\\\\ldots,G_{N}\\\\} , where each graph G i = ( V i , E i , A V i , A E i ) G_{i}=(V_{i},E_{i},A_{V_{i}},A_{E_{i}}) consists of a set of nodes V i V_{i} , a set of edges E i ⊂ V i × V i E_{i}\\\\subset V_{i}\\\\times V_{i} , a node attribute function A V i : V i → ℝ q V A_{V_{i}}\\\\colon V_{i}\\\\to\\\\mathbb{R}^{q_{V}} mapping each node to a vector of q V q_{V} attributes, and an edge attribute function A E i : E i → ℝ q E A_{E_{i}}\\\\colon E_{i}\\\\to\\\\mathbb{R}^{q_{E}} mapping each edge to a vector of q E q_{E} attributes. Moreover, each graph is associated with a target value y i ∈ 𝒴 y_{i}\\\\in\\\\mathcal{Y} . The goal is to learn a function f : 𝒢 → 𝒴 f\\\\colon\\\\mathcal{G}\\\\to\\\\mathcal{Y} that predicts y i y_{i} from G i G_{i} . The package supports regression, where 𝒴 = ℝ \\\\mathcal{Y}=\\\\mathbb{R} , and binary classification, where 𝒴 = { 0 , 1 } \\\\mathcal{Y}=\\\\{0,1\\\\} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A key requirement is that at least one node attribute is categorical , which means that it takes finitely many distinct values. One such attribute is designated as the anchor attribute : its distinct values, or a chosen subset of them, form the set of anchor labels , the values from which path exploration is allowed to start. A labeled path is then a sequence of values of this anchor attribute. Apart from this role, the anchor attribute is treated like any other — every attribute contributes its averaged value to the EBM (Section 4.3 ). For example, in molecular graphs where nodes represent atoms and edges represent bonds, the chemical element is the anchor attribute, and each element of the periodic table (e.g., carbon, nitrogen, oxygen) is an anchor label. In the implementation, elements are identified by their atomic number, so these categorical labels appear as integers (e.g., 6, 7, 8 for carbon, nitrogen, oxygen).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The method is particularly suited to problems where the target depends on local structural patterns within the graphs; however, additional numerical node and edge attributes (e.g., atomic mass, bond strength) can be incorporated into the feature space to enrich the path-based representation, as described in Section 3 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Algorithm\"\n },\n {\n \"type\": \"para\",\n \"text\": \"PathBoost constructs a feature space from graph-structured data that can be used by standard supervised learning algorithms. It does so by iteratively discovering the most informative labeled paths within graphs via gradient boosting.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 The boosting procedure\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The core algorithm, summarized in Algorithm 1 , proceeds as follows to minimize a given loss function, ℓ \\\\ell . Given training graphs and associated target values, the boosting matrix (BM) is initialized with single-node paths corresponding to the anchor labels. At each iteration: (i) the selector identifies the most predictive path from the BM, (ii) the extended boosting matrix (EBM) is constructed for that path along with its prefixes (if additional attributes are available), (iii) a base learner is fitted to the pseudo-residuals using the EBM features, and (iv) the model predictions are updated. When a path is selected for the first time, the BM is expanded with all one-step extensions of that path found in the training data. This lazy expansion avoids the combinatorial cost of enumerating all possible paths upfront.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A key design choice is the separation between path selection and path fitting . The selector operates only on the BM, while the base learner operates on the full EBM. This separation is motivated by two primary considerations. First, computational efficiency : the BM is a compact matrix of integer counts (one column per candidate path, one row per graph) and can be evaluated very quickly by a simple model such as a decision stump. The EBM, by contrast, includes all averaged node and edge attributes for each path and its prefixes, making it much larger. Using the base learner directly for path selection would require fitting this richer model once per candidate path at each iteration, which would be prohibitively expensive. Second, flexibility : because path selection depends only on structural frequency counts and not on attribute values, the selector and the base learner can be chosen independently. The selector can stay a fast, simple model, while the base learner that fits the attribute-rich EBM can be any compatible estimator, from a shallow tree to a linear or gradient-boosted model, without changing how paths are selected. This modularity also preserves interpretability: the selected path provides a clear structural interpretation regardless of how the base learner uses the associated attributes.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In practice, the number of iterations, T T , is set through the n_iter parameter. This quantity is the most important hyperparameter: too few iterations and the model underfits, too many and it overfits. The package supports two possible strategies for selecting it. The first is to treat n_iter as any other hyperparameter and tune it via cross-validation. The second is to fix a upper bound for n_iter and let the algorithm decide when to stop. To do so, two criteria are available: patience stops training after a given number of consecutive iterations without improvement, and target_error stops training once the evaluation MSE falls below a user-defined threshold. Section 4.9 discusses how to use them in practice.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2 Illustration: boosting iteration on a molecular graph\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To illustrate how PathBoost discovers paths, we present a step-by-step walkthrough using the three toy molecules shown in Figure 1 , with platinum and zirconium ( (Pt) and (Zr) ) as the anchor labels. In these molecular graphs, nodes correspond to atoms and edges to bonds; for readability, atoms are labelled by their atomic symbol rather than their atomic number. The algorithm proceeds as follows:\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Step 1: Initialization.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given the platinum anchor (Pt) , the algorithm initializes the BM with one column per anchor label present in the training data (here columns for (Pt) and (Zr) ) recording how many times each anchor atom appears in each molecule.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Step 2: Path selection and exploration.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The selector evaluates the BM to identify the most predictive path. As shown in Figure 1 , at the first iteration the selector identifies (Pt, C) as the most informative path. Since this path has not been selected before, the algorithm adds all one-step extensions found in the training graphs as new BM columns. Note that Figure 2 depicts the state at the second iteration, after (Pt, C) was already selected in iteration 1: the shaded columns (Pt, C, N) and (Pt, C, C) are the extensions added upon that selection.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Step 3: EBM expansion.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Once a path is selected, the EBM is expanded with features for that path and all its prefixes. As shown in the bottom panel of Figure 2 , the EBM for (Pt, C) includes the occurrence count and averaged atomic mass of the terminal C atom, together with the averaged bond order of the Pt–C edge; the prefix (Pt) columns were already present from the initialization step.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Step 4: Base learner training.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The base learner is trained on the EBM columns corresponding to the selected path and its prefixes, fitting to the negative gradient of the current predictions.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Step 5: Repeat.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Steps 2–4 are repeated for the next boosting iteration, with the updated BM now including the extensions discovered in the previous step. This continues until the specified number of iterations T T is reached.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.3 Loss functions and boosting details\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Section 3.1 described the generic boosting procedure. Here we specify how it is instantiated for the two supported tasks, how the package handles single versus multiple anchor labels, and how the learning rate is applied externally.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.3.1 Regression\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For regression tasks, the package minimizes the squared error loss:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The model is initialized with the mean of the training targets, and pseudo-residuals are the ordinary residuals r t , i = y i − y ^ t − 1 , i r_{t,i}=y_{i}-\\\\hat{y}_{t-1,i} . After T T boosting iterations, the final prediction is the sum of all base learner outputs, scaled by the learning rate η \\\\eta .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.3.2 Classification\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For binary classification, the package minimizes the binary cross-entropy loss:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where p ^ i \\\\hat{p}_{i} is the predicted probability of Y i = 1 Y_{i}=1 for graph G i G_{i} . Boosting updates accumulate in log-odds space: the model maintains a score F ( G i ) = ∑ t η ⋅ h ( t ) ( G i ) F(G_{i})=\\\\sum_{t}\\\\eta\\\\cdot h^{(t)}(G_{i}) , which is converted to a probability via the sigmoid function p ^ i = σ ( F ( G i ) ) = ( 1 + e − F ( G i ) ) − 1 \\\\hat{p}_{i}=\\\\sigma(F(G_{i}))=(1+e^{-F(G_{i})})^{-1} . The model is initialized with the log-odds of the positive class proportion in the training set. The negative gradient (pseudo-residuals) at each iteration is r t , i = y i − p ^ t − 1 , i r_{t,i}=y_{i}-\\\\hat{p}_{t-1,i} , i.e., the difference between the true label and the current predicted probability.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Standard mode.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"By default, the tree is fitted to the pseudo-residuals, and its leaf values are used directly as the boosting update. This is the standard gradient boosting approach and works well in practice for most tasks.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Leaf-optimized mode ( use_tree_boost=True , tree base learners only).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"When use_tree_boost=True is set on SequentialPathBoostClassifier , the package activates the BoostedTreeBaselearner , which follows the TreeBoost approach of friedman2001greedy . The tree is still fitted to the pseudo-residuals to determine the partition of graphs into leaves, but the leaf output values are then re-optimized via a line search that directly minimizes the logistic loss in the log-odds space. For each leaf z z , let R z R_{z} denote the set of training graphs assigned to that leaf. The optimal scalar γ z \\\\gamma_{z} is found by minimizing\"\n },\n {\n \"type\": \"para\",\n \"text\": \"over γ ∈ [ − 10 , 10 ] \\\\gamma\\\\in[-10,10] . At each boosting iteration, the update for each graph uses the optimized γ z \\\\gamma_{z} of its leaf in the current base learner, rather than the tree’s original leaf value. This approach produces better-calibrated probabilities than the standard mode, since the leaf values are optimized directly for logistic loss rather than squared error with respect to the residuals.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"After T T boosting iterations, the final class prediction is obtained by thresholding the predicted probability: y ^ i = 𝟏 [ p ^ i ≥ 0.5 ] \\\\hat{y}_{i}=\\\\mathbf{1}[\\\\hat{p}_{i}\\\\geq 0.5] .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.3.3 PathBoost and SequentialPathBoost\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The package provides two main classes. SequentialPathBoost implements the algorithm described above for a single anchor label. PathBoost extends this to multiple anchor labels by training independent SequentialPathBoost instances (one per anchor label) and aggregating their predictions. Since paths starting from different anchor labels cannot discover common extensions, these sub-models are independent and can be trained in parallel. The choice between the two depends on how many anchor nodes of each label a graph contains. When each graph has exactly one anchor node per label (as in molecular graphs with a unique metal center), the per-label sub-models are fully independent, and PathBoost is recommended. When a graph contains several anchor nodes of the same label, their contributions must be averaged within the sub-model, which can make its output less stable; in this case SequentialPathBoost applied to a single anchor set combining all labels is recommended.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 3,\n \"text\": \"3.3.4 Learning rate handling\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The package does not assume that the base learner has an internal learning rate. Instead, the learning rate η \\\\eta is applied externally (the default value is 0.1). To ensure that this external scaling behaves correctly, the negative gradient is centered (its mean is subtracted) before fitting the base learner. The mean is then added back after scaling the predictions by η \\\\eta . This guarantees that the base learner’s predictions are centered around zero, so that multiplication by η \\\\eta produces a proper shrinkage toward zero.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.4 Variable importance\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The package provides two criteria for computing path importance, both implemented in VariableImportance_ForSequentialPathBoost and controlled via the parameters_variable_importance dictionary. Variable importance is also available for PathBoost , where per-anchor importances are combined into a single global score as described in the cross-anchor combination paragraph below. Both criteria can optionally apply a correlation adjustment and normalization. The error metric used ( ’mse’ or ’mae’ ) is set via the ’error_used’ key (this controls importance scoring only, it does not affect how base learners are fitted).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Absolute importance (default).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A path’s importance is the sum of all the improvements in training error over the iterations in which the path was selected. At each boosting iteration t ≥ 1 t\\\\geq 1 , the error improvement is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where ℓ t \\\\ell_{t} is the training loss after iteration t t . This improvement is accumulated for each path p p selected at iteration t t :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As a special case, the path selected at iteration 0 is assigned the improvement with respect to the mean of the target variables y y .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Relative importance.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Relative importance measures, at each iteration, how much better the selected path is compared to the best available alternative. This complements absolute importance: a path can accumulate high absolute importance simply because it is selected early and often, even when other paths would have reduced the error almost as much. Relative importance instead rewards paths that are irreplaceable ,those for which no comparably predictive alternative was available at the time of selection, which is especially informative when many candidate paths are correlated (for example nested paths sharing a common prefix) and one wants to identify the substructures that uniquely drive predictions. At iteration t t , the frequency column of the selected path p t ∗ p^{*}_{t} is temporarily removed from the BM, the selector is re-run on the remaining columns to identify the second-best path p t ( 2 ) p^{(2)}_{t} , and an additional base learner is fitted on p t ( 2 ) p^{(2)}_{t} . The importance contribution is the error gap\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where ℓ t ( 2 ) \\\\ell^{(2)}_{t} is the training error that would have resulted from selecting p t ( 2 ) p^{(2)}_{t} instead. If a path has high relative importance it means it is substantially better than any available alternative at the time of selection. Because the selector operates on the BM while the base learner operates on the full EBM, it is possible (though rare) for the second-best path to lead to a lower error, producing a negative contribution. Relative importance requires re-running the selector and fitting an additional base learner at every iteration, making it significantly more expensive than absolute importance. In the Python function, it must be enabled explicitly by setting ’criterion’: ’relative’ .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Correlation adjustment.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Because longer paths extend shorter ones as a prefix, their occurrence in a graph is correlated by construction. This means that if the shorter path drove most of the error reduction early in training, the longer extension may appear unimportant even though its presence is linked to the shorter path. When ’use_correlation’: True is set, the importance of each path p p is given by the sum of correlation-weighted importances of all its prefixes (see Section 2.1 ):\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where ρ ( p , q ) \\\\rho(p,q) is the Pearson correlation between the occurrence counts of p p and q q across all training graphs.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Normalization.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"When ’normalize’: True (the default), importances are rescaled to sum to normalization_value (default: 100):\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For the absolute criterion without correlation adjustment, this makes importances directly interpretable as percentage contributions to total explained error. For other criteria, normalization provides a common scale for comparing path importances across runs, but the percentage-of-error interpretation is not as straightforward.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Cross-anchor combination.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"When PathBoost trains one SequentialPathBoost model per anchor label, the per-anchor importances are combined via a weighted average, where the weight for anchor a a is proportional to the number of training graphs n a n_{a} in which that anchor appears. The combined importances are then re-normalized to normalization_value . The final importance scores are stored in model.variable_importance_ , mapping each selected path (e.g., (Mn, N, O) ) to its scalar importance.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Software design and implementation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section describes the practical aspects of using the path_boost package.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 Data requirements\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Attribute types.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All attributes are treated uniformly: any attribute whose values are numbers (instances of numbers.Number ) is added to the EBM, while attributes with non-numeric values are ignored. This means categorical attributes that the user wants to include must be encoded numerically beforehand, for example via one-hot encoding or an integer representation. This applies to the anchor attribute as well, which both defines the paths and contributes its own averaged value as a feature. Finally if a node and an edge share an attribute name, the edge attribute takes precedence.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Graph types.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The package supports directed and undirected graphs from networkx . In directed graphs, edge orientation matters both when a path is extended and when paths are compared: two paths are considered identical only if they visit the same node labels and agree on the orientation of every edge. For example, a → b → c a\\\\rightarrow b\\\\rightarrow c and a → b ← c a\\\\rightarrow b\\\\leftarrow c are distinct paths. Multigraphs are also accepted, but in this case each edge is treated as bidirectional.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 Multiple paths handling\"\n },\n {\n \"type\": \"para\",\n \"text\": \"When multiple instances of the same labeled path exist in a graph, their attribute values are aggregated by averaging over all occurrences. To use a different aggregation function (e.g., sum or max), the the user can create a subclass of EBM and override the static method combine_attribute_value_of_multiple_paths_in_the_same_graph .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3 Extended Boosting Matrix structure\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The EBM is a matrix where rows correspond to individual graphs and columns correspond to path-based features, following the naming convention (path)_attribute . For example, a column named (6, 7, 8)_atomic_mass contains the averaged atomic mass of oxygen atoms (atomic number 8) reached via the path carbon → \\\\rightarrow nitrogen → \\\\rightarrow oxygen. After fitting, the full EBM can be inspected via the train_ebm_dataframe_ attribute:\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.4 Path length and feature-space growth\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The parameter with the largest impact on the cost of training is max_path_length . The number of distinct labelled paths that can be discovered grows combinatorially with both the path length and the diversity of node labels: at each additional step a path may branch into as many extensions as there are distinct neighbouring labels, so the space of candidate paths grows roughly exponentially in max_path_length and in the branching factor of the label set. Because every discovered path contributes several columns to the Extended Boosting Matrix (EBM, one per node and edge attribute of the path and its prefixes, the width of the EBM, and hence the memory and time needed to build and fit it, scales with the number of paths actually realized during training.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Two mechanisms keep this growth manageable. First, paths are expanded lazily : the boosting matrix is extended only when a path is selected, so the realized feature space is far smaller than the full combinatorial enumeration and is concentrated on structurally informative paths (Section 3.1 ). Second, max_path_length caps the depth of exploration directly. The realized EBM can be inspected after fitting through the train_ebm_dataframe_ attribute (Section 4.3 ), which is useful for diagnosing how large the feature space has actually become on a given dataset.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In practice we recommend treating max_path_length as a primary hyperparameter and tuning it by cross-validation, starting from small values (e.g. 3–4) and increasing until the validation error stops improving. Short paths may miss long-range structural signal, whereas overly long paths inflate the EBM, raise training time and memory, and add many weakly informative features that increase the risk of overfitting. The appropriate value is dataset-dependent: graphs with diverse node labels (for example molecular datasets spanning many chemical elements) branch more quickly and reach a large EBM at smaller path lengths than graphs with few distinct labels.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.5 Selector\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The default selector for both regression and classification is sklearn.tree.DecisionTreeRegressor with max_depth=1 (a decision stump), passed via the SelectorClass constructor parameter.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The selector’s sole purpose is path selection: at each boosting iteration it identifies which candidate path in the BM is most predictive of the current negative gradient. Concretely, it is fitted on the frequency-only columns of the BM (one column per known path, recording how many times that path appears in each graph) and the path corresponding to the highest entry in feature_importances_ is selected. The selector is never called during prediction. This design is intentional: keeping the selector focused on structural frequency counts rather than attribute values makes path selection fast and robust to overfitting, as discussed in Section 3.1 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Custom selectors.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The selector interface is duck-typed via SelectorClassInterface : explicit inheritance is not required, and any class implementing the expected methods is automatically recognized as compatible. A custom selector must implement fit(X, y) , predict(X) (required by the interface but never called during training), and expose a feature_importances_ attribute after fitting, namely an array of non-negative scores, one per column of X . Only the ordering of scores matters, not their scale. In principle, any scikit-learn estimator that exposes feature_importances_ can be used directly as a selector.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.6 Available base learners\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The default base learner for both SequentialPathBoost and PathBoost (see Section 3.3.3 ) is sklearn.tree.DecisionTreeRegressor with max_depth=3 , passed via the BaseLearnerClass constructor parameter. Shallow decision trees handle NaN values natively, which arise in the EBM when a path is absent from a graph, and are well suited to the small number of columns (typically 3–10) that the base learner receives at each iteration, corresponding to the EBM features of the selected path and its prefixes.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Custom base learners.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As with the selector, base learners use a duck-typed interface, defined in BaseLearnerClassInterface : explicit inheritance is not required, and any class implementing fit(X, y) and predict(X) is automatically recognized as compatible. The X argument is always a pd.DataFrame containing only the EBM columns for the currently selected path; it is not the full feature matrix. Any scikit-learn estimator with fit and predict (e.g., LinearRegression , XGBRegressor ) works without modification. Table 1 summarizes the base learners that have been tested for compatibility with the package.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"NaN handling.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"By default, EBM entries are np.nan when a path is absent from a graph. Tree-based models handle this natively. Linear models do not: when using LinearRegression , Ridge , or similar, the replace_nan_with parameter must be set to a numeric value:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The choice of replacement value matters: 0 is a neutral default, while a large negative value such as -100 signals path absence explicitly to the model. For most use cases 0 is the safer choice, unless missing paths need to be distinguished from paths with a true zero attribute value.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Classification variant.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The SequentialPathBoostClassifier class provides an additional option via the use_tree_boost flag, which activates an internal base learner that optimizes per-leaf values by minimizing the logistic loss directly, following the approach of friedman2001greedy :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"When use_tree_boost=True , the BaseLearnerClass parameter is ignored.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.7 Parallelization\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The PathBoost class supports parallel training across anchor labels by setting n_of_cores > 1 . Thread limits are configured at module load time to prevent over-subscription. For cross-platform compatibility, the main script should be wrapped with:\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.8 Model persistence\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Trained models can be saved and loaded for reproducibility:\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.9 Tuning the number of iterations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As anticipated in Section 3.1 , the number of boosting iterations T T is one of the most influential hyperparameters of the model. The package offers two complementary ways of choosing it: early stopping on a held-out evaluation set, and cross-validation over a grid of candidate values. This section describes each mechanism in turn and explains how they should (and should not) be combined.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Early stopping and the patience parameter.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Early stopping requires an evaluation set, passed to fit() through the eval_set argument. After every boosting iteration, the model predicts on this set, records the evaluation MSE, and checks whether the active stopping criterion has been triggered. The patience parameter is an integer that controls how tolerant the procedure is to short-term fluctuations in the evaluation curve: training is halted once the evaluation MSE has failed to improve on its best value for patience consecutive iterations. This tolerance is needed because the evaluation MSE does not always decrease monotonically. A small patience (e.g. 5) reacts quickly to real overfitting, but it can also stop on one of these temporary upticks and treat it as overfitting when it is not, halting in a local minimum of the evaluation curve before training reaches a deeper minimum a few iterations later. A larger value (e.g. 20–50) ignores these small fluctuations and is more likely to find the best model, at the cost of more iterations. Stopping too early is partly offset by restore_best_model (discussed below), which rolls the model back to the lowest-MSE checkpoint; patience still controls how far past the current best the search keeps going. The alternative criterion, target_error , halts training as soon as the evaluation MSE drops to or below a user-specified threshold; it is useful when a problem-specific error target is known a priori. Without an eval_set both criteria are silently ignored and training always runs for the full n_iter iterations. When restore_best_model=True (the default), after stopping the model is rolled back to the checkpoint with the lowest evaluation MSE, so the returned estimator is not the (possibly overfitted) one from the final iteration. To be able to use restore_best_model an evaluation dataset (typically a holdout of the training data) must be passed. Choosing the best checkpoint on it is therefore a model-selection step on validation data, just as in ordinary early stopping for gradient boosting or neural networks. Because this set decides when to stop, its MSE is an optimistic measure of accuracy and should not be reported as the final result: that must come from a separate test set that took no part in training or early stopping, as done in the tutorial (Section 5 ) and benchmark (Appendix C ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Cross-validation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Cross-validation is supported through the standard sklearn interface: PathBoost can be passed directly to utilities such as GridSearchCV together with a grid over the hyperparameters of interest (typically learning_rate , max_path_length , and, when desired, n_iter ). For each candidate configuration the model is refit on every fold and the validation score is averaged across folds, yielding a fair estimate of generalisation error.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Combining the two strategies.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Early stopping and cross-validation should not be combined in a single GridSearchCV call: forwarding an eval_set through fit_params reuses the same validation set across every fold, and different folds stop at different iterations, so the cross-validated score no longer corresponds to a single, well-defined model. We recommend keeping n_iter fixed during cross-validation and applying early stopping only in the final refit.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Tutorial\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.1 Installation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The package can be installed from the Python Package Index via:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Dependencies can be installed with:\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.2 Molecular property prediction on the tmQMg dataset\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We demonstrate the package on the tmQMg dataset [ D2DD00129B ] , a collection of transition metal compounds represented as molecular graphs. In this dataset, atoms are nodes and bonds are edges. Each node carries numerical attributes derived from the periodic table (e.g., atomic number, atomic mass), and edges carry bond-related attributes (e.g., bond strength). The target variable is the HOMO–LUMO gap, a continuous molecular property. The atomic number serves as the anchor attribute.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We initialize PathBoost with the parameters that will remain constant during cross-validation, for example the fixed number of iterations n_iter , together with the variable-importance settings. The grid search will then vary only learning_rate , max_path_length , and kwargs_for_base_learner . A full description of each parameter is available in Appendix A .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Note that error_used controls only how variable importance scores are computed after fitting (MSE vs. MAE when measuring a path’s contribution to error reduction); it has no effect on how the base learners are trained.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To perform cross validation we use the standard sklearn procedure.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A compact, self-contained version of the same workflow, without the surrounding commentary, is provided in Appendix B for quick reference.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As discussed in Section 4.9 , we recommend leaving patience and target_error unset during GridSearchCV , so that each fold is trained for the same fixed number of iterations and the cross-validated scores remain directly comparable. Early stopping is reintroduced in the final training run below.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.3 Training\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For the final training run, the model should be re-initialized with the best hyperparameters and a patience value for early stopping. Here n_iter serves as the maximum number of iterations for each sub-model. This is the strategy we adopt here, but it is not the only one: a common alternative is to fix the other hyperparameters in advance and instead cross-validate the number of iterations directly (Section 4.9 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The best hyperparameters from cross-validation should be included in the initialization. After training, the model’s performance and feature importance can be inspected:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The result is shown in Figure 3\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.4 Prediction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To perform a prediction, input a list of nx.Graph objects:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"When working directly with SequentialPathBoost , the EBM can be pre-computed once and reused across multiple calls to avoid recomputation:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Note that PathBoost.predict() does not expose this shortcut; it always recomputes the EBM internally.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6 Benchmark comparison\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We compare PathBoost against two established baselines across six molecular regression tasks. The baselines are the Graph Isomorphism Network with Edge features (GINE) [ hu2020strategies ] and the Weisfeiler–Leman graph kernel paired with support vector regression (WL + SVR) [ Kriege2020 ] . All methods are evaluated using 5-fold cross-validation repeated three times, yielding 15 train/test evaluations per method and dataset. Within each fold the training set is further split 90/10 into a training and a validation subset used for early stopping and, where applicable, hyperparameter selection; the validation subset is never used for final evaluation. Reported metrics (MAE, RMSE, R 2 ) are the mean and standard deviation across the 15 folds, together with the average training time per fold. All experiments were run on CPU.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"PathBoost was applied with fixed hyperparameters, with anchor labels automatically extracted from each training fold as the set of unique atomic numbers present in that fold. The two baselines were instead tuned by grid search on the validation subset of each fold (minimising validation MAE) and then refit on the combined train and validation set using the best configuration. The full hyperparameter grids and fixed settings for all three methods are reported in Appendix C . This benchmark focuses on regression tasks; a comparative study that also covers classification on a broader range of graph datasets is presented in meggio_2nd .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"6.1 Datasets\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"ESOL.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"ESOL [ wu2018moleculenet ] is a small dataset of 1,128 organic compounds with experimentally measured water solubility expressed as log(mol/L). It is part of the MoleculeNet benchmark collection [ wu2018moleculenet ] .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"FreeSolv.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"FreeSolv [ wu2018moleculenet ] provides experimental hydration free energies (kcal/mol) for 643 small molecules in water, derived from alchemical free energy calculations.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"QM9.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"QM9 [ ramakrishnan2014quantum ] contains quantum-mechanical properties of approximately 134k stable small organic molecules composed of C, H, O, N, and F atoms with up to 9 heavy atoms, computed at the B3LYP/6-31G(2df,p) level of theory. The target property is the HOMO–LUMO gap. Due to its size, we use a random subsample of 10,000 graphs.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"tmQMg.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The tmQMg dataset [ D2DD00129B ] is a collection of mononuclear transition metal complexes extracted from the Cambridge Structural Database, represented as molecular graphs where atoms are nodes and bonds are edges.From the u-NatQG variant we consider three regression targets, each treated as a separate task:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"dipole moment;\"\n },\n {\n \"type\": \"para\",\n \"text\": \"HOMO energy;\"\n },\n {\n \"type\": \"para\",\n \"text\": \"polarizability.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Due to the size of the dataset, each target uses a random subsample of 5,000 graphs.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"6.2 Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 2 reports the benchmark results. PathBoost achieves the best performance on five of the six tasks. On the two small organic datasets, ESOL and FreeSolv, PathBoost outperforms both baselines by a clear margin across all metrics, suggesting that path-based features are particularly effective when training data is limited. On QM9, GINE performs best, which is consistent with the known strength of graph neural networks on large, homogeneous organic datasets where message-passing can exploit rich local chemical environments. On the three tmQMg targets, PathBoost performs best on all the considered tasks, with especially strong results on polarizability ( R 2 = 0.928 R^{2}=0.928 ) and HOMO energy ( R 2 = 0.584 R^{2}=0.584 ), while all methods struggle on the dipole moment target, reflecting the intrinsic difficulty of predicting this property from graph topology alone.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Regarding computational cost, GINE is substantially slower than both PathBoost and WL + SVR across all datasets, with training times up to an order of magnitude larger on the tmQMg tasks. Note that all methods were run on CPU; GPU acceleration would be expected to reduce GINE’s training time considerably. PathBoost occupies a middle ground, being faster than GINE while achieving better predictive performance on most tasks.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"7 Summary and discussion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"PathBoost, introduced in meggio2024pathboost and meggio_2nd , is a gradient boosting framework for learning from graph-structured data. Its distinguishing feature is interpretability: predictions are decomposed into contributions from labelled paths through the input graphs, and the importance of each path can be quantified after training. While previous works introduced the method and released code for their own experiments, they did not provide a general-purpose easy-to-use implementation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The path_boost package presented in this paper fills this gap. It provides a tested, open-source, scikit-learn -compatible reference implementation supporting both regression and binary classification, multiple anchor labels, parallel training across sub-models, and early stopping. Two variable importance criteria, absolute and relative with an optional correlation adjustment, let users identify which labelled paths drive the model’s predictions; this matters in scientific applications such as computational chemistry, where understanding why a prediction was made is often as important as the prediction itself. By following scikit-learn conventions and using networkx for graph handling, the package also integrates naturally with existing data science workflows, enabling cross-validation via GridSearchCV and custom scores, without requiring any custom plumbing on the user’s side.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Computational details\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The results in this paper were obtained using Python 3.12 with the following key packages: path_boost 2.1.0, scikit-learn 1.6.1 [ pedregosa2011scikit ] , networkx 3.4.2 [ hagberg2008exploring ] , pandas 2.2.3 [ mckinney2010data ] , numpy 2.1.3 [ harris2020array ] , and matplotlib 3.9.3 [ hunter2007matplotlib ] .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Statements and Declarations\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Ethics approval and consent to participate.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Not applicable.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Code availability.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The path_boost package is open-source and freely available. The source code is hosted on GitHub at https://github.com/Claudio-Me/extended_path_boost , and releases are distributed on the Python Package Index (PyPI) at https://pypi.org/project/path-boost/ . The version used to produce the results in this paper is reported in the Computational details section.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This appendix provides a summary of the main classes and their parameters.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"A.1 PathBoost Parameters\"\n },\n {\n \"type\": \"para\",\n \"text\": \"n_iter : Maximum number of boosting iterations (default: 100)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"max_path_length : Maximum length of discovered paths (default: 5)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"learning_rate : Shrinkage parameter for boosting updates (default: 0.1)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"n_of_cores : Number of CPU cores for parallel training (default: 1)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"patience : Early stopping patience; halts training if validation loss does not improve for this many consecutive iterations (default: None)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"restore_best_model : If True , the model is restored to the checkpoint with the best validation performance after early stopping, rather than retaining the weights from the final iteration (default: True)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"verbose : Whether to print progress information (default: False)\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"A.2 SequentialPathBoost Parameters\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The SequentialPathBoost class accepts the same parameters as PathBoost , except for n_of_cores which is not applicable to single-anchor training.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A complete workflow for using PathBoost on molecular data:\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"C.1 PathBoost hyperparameters\"\n },\n {\n \"type\": \"para\",\n \"text\": \"PathBoost was run with fixed hyperparameters without grid search. Anchor labels are automatically extracted from each training fold as the set of unique atomic numbers present. The parameters used are listed in Table 3 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"C.2 WL Kernel + SVR hyperparameters\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The number of WL iterations and the SVR regularisation parameter C C were selected by grid search minimising validation MAE, and then the model was refit on the combined train and validation set using the best configuration. The target y y was standardised with StandardScaler . The search grid and fixed settings are listed in Table 4 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"C.3 GINE hyperparameters\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Six GINE configurations (combinations of hidden_dim and num_layers ) were evaluated by grid search minimising validation MAE; the best configuration was then refit on the combined train and validation set with early stopping. Node features are one-hot encoded atomic numbers (11 classes: H, C, N, O, F, P, S, Cl, Br, I, and other); edge features are bond type encoded as a scalar. The architecture consists of a linear embedding layer followed by num_layers GINEConv blocks (with batch normalisation and ReLU activations) and a global mean pooling layer feeding an MLP prediction head. The hyperparameter grid and fixed settings are listed in Table 5 .\"\n }\n]","isConversational":false}}},{"rowIdx":92,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.07847"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"When Does Continual Learning Require Learning"},"authors":{"kind":"list like","value":["Anne Harrington","Nayan Saxena","Michael Murphy","Anastasia Borovykh","Zeyu Yun","Sridhar Kamath","Ara Eindra Kyi","Trevor Darrell","Jitendra Malik","Yutong Bai"],"string":"[\n \"Anne Harrington\",\n \"Nayan Saxena\",\n \"Michael Murphy\",\n \"Anastasia Borovykh\",\n \"Zeyu Yun\",\n \"Sridhar Kamath\",\n \"Ara Eindra Kyi\",\n \"Trevor Darrell\",\n \"Jitendra Malik\",\n \"Yutong Bai\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.07847v1"},"published":{"kind":"timestamp","value":"2026-07-08T00:00:00","string":"2026-07-08T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Traditionally, continual learning has been defined as mitigating catastrophic forgetting [ 40 ] : how do we learn task B B without forgetting task A A ? One approach to this problem is to augment or compose model parameters as proposed in superposition [ 6 ] , adversarial networks [ 9 ] and side networks [ 57 ] . However, this setup is not realistic in the current paradigm of large language models (LLMs), which often rely on large scale pre-, mid-, and post-training to acquire general capability and after deployment need to be adapted to a broad range of potential use cases. At the same time, standard LLM evaluations focus on measuring capabilities on static tasks such as general knowledge [ 51 ] , domain specific abilities [ 17 , 23 , 45 , 16 ] , and challenging frontier problems [ 14 , 38 ] ; these however fail to measure the ability to become more competent under conditions that evolve through time – which is critical for practical deployment."},{"type":"para","text":"Without a unified framework, the community approached continual learning as a memory problem solved with retrieval-augmented generation [ 27 ] , in-context learning [ 3 ] , smart context management [ 29 , 56 ] and harnesses [ 59 , 54 , 26 ] ; as a steering problem solved with prompt optimization [ 43 , 47 , 2 ] ; as a reasoning problem solved with reinforcement learning and distillation [ 35 , 60 , 19 , 48 ] ; and as a compression problem solved with architectural changes [ 10 , 30 , 49 , 11 ] . Each of these takes a different bet on what continual learning actually requires, but the field still lacks unified insights across these methods to direct progress."},{"type":"para","text":"In our work, we provide a unified framework to define and measure continual learning in modern LLMs. We define continual learning as the problem of increasing competence as the world changes. We consider change along two axes: space and time , as depicted in Figure 1 . Space corresponds to the classical notion of domain or task shift, where the model is asked to acquire a new skill on a different distribution without losing the previous one. Time corresponds to a different and, more common form of change in real deployed systems: the task stays the same, but the underlying world drifts. Time itself can be further split into two sub-regimes that the literature usually conflates. Slow trends (e.g. yearly financial filings) ask the model to stay aligned; the best strategy here is to keep extracting whatever structure survives across years and ignore year-specific noise. Discrete fact changes (e.g. Wikipedia revisions) ask the opposite – the right move is to actually rewrite the affected belief while leaving the rest of the model’s knowledge untouched. A third sub-regime is agentic accumulation: the task family itself stays fixed (e.g., actions inside a web app), but the environment’s state drifts as a direct consequence of the model’s own prior actions, so that what one step leaves behind becomes part of the task the model faces at the next step. This second sub-regime is the LLM analogue of the frame problem [ 39 ] : when the world updates, a learner must decide which of its beliefs are now outdated and which can remain fixed."},{"type":"para","text":"To measure continual learning under our definition, we cast widely used LLM evaluations as problems that unfold sequentially under a single mechanism-agnostic protocol, so prompt, weight, and architectural updates can be compared on equal ground. On a common backbone (Qwen3-8B) [ 53 ] , we evaluate eight methods across four families: prompt optimization (GEPA [ 2 ] , ACE [ 59 ] ), offline supervised updates (SFT, SDFT [ 48 ] ), online reinforcement learning (GRPO [ 46 ] , SDPO [ 19 ] ), and context-compression (Cartridges [ 10 ] , In-place TTT [ 11 ] )."},{"type":"para","text":"Interestingly, we observe a consistent set of tradeoffs across methods: (i) Prompting-based approaches such as GEPA and ACE fit quickly and heavily to past trajectories, achieving strong backward accuracy, but encounter sharp degradation on future tasks. (ii) Distillation-based methods such as SDFT and SDPO accumulate knowledge more stably over time, yet struggle to rapidly incorporate new information or update outdated knowledge. (iii) Context compression approaches, including Cartridges and In-place TTT, improve efficiency and memory management but do not substantially improve the ability to learn new tasks. (iv) In contrast, online reinforcement learning methods such as GRPO adapt more effectively to knowledge updating, though they remain highly sensitive to noisy or unstable reward signals. (v) On the agentic axis, where the stage ordering is generated by the agent’s own actions rather than imposed by us, both a prompt-based playbook (ACE) and weight-based fine-tuning (SFT) compound experience and beat their zero-shot baselines at every chain length, though absolute success still decays as the chain grows."},{"type":"para","text":"All together, we make the following contributions:"},{"type":"para","text":"We formalize continual learning in LLMs as the problem of increasing competence over time. Through our protocol, we provide a unified perspective that allows us to consider both parametric and non-parametric update methods for continual learning. We adapt these methods to the continually learning setting and align their evaluation metrics."},{"type":"para","text":"We introduce a broad suite of sequential LLM tasks to evaluate continual learning. These including domain adaptation, agentic tasks, financial analysis, and temporally-dependent knowledge updates."},{"type":"para","text":"We find that prompt-based methods alone are insufficient across most regimes, calling for actual learning. Different learning methods each have their own strengths depending on how the task and data shift over time."},{"type":"heading","level":1,"text":"2 Related Works"},{"type":"para","text":"Continual Learning. Traditionally, the field has considered continual learning as preventing the problem of catastrophic forgetting [ 40 , 13 ] . In this context, the problem is that sequential parameter updates can overwrite knowledge needed for previous tasks; this led to a focus on designing methods that reduce task interference [ 22 , 15 , 9 , 57 , 6 ] . From another angle, continual learning has been positioned as maintaining plasticity [ 8 ] , where models must not only remember old tasks, but also have the ability to learn new ones [ 36 , 42 ] ."},{"type":"para","text":"Non-parametric updates. At deployment, we can adapt LLMs without weight changes by changing the inference-time environment around a fixed model. One family of methods changes the information available to the model, for example through retrieval-augmented generation (RAG) [ 27 ] , external memory, reusable long-context representations [ 10 ] , context compression [ 29 ] , or recursive access to large external contexts [ 56 ] . A closely related family changes the behavioral scaffold through which that information is used, including demonstrations [ 3 ] , prompt instructions and agent contexts [ 2 , 59 , 55 ] and multi-stage LM programs [ 44 , 25 , 28 ] . More broadly, agentic systems can adapt by modifying their surrounding harnesses [ 54 , 26 ] , learn component-level reward specifications [ 52 ] or improve through external programs and agent code modifications [ 4 , 43 , 58 , 47 ] ."},{"type":"para","text":"Gradient-based (parametric) updates. Another adaption approach is to update weights or activations. At one extreme, full fine-tuning modifies all parameters of the model; while effective, this regime is often computationally expensive and unrealistic for frequent or online adaptation. Activation editing [ 20 ] uses task vectors applied to all parameters. More commonly, work studies parameter-efficient methods, such as LoRA [ 18 ] and adapter-based tuning [ 30 , 33 ] , which restrict updates to small subsets of parameters. An important but underexplored regime involves severely constrained updates—very few gradient steps and tightly limited compute—closer to the conditions under which continual adaptation would need to occur in deployed systems. Continual learning methods differ not only in where they store updates, but also in what learning signal drives the update. OPSD [ 60 , 35 ] , SDFT [ 48 ] and SDPO [ 19 ] learn from self-distillation, GRPO [ 46 ] and h1 [ 41 ] optimize end-of-task rewards, LoRD [ 32 ] uses teacher–student distillation, sparse memory finetuning [ 30 ] localizes the next-token objective to sparsely accessed memory slots, test-time training [ 11 ] performs input-conditioned weight updates during inference."},{"type":"heading","level":1,"text":"3 Method"},{"type":"heading","level":2,"text":"3.1 A Protocol for Continual Learning"},{"type":"para","text":"Setup. Typically, continual-learning evaluations are tightly coupled to a particular method: weight-update methods use task accuracy with a fixed fine-tuning budget; prompt-optimization often uses few-shot generalization with a fixed token budget; agentic frameworks rely on end-to-end pass rate. As a result, prompt, weight, and architectural updates are rarely compared fairly against one another. To gather broader insights about what methods are necessary for continual learning, we define a simple protocol that is compatible with any update operator. To do so, we consider all forms of updates (prompting, full weight, sparse weight/activation) as potential strategies, and group methods into families based on which aspect of the model they change during optimization (Tab. 1 ). Formally:"},{"type":"para","text":"A learner sees stages 𝒯 1 , … , 𝒯 K \\mathcal{T}_{1},\\dots,\\mathcal{T}_{K} in fixed order. Each 𝒯 k \\mathcal{T}_{k} specifies a training distribution 𝒟 k tr \\mathcal{D}_{k}^{\\text{tr}} and an evaluation set 𝒟 k eval \\mathcal{D}_{k}^{\\text{eval}} . Between stages the learner applies an update operator"},{"type":"para","text":"𝒰 k \\mathcal{U}_{k} is unrestricted: it may modify model weights, edit a system prompt, write to or read from external memory, attach an adapter, or perform no update. A method is fully specified by its family of 𝒰 k \\mathcal{U}_{k} and a per-stage compute budget C C (held constant across methods within a benchmark)."},{"type":"para","text":"Forgetting matrix. After each stage we evaluate on every stage’s eval set, producing"},{"type":"para","text":"with row 0 the base model before any update. We report two transfer quantities:"},{"type":"para","text":"BWT [ 33 ] measures whether later stages improve or damage earlier ones (negative BWT is catastrophic forgetting). FWT measures whether prior stages prepare the model for stages it has not yet trained on, and is the sharper signal: it tests acquisition of structure that transfers, rather than fitting the current stage at the cost of the rest."},{"type":"heading","level":2,"text":"3.2 Update Methods"},{"type":"para","text":"We evaluate eight methods across four families: prompt optimization, supervised weight updates, reinforcement learning, and per-stage architectural updates. All use Qwen3-8B (non-thinking) [ 53 ] as θ 0 \\theta_{0} and run under the protocol of § 3 . Table 1 states what each method carries across the stage boundary; the per-family paragraphs that follow specify each 𝒰 k \\mathcal{U}_{k} and its sequential adaptation."},{"type":"para","text":"Prompt optimization (GEPA, ACE) Both methods leave model weights unchanged. GEPA [ 2 ] optimizes the system prompt through a reflect-and-mutate loop scored on a validation minibatch; the best-scoring mutations propagate forward. ACE [ 59 ] maintains a markdown playbook edited incrementally through explicit add/keep/drop operations, which the authors show resists brevity-bias and context collapse. In our protocol, 𝒰 k \\mathcal{U}_{k} initializes optimization for stage k k from the optimized state at the end of stage k − 1 k{-}1 (the evolved prompt for GEPA, the playbook for ACE), prepended to the default instruction for 𝒯 k \\mathcal{T}_{k} . No replay of prior data is performed. Apparent forward transfer is therefore attributable to the carried prompt or playbook, not to revisiting training examples."},{"type":"para","text":"Supervised / Offline Learning (SFT, SDFT) SFT fine-tunes all parameters on 𝒟 k tr \\mathcal{D}_{k}^{\\text{tr}} , resuming from θ k − 1 \\theta_{k-1} . SDFT [ 48 ] is self-distillation under a forward-KL loss against soft targets from a teacher; in the original work the teacher is the model itself on demonstration-conditioned inputs. Our sequential adaptation: at stage k k , the teacher is θ k − 1 \\theta_{k-1} , so the student learns the new stage under a regularizer pulling it toward its own previous-stage distribution. This is the smallest modification that lifts SDFT to a continual-learning operator and is the only modification we make. SFT and SDFT share a per-stage update budget within C C and differ only in the loss applied at each step."},{"type":"para","text":"Reinforcement / Online Learning (GRPO, SDPO) GRPO [ 46 ] samples rollouts in groups, computes within-group advantages from a verifiable reward, and updates the policy under a KL constraint. For benchmarks without a verifiable reward (10-K, TempWiki) we use a binary correctness signal from the ground-truth label. SDPO [ 19 ] is sequential DPO. Our adaptation: at stage k k , θ k − 1 \\theta_{k-1} generates both the chosen and rejected continuation pools, so the preference signal is implicitly relative to the model’s accumulated prior rather than to a fresh reference. Both methods consume C C as policy-update steps."},{"type":"para","text":"Architectural / Context Compression (Cartridges, In-place TTT) These methods change the model’s update interface rather than the loss applied to its parameters. Cartridges [ 10 ] freezes the backbone and learns a small per-stage component, distilled from a teacher trace and stitched in at inference. The backbone is θ 0 \\theta_{0} for every stage; each stage’s update is structurally local. In-place TTT [ 11 ] performs an input-conditioned weight update at inference using a self-supervised auxiliary loss, reset between inputs. We include In-place TTT as a locality reference: its update is per-input rather than per-stage, which lets us isolate the locality axis from the accumulation axis in the experiment."},{"type":"heading","level":1,"text":"4 Experiments"},{"type":"para","text":"For each task evaluation, we apply our general continual learning protocol and vary only how the world changes between optimization stages. We work with the two axes of change: space , where the model moves across substantively different domains, and time , where the model stays in one domain but the underlying data drifts across stages. Within these axes we instantiate four settings: independent domains (space), where each stage is a different task; selective fact change (time), where some labels change between stages while others remain stable; continuous temporal drift (time), where the task is fixed but the data slides smoothly across stages; and and agentic accumulation (time), where the task family is fixed but the environment’s state drifts as a direct consequence of the agent’s own prior actions rather than an external clock or editor. More details on task design choices can be found in Appendix A.2 ."},{"type":"heading","level":2,"text":"4.1 Domain Transfer"},{"type":"para","text":"When a model is exposed to a new domain, it will need to acquire new skills or capabilities without catastrophically forgetting. This is our evaluation setting and the classic test of continual learning and covers the axis of space."},{"type":"para","text":"Setup. Three unrelated supervised tasks are presented in a fixed order — ToolUse [ 50 ] , FinQA [ 5 ] , SciKE-Bio [ 12 ] with each phase training a Qwen-3-8B on a 500-example subsample of one task’s training set and validating against all three tasks at every step. The subset of ToolUse and SciKE-Bio are taken from [ 48 ] . The task chain produces a forgetting matrix that answers \"after seeing tasks 1 … i 1...i , how well does the model still do task j j ?\" The three tasks deliberately share no surface-level structure, so a method that nominally improves task acquisition can still fail by drifting the model’s output format or by overwriting earlier formats once it learns a later one. For continual training, the best performing checkpoint on the current task is carried over to the next one. Further details about the experimental setup including relevant hyperparameters are included in Appendix A.2.1 .9"},{"type":"para","text":"Prompt and harness optimization surfaces a speed-vs-stability trade-off. We observe that GEPA fits very quickly to some training stages such as FinQA, but has the most severe catastrophic forgetting when moving on to the next stage. In Fig. 2 we observe that despite an almost 40% increase in performance during the FinQA training stage, after the training on biology accuracy drops almost entire back down the baseline accuracy before any training. While ACE appears to be slight more stable than GEPA, it similarly improves and then immediately degrades at this same phase transition."},{"type":"para","text":"Distillation-based methods manage to accumulate knowledge. We observe that SDFT and SDPO both finish the sequential training with FinQA and Bio above their zero-shot baselines while keeping ToolUse near 55–60%; SDFT is the only method in our table whose three final-stage scores are all at or above their respective zero-shot levels."},{"type":"para","text":"Context compression does not help learn new tasks. Cartridges and In-place TTT are less sensitive to than prompt methods to domain shift, but at the same time they do not show steady improvement across all tasks over training phases like SDFT. Performance is largely flat or even simply degrades per task at each progressive training phase. These methods never perform a full weight update, and thus appear to struggle to accumulate new capabilities under the domain shift task."},{"type":"heading","level":2,"text":"4.2 Catastrophic Memorizing: Update Truly New Knowledge"},{"type":"para","text":"In the previous experiment, we tested methods on their ability to stay aligned and accumulate knowledge across time. Here, we ask the opposite: when presented with discrete fact changes, can a model rewrite the affected belief while leaving the rest of its knowledge untouched? Avoiding catastrophic forgetting is necessary, but so is knowing when and how to update. The failure to do so is what we call ’catastrophic memorizing’."},{"type":"para","text":"Data. We construct the chain from four monthly Wikipedia mirrors covering the post-pretraining-cutoff window: 2025-11-20, 2025-12-01, 2026-01-01, and 2026-02-01. For each adjacent pair of snapshots we extract a drift set of (subject, relation, object) triples whose object value changed between the snapshots, treating these as new world-knowledge facts the model is being asked to acquire. We also extract a stable set of (subject, relation) pairs whose object did not change across the entire window, and hold this set out across all phases as a forgetting probe: a method that improves drift accuracy by overwriting the model’s general world knowledge will pay for it on the stable probe."},{"type":"para","text":"Setup. To test for small volume adversarial updates evolving over time on Qwen-3-8B, Wikipedia is mirrored at four monthly snapshots, where every method updates from the same Wikipedia diff snapshot, in its method-appropriate format. Weight-update and RL methods (SFT, SDFT, SDPO, GRPO) consume the 500 drift triples directly as Q/A pairs. Cartridges and In-place TTT consume the corresponding Wikipedia article bodies pulled from per-snapshot – approximately 528 articles per slice. Further details about the experimental setup including relevant hyperparameters are included in Appendix A.2.2 . A filtered, continuously-scored variant of this benchmark (TempWiki-Easy), which isolates the update behavior from noisy diff categories and binary scoring, is analyzed in Appendix A.2.3 ."},{"type":"para","text":"Stability-anchored methods fail to learn how to change. Interestingly, SDFT and SDPO, which perform well on accumulating knowledge in noisy signal, struggle to learn changes in this setting. SDFT’s stable-fact F1 falls from ≈ \\approx 0.30 0.30 at s 1 s_{1} to ≈ \\approx 0.15 0.15 at s 3 s_{3} : by trying to interpolate the new slice against its own previous beliefs, the model ends up corrupting facts that were never supposed to move. SDPO shows the same effect at smaller magnitude. In contrast, GRPO shows strong performance in learning how to change while preserving performance averaging above baseline across the chain + 1.6 +1.6 points. GRPO witnesses the sharpest jump at s 2 s_{2} of + 6.2 +6.2 points which is the largest measured gain of any weight update method."},{"type":"para","text":"Prompt evolution shows strong optimization ability on adapting to new knowledge at the cost of forgetting stable knowledge. GEPA reaches the highest drift F1 in the comparison (0.18 on s 1 s_{1} , 0.24 on s 3 s_{3} ) by writing the new fact directly into the playbook in a single optimization step. The optimizer is rewarded on the validation set, which contains drift facts; it is not rewarded on stable facts, which it never sees. So GEPA edits the playbook freely, and the same edits overwrite stable entries that depend on the same playbook tokens. Stable F1 falls from 0.32 at s 1 s_{1} to 0.26 at s 3 s_{3} . ACE shows the same profile although at a smaller magnitude."},{"type":"para","text":"Context compression barely influences the base model. For cartridges and In-place TTT drift F1 stays within a few points of baseline across all three slices, with no separation in either direction. Cartridges averages − 0.8 -0.8 on drift, with one + 3.2 +3.2 inflection on s 2 s_{2} and despite performing better than SDFT and SDPO, but shows no improvement compared to baseline, even harming the performance, belonging to the same category at times."},{"type":"heading","level":2,"text":"4.3 Noisy Temporal Drift"},{"type":"para","text":"From the previous experiments, we observe over longer time horizons, performance steadily degrades. In our following temporal drift experiments, we dig deeper into the problem by setting up a task where the model can only improve if it manages to accumulate weak and noisy signals through time."},{"type":"para","text":"Setup. We construct a challenging temporal data task by taking a realistic case study in finance: predicting whether a stock moves up or down in the 30 days after its 10-K filing. This setting forces a model to operate inside a two-sided time drift. Some patterns in how firms describe risk, growth, and capital plans should be retained, while other signals are regime-specific (e.g. before and after Covid) and stop working as markets evolve. Doing well now does not guarantee doing well in the future. A model that loses past knowledge cannot tell whether a new filing is unusual relative to history, and a model that holds too tightly to past years may overfit on spurious or transient regimes."},{"type":"para","text":"We use full-length SEC EDGAR filings (typically 10–13K tokens), with each filing labeled up or down based on its 30-day forward return [ 34 , 37 ] . Additional details on data can be found in Appendix A.2.4 . We train sequentially on filing years 2015 through 2020 in chronological order. After each year, we evaluate on every year in the range to measure both backward transfer (did the model actually learn during training) and forward transfer (did it learn something general it can apply to future years). The signal in any individual filing is known to be weak, but in aggregate a small, tradeable edge can be accumulated [ 37 ] (see also Figure 8 ). At baseline before training, Qwen3-8B sits around 50% and accuracy is noisy across years. Further details about the experimental setup including relevant hyperparameters are included in Appendix A.2.4"},{"type":"para","text":"RL struggles when the reward signal is weak and noisy. GRPO shows a significant accuracy drop both on forward- and backward-looking sentiment; its success depends heavily on the reward ranking inside each group. If the reward is noisy, the algorithm may reinforce the wrong answer."},{"type":"para","text":"Stability-anchored methods show the strongest forward transfer with minimal past degradation. SDFT’s past-year accuracy stays flat around 0.50 0.50 while its future-year accuracy climbs to ≈ \\approx 0.62 0.62 by 2020: distilling against its own previous state avoids both drift and over-fitting to the current year. Cartridges is the most stable method we tested, with past and future hovering around 0.55 0.55 – 0.58 0.58 throughout the chain. In-place TTT is the locality reference: its accuracy stays in a tight 34.5–51.5% band across the chain (mean 46.9%) and never separates from baseline. The reset-per-input update geometry is incompatible with absorbing a slow-moving trend, but it pays no price for trying – the method neither acquires nor degrades."},{"type":"para","text":"Prompting methods fail to extract useful knowledge for forward transfer. On this task, similar to RL, GEPA also struggles with learning meaningful information from noisy signals. ACE fits better on past-year data, but also completely fails to predict future years’ trend, which suggests that it does not accumulate meaningful knowledge."},{"type":"para","text":"Qualitatively, GEPA discovers generic financial-analysis heuristics (Figure 5 ). These edits fail to capture characteristics of an ideal adaptive analyst e.g. learn to discard irrelevant sections, normalize sentiment relative to sector or company baselines, or distinguish between newly informative elements and those that are already priced in by the market. While it is possible to construct a strategy with a positive Profit and Loss (PnL) with a rule as simple as focusing on relevant sections in the 10-K (see Figure 8 ), consistent with the observed drop in forward accuracy after optimization, the optimized prompt reduces even this forward PnL. We conclude that in this noisy, time-changing regime, GEPA overfits to transient textual heuristics while failing to preserve exposure to economically meaningful signals that remain predictive out-of-sample."},{"type":"heading","level":2,"text":"4.4 Agentic Task"},{"type":"para","text":"The previous two experiments test the time axis under drift that comes from outside the model. Here we test a third sub-regime of time introduced in § 3 : the task family stays fixed (actions inside a web app), but the environment’s state drifts only because the agent itself acted on it, so state left behind by one step becomes part of the task the model faces at the next."},{"type":"para","text":"Data. We build the agentic suite on top of WebArena-Infinity [ 61 ] , seeding chain generation from each app’s description (which enumerates the available features) and its underlying state files (which fix the entities those features can act on). From these we generate sequential chains in which step i + 1 i{+}1 depends on a concrete piece of state that step i i leaves behind in the environment: in Gmail, create a label, rename it, then apply the renamed label to a specific email. Each step is paired with a programmatic verifier that reads the app’s state directly, so chain success is checked against the environment rather than the agent’s narration. Because the sequence arises from the agent’s own actions rather than an ordering we impose, we treat it as the most realistic instantiation of continual learning over time."},{"type":"para","text":"Setup. Beyond zero-shot evaluation, we ask whether experience compounds across a chain: whether learning from earlier trajectories improves later chain success. We add learning two ways on top of a browser-use agent driving a headless Chromium against the live app. First, an ACE playbook [ 59 ] on Qwen-32B, which accumulates reusable procedural guidance distilled from prior rollouts. Second, supervised fine-tuning of Qwen-8B on successful traces from a stronger teacher agent, distilling the larger agent’s behavior into the smaller model (per-level teachers are given in Appendix A.2.5 ). Within a chain we keep a single agent instance alive across all steps: the browser session is not reset, and each follow-up instruction is appended to the same conversation rather than starting a fresh context, so the model sees its own prior trajectory when acting on step i + 1 i{+}1 . Each step is given a fixed action budget and is scored independently by its verifier. Further details about the experimental setup including relevant hyperparameters are included in Appendix A.2.5 ."},{"type":"para","text":"Results. Both update mechanisms raise end-to-end chain success over their no-learning baseline at every horizon length (Figure 6 ). The ACE playbook lifts Qwen-32B over its zero-shot baseline at every length, most on the three-step (L3) chains ( 32.4 % 32.4\\% to 46.5 % 46.5\\% ). Without training, Qwen-8B collapses to zero past L1; supervised fine-tuning on the teacher’s traces recovers substantial success ( 40.0 % 40.0\\% at L3, 20.0 % 20.0\\% at L5, and 6.7 % 6.7\\% even at L10), and overtakes both Qwen-32B lines at L1 ( 60.0 % 60.0\\% ). Absolute success still decays with chain length for every configuration, as accumulated trajectory state grows and later steps must be grounded in increasingly distant environmental cues, converging near the floor by L10. This is the closest setting to deployed use, since the chain comes from the agent’s own actions rather than an ordering we picked, and future work could go deeper on this realistic agentic setting."},{"type":"heading","level":1,"text":"5 Discussion"},{"type":"heading","level":2,"text":"5.1 Conclusion"},{"type":"para","text":"We introduced a unified framework for evaluating continual learning in large language models, defining it as the problem of increasing competence as the world changes along two axes: space (domain shift) and time (temporal drift). By recasting widely used LLM benchmarks as sequential problems and evaluating eight methods under a common mechanism-agnostic protocol, we surface a consistent set of tradeoffs that span across method families. No single method handles all task regimes well: prompt-based methods fit quickly but overwrite prior capabilities; distillation-based methods accumulate knowledge stably but resist new updates; context compression improves efficiency without improving task acquisition; and online reinforcement learning adapts most effectively to factual change but degrades under noisy reward signals. Overall, our results suggest that continual learning is not a single capability: different patterns of environmental change require fundamentally different update behaviors. We hope that understanding where each method succeeds and fails will help guide the design of more capable continual learning systems."},{"type":"heading","level":2,"text":"5.2 Limitations and Challenges"},{"type":"para","text":"To draw unified conclusions, we use a single model across all evaluations, Qwen3-8B; the relative behavior of the tested methods may change for larger models or models in reasoning mode. In addition, our benchmark suite captures only a subset of realistic environmental change: domain shifts, agentic sequences, noisy temporal drift, and discrete factual updates. While this is a step beyond fixed benchmarks and already reveals that different regimes require different update behaviors, deployed LLMs will face many more open-ended, changing, and uncertain environments and diverse use cases. The results should therefore be read as evidence about which update behaviors are required in different regimes, not as a definitive ranking of continual-learning algorithms."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Traditionally, continual learning has been defined as mitigating catastrophic forgetting [ 40 ] : how do we learn task B B without forgetting task A A ? One approach to this problem is to augment or compose model parameters as proposed in superposition [ 6 ] , adversarial networks [ 9 ] and side networks [ 57 ] . However, this setup is not realistic in the current paradigm of large language models (LLMs), which often rely on large scale pre-, mid-, and post-training to acquire general capability and after deployment need to be adapted to a broad range of potential use cases. At the same time, standard LLM evaluations focus on measuring capabilities on static tasks such as general knowledge [ 51 ] , domain specific abilities [ 17 , 23 , 45 , 16 ] , and challenging frontier problems [ 14 , 38 ] ; these however fail to measure the ability to become more competent under conditions that evolve through time – which is critical for practical deployment.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Without a unified framework, the community approached continual learning as a memory problem solved with retrieval-augmented generation [ 27 ] , in-context learning [ 3 ] , smart context management [ 29 , 56 ] and harnesses [ 59 , 54 , 26 ] ; as a steering problem solved with prompt optimization [ 43 , 47 , 2 ] ; as a reasoning problem solved with reinforcement learning and distillation [ 35 , 60 , 19 , 48 ] ; and as a compression problem solved with architectural changes [ 10 , 30 , 49 , 11 ] . Each of these takes a different bet on what continual learning actually requires, but the field still lacks unified insights across these methods to direct progress.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In our work, we provide a unified framework to define and measure continual learning in modern LLMs. We define continual learning as the problem of increasing competence as the world changes. We consider change along two axes: space and time , as depicted in Figure 1 . Space corresponds to the classical notion of domain or task shift, where the model is asked to acquire a new skill on a different distribution without losing the previous one. Time corresponds to a different and, more common form of change in real deployed systems: the task stays the same, but the underlying world drifts. Time itself can be further split into two sub-regimes that the literature usually conflates. Slow trends (e.g. yearly financial filings) ask the model to stay aligned; the best strategy here is to keep extracting whatever structure survives across years and ignore year-specific noise. Discrete fact changes (e.g. Wikipedia revisions) ask the opposite – the right move is to actually rewrite the affected belief while leaving the rest of the model’s knowledge untouched. A third sub-regime is agentic accumulation: the task family itself stays fixed (e.g., actions inside a web app), but the environment’s state drifts as a direct consequence of the model’s own prior actions, so that what one step leaves behind becomes part of the task the model faces at the next step. This second sub-regime is the LLM analogue of the frame problem [ 39 ] : when the world updates, a learner must decide which of its beliefs are now outdated and which can remain fixed.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To measure continual learning under our definition, we cast widely used LLM evaluations as problems that unfold sequentially under a single mechanism-agnostic protocol, so prompt, weight, and architectural updates can be compared on equal ground. On a common backbone (Qwen3-8B) [ 53 ] , we evaluate eight methods across four families: prompt optimization (GEPA [ 2 ] , ACE [ 59 ] ), offline supervised updates (SFT, SDFT [ 48 ] ), online reinforcement learning (GRPO [ 46 ] , SDPO [ 19 ] ), and context-compression (Cartridges [ 10 ] , In-place TTT [ 11 ] ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Interestingly, we observe a consistent set of tradeoffs across methods: (i) Prompting-based approaches such as GEPA and ACE fit quickly and heavily to past trajectories, achieving strong backward accuracy, but encounter sharp degradation on future tasks. (ii) Distillation-based methods such as SDFT and SDPO accumulate knowledge more stably over time, yet struggle to rapidly incorporate new information or update outdated knowledge. (iii) Context compression approaches, including Cartridges and In-place TTT, improve efficiency and memory management but do not substantially improve the ability to learn new tasks. (iv) In contrast, online reinforcement learning methods such as GRPO adapt more effectively to knowledge updating, though they remain highly sensitive to noisy or unstable reward signals. (v) On the agentic axis, where the stage ordering is generated by the agent’s own actions rather than imposed by us, both a prompt-based playbook (ACE) and weight-based fine-tuning (SFT) compound experience and beat their zero-shot baselines at every chain length, though absolute success still decays as the chain grows.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All together, we make the following contributions:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We formalize continual learning in LLMs as the problem of increasing competence over time. Through our protocol, we provide a unified perspective that allows us to consider both parametric and non-parametric update methods for continual learning. We adapt these methods to the continually learning setting and align their evaluation metrics.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We introduce a broad suite of sequential LLM tasks to evaluate continual learning. These including domain adaptation, agentic tasks, financial analysis, and temporally-dependent knowledge updates.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We find that prompt-based methods alone are insufficient across most regimes, calling for actual learning. Different learning methods each have their own strengths depending on how the task and data shift over time.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Related Works\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Continual Learning. Traditionally, the field has considered continual learning as preventing the problem of catastrophic forgetting [ 40 , 13 ] . In this context, the problem is that sequential parameter updates can overwrite knowledge needed for previous tasks; this led to a focus on designing methods that reduce task interference [ 22 , 15 , 9 , 57 , 6 ] . From another angle, continual learning has been positioned as maintaining plasticity [ 8 ] , where models must not only remember old tasks, but also have the ability to learn new ones [ 36 , 42 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Non-parametric updates. At deployment, we can adapt LLMs without weight changes by changing the inference-time environment around a fixed model. One family of methods changes the information available to the model, for example through retrieval-augmented generation (RAG) [ 27 ] , external memory, reusable long-context representations [ 10 ] , context compression [ 29 ] , or recursive access to large external contexts [ 56 ] . A closely related family changes the behavioral scaffold through which that information is used, including demonstrations [ 3 ] , prompt instructions and agent contexts [ 2 , 59 , 55 ] and multi-stage LM programs [ 44 , 25 , 28 ] . More broadly, agentic systems can adapt by modifying their surrounding harnesses [ 54 , 26 ] , learn component-level reward specifications [ 52 ] or improve through external programs and agent code modifications [ 4 , 43 , 58 , 47 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Gradient-based (parametric) updates. Another adaption approach is to update weights or activations. At one extreme, full fine-tuning modifies all parameters of the model; while effective, this regime is often computationally expensive and unrealistic for frequent or online adaptation. Activation editing [ 20 ] uses task vectors applied to all parameters. More commonly, work studies parameter-efficient methods, such as LoRA [ 18 ] and adapter-based tuning [ 30 , 33 ] , which restrict updates to small subsets of parameters. An important but underexplored regime involves severely constrained updates—very few gradient steps and tightly limited compute—closer to the conditions under which continual adaptation would need to occur in deployed systems. Continual learning methods differ not only in where they store updates, but also in what learning signal drives the update. OPSD [ 60 , 35 ] , SDFT [ 48 ] and SDPO [ 19 ] learn from self-distillation, GRPO [ 46 ] and h1 [ 41 ] optimize end-of-task rewards, LoRD [ 32 ] uses teacher–student distillation, sparse memory finetuning [ 30 ] localizes the next-token objective to sparsely accessed memory slots, test-time training [ 11 ] performs input-conditioned weight updates during inference.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Method\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 A Protocol for Continual Learning\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Setup. Typically, continual-learning evaluations are tightly coupled to a particular method: weight-update methods use task accuracy with a fixed fine-tuning budget; prompt-optimization often uses few-shot generalization with a fixed token budget; agentic frameworks rely on end-to-end pass rate. As a result, prompt, weight, and architectural updates are rarely compared fairly against one another. To gather broader insights about what methods are necessary for continual learning, we define a simple protocol that is compatible with any update operator. To do so, we consider all forms of updates (prompting, full weight, sparse weight/activation) as potential strategies, and group methods into families based on which aspect of the model they change during optimization (Tab. 1 ). Formally:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A learner sees stages 𝒯 1 , … , 𝒯 K \\\\mathcal{T}_{1},\\\\dots,\\\\mathcal{T}_{K} in fixed order. Each 𝒯 k \\\\mathcal{T}_{k} specifies a training distribution 𝒟 k tr \\\\mathcal{D}_{k}^{\\\\text{tr}} and an evaluation set 𝒟 k eval \\\\mathcal{D}_{k}^{\\\\text{eval}} . Between stages the learner applies an update operator\"\n },\n {\n \"type\": \"para\",\n \"text\": \"𝒰 k \\\\mathcal{U}_{k} is unrestricted: it may modify model weights, edit a system prompt, write to or read from external memory, attach an adapter, or perform no update. A method is fully specified by its family of 𝒰 k \\\\mathcal{U}_{k} and a per-stage compute budget C C (held constant across methods within a benchmark).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Forgetting matrix. After each stage we evaluate on every stage’s eval set, producing\"\n },\n {\n \"type\": \"para\",\n \"text\": \"with row 0 the base model before any update. We report two transfer quantities:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"BWT [ 33 ] measures whether later stages improve or damage earlier ones (negative BWT is catastrophic forgetting). FWT measures whether prior stages prepare the model for stages it has not yet trained on, and is the sharper signal: it tests acquisition of structure that transfers, rather than fitting the current stage at the cost of the rest.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2 Update Methods\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate eight methods across four families: prompt optimization, supervised weight updates, reinforcement learning, and per-stage architectural updates. All use Qwen3-8B (non-thinking) [ 53 ] as θ 0 \\\\theta_{0} and run under the protocol of § 3 . Table 1 states what each method carries across the stage boundary; the per-family paragraphs that follow specify each 𝒰 k \\\\mathcal{U}_{k} and its sequential adaptation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prompt optimization (GEPA, ACE) Both methods leave model weights unchanged. GEPA [ 2 ] optimizes the system prompt through a reflect-and-mutate loop scored on a validation minibatch; the best-scoring mutations propagate forward. ACE [ 59 ] maintains a markdown playbook edited incrementally through explicit add/keep/drop operations, which the authors show resists brevity-bias and context collapse. In our protocol, 𝒰 k \\\\mathcal{U}_{k} initializes optimization for stage k k from the optimized state at the end of stage k − 1 k{-}1 (the evolved prompt for GEPA, the playbook for ACE), prepended to the default instruction for 𝒯 k \\\\mathcal{T}_{k} . No replay of prior data is performed. Apparent forward transfer is therefore attributable to the carried prompt or playbook, not to revisiting training examples.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Supervised / Offline Learning (SFT, SDFT) SFT fine-tunes all parameters on 𝒟 k tr \\\\mathcal{D}_{k}^{\\\\text{tr}} , resuming from θ k − 1 \\\\theta_{k-1} . SDFT [ 48 ] is self-distillation under a forward-KL loss against soft targets from a teacher; in the original work the teacher is the model itself on demonstration-conditioned inputs. Our sequential adaptation: at stage k k , the teacher is θ k − 1 \\\\theta_{k-1} , so the student learns the new stage under a regularizer pulling it toward its own previous-stage distribution. This is the smallest modification that lifts SDFT to a continual-learning operator and is the only modification we make. SFT and SDFT share a per-stage update budget within C C and differ only in the loss applied at each step.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Reinforcement / Online Learning (GRPO, SDPO) GRPO [ 46 ] samples rollouts in groups, computes within-group advantages from a verifiable reward, and updates the policy under a KL constraint. For benchmarks without a verifiable reward (10-K, TempWiki) we use a binary correctness signal from the ground-truth label. SDPO [ 19 ] is sequential DPO. Our adaptation: at stage k k , θ k − 1 \\\\theta_{k-1} generates both the chosen and rejected continuation pools, so the preference signal is implicitly relative to the model’s accumulated prior rather than to a fresh reference. Both methods consume C C as policy-update steps.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Architectural / Context Compression (Cartridges, In-place TTT) These methods change the model’s update interface rather than the loss applied to its parameters. Cartridges [ 10 ] freezes the backbone and learns a small per-stage component, distilled from a teacher trace and stitched in at inference. The backbone is θ 0 \\\\theta_{0} for every stage; each stage’s update is structurally local. In-place TTT [ 11 ] performs an input-conditioned weight update at inference using a self-supervised auxiliary loss, reset between inputs. We include In-place TTT as a locality reference: its update is per-input rather than per-stage, which lets us isolate the locality axis from the accumulation axis in the experiment.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Experiments\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For each task evaluation, we apply our general continual learning protocol and vary only how the world changes between optimization stages. We work with the two axes of change: space , where the model moves across substantively different domains, and time , where the model stays in one domain but the underlying data drifts across stages. Within these axes we instantiate four settings: independent domains (space), where each stage is a different task; selective fact change (time), where some labels change between stages while others remain stable; continuous temporal drift (time), where the task is fixed but the data slides smoothly across stages; and and agentic accumulation (time), where the task family is fixed but the environment’s state drifts as a direct consequence of the agent’s own prior actions rather than an external clock or editor. More details on task design choices can be found in Appendix A.2 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 Domain Transfer\"\n },\n {\n \"type\": \"para\",\n \"text\": \"When a model is exposed to a new domain, it will need to acquire new skills or capabilities without catastrophically forgetting. This is our evaluation setting and the classic test of continual learning and covers the axis of space.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Setup. Three unrelated supervised tasks are presented in a fixed order — ToolUse [ 50 ] , FinQA [ 5 ] , SciKE-Bio [ 12 ] with each phase training a Qwen-3-8B on a 500-example subsample of one task’s training set and validating against all three tasks at every step. The subset of ToolUse and SciKE-Bio are taken from [ 48 ] . The task chain produces a forgetting matrix that answers \\\"after seeing tasks 1 … i 1...i , how well does the model still do task j j ?\\\" The three tasks deliberately share no surface-level structure, so a method that nominally improves task acquisition can still fail by drifting the model’s output format or by overwriting earlier formats once it learns a later one. For continual training, the best performing checkpoint on the current task is carried over to the next one. Further details about the experimental setup including relevant hyperparameters are included in Appendix A.2.1 .9\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prompt and harness optimization surfaces a speed-vs-stability trade-off. We observe that GEPA fits very quickly to some training stages such as FinQA, but has the most severe catastrophic forgetting when moving on to the next stage. In Fig. 2 we observe that despite an almost 40% increase in performance during the FinQA training stage, after the training on biology accuracy drops almost entire back down the baseline accuracy before any training. While ACE appears to be slight more stable than GEPA, it similarly improves and then immediately degrades at this same phase transition.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Distillation-based methods manage to accumulate knowledge. We observe that SDFT and SDPO both finish the sequential training with FinQA and Bio above their zero-shot baselines while keeping ToolUse near 55–60%; SDFT is the only method in our table whose three final-stage scores are all at or above their respective zero-shot levels.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Context compression does not help learn new tasks. Cartridges and In-place TTT are less sensitive to than prompt methods to domain shift, but at the same time they do not show steady improvement across all tasks over training phases like SDFT. Performance is largely flat or even simply degrades per task at each progressive training phase. These methods never perform a full weight update, and thus appear to struggle to accumulate new capabilities under the domain shift task.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 Catastrophic Memorizing: Update Truly New Knowledge\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In the previous experiment, we tested methods on their ability to stay aligned and accumulate knowledge across time. Here, we ask the opposite: when presented with discrete fact changes, can a model rewrite the affected belief while leaving the rest of its knowledge untouched? Avoiding catastrophic forgetting is necessary, but so is knowing when and how to update. The failure to do so is what we call ’catastrophic memorizing’.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Data. We construct the chain from four monthly Wikipedia mirrors covering the post-pretraining-cutoff window: 2025-11-20, 2025-12-01, 2026-01-01, and 2026-02-01. For each adjacent pair of snapshots we extract a drift set of (subject, relation, object) triples whose object value changed between the snapshots, treating these as new world-knowledge facts the model is being asked to acquire. We also extract a stable set of (subject, relation) pairs whose object did not change across the entire window, and hold this set out across all phases as a forgetting probe: a method that improves drift accuracy by overwriting the model’s general world knowledge will pay for it on the stable probe.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Setup. To test for small volume adversarial updates evolving over time on Qwen-3-8B, Wikipedia is mirrored at four monthly snapshots, where every method updates from the same Wikipedia diff snapshot, in its method-appropriate format. Weight-update and RL methods (SFT, SDFT, SDPO, GRPO) consume the 500 drift triples directly as Q/A pairs. Cartridges and In-place TTT consume the corresponding Wikipedia article bodies pulled from per-snapshot – approximately 528 articles per slice. Further details about the experimental setup including relevant hyperparameters are included in Appendix A.2.2 . A filtered, continuously-scored variant of this benchmark (TempWiki-Easy), which isolates the update behavior from noisy diff categories and binary scoring, is analyzed in Appendix A.2.3 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Stability-anchored methods fail to learn how to change. Interestingly, SDFT and SDPO, which perform well on accumulating knowledge in noisy signal, struggle to learn changes in this setting. SDFT’s stable-fact F1 falls from ≈ \\\\approx 0.30 0.30 at s 1 s_{1} to ≈ \\\\approx 0.15 0.15 at s 3 s_{3} : by trying to interpolate the new slice against its own previous beliefs, the model ends up corrupting facts that were never supposed to move. SDPO shows the same effect at smaller magnitude. In contrast, GRPO shows strong performance in learning how to change while preserving performance averaging above baseline across the chain + 1.6 +1.6 points. GRPO witnesses the sharpest jump at s 2 s_{2} of + 6.2 +6.2 points which is the largest measured gain of any weight update method.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prompt evolution shows strong optimization ability on adapting to new knowledge at the cost of forgetting stable knowledge. GEPA reaches the highest drift F1 in the comparison (0.18 on s 1 s_{1} , 0.24 on s 3 s_{3} ) by writing the new fact directly into the playbook in a single optimization step. The optimizer is rewarded on the validation set, which contains drift facts; it is not rewarded on stable facts, which it never sees. So GEPA edits the playbook freely, and the same edits overwrite stable entries that depend on the same playbook tokens. Stable F1 falls from 0.32 at s 1 s_{1} to 0.26 at s 3 s_{3} . ACE shows the same profile although at a smaller magnitude.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Context compression barely influences the base model. For cartridges and In-place TTT drift F1 stays within a few points of baseline across all three slices, with no separation in either direction. Cartridges averages − 0.8 -0.8 on drift, with one + 3.2 +3.2 inflection on s 2 s_{2} and despite performing better than SDFT and SDPO, but shows no improvement compared to baseline, even harming the performance, belonging to the same category at times.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3 Noisy Temporal Drift\"\n },\n {\n \"type\": \"para\",\n \"text\": \"From the previous experiments, we observe over longer time horizons, performance steadily degrades. In our following temporal drift experiments, we dig deeper into the problem by setting up a task where the model can only improve if it manages to accumulate weak and noisy signals through time.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Setup. We construct a challenging temporal data task by taking a realistic case study in finance: predicting whether a stock moves up or down in the 30 days after its 10-K filing. This setting forces a model to operate inside a two-sided time drift. Some patterns in how firms describe risk, growth, and capital plans should be retained, while other signals are regime-specific (e.g. before and after Covid) and stop working as markets evolve. Doing well now does not guarantee doing well in the future. A model that loses past knowledge cannot tell whether a new filing is unusual relative to history, and a model that holds too tightly to past years may overfit on spurious or transient regimes.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We use full-length SEC EDGAR filings (typically 10–13K tokens), with each filing labeled up or down based on its 30-day forward return [ 34 , 37 ] . Additional details on data can be found in Appendix A.2.4 . We train sequentially on filing years 2015 through 2020 in chronological order. After each year, we evaluate on every year in the range to measure both backward transfer (did the model actually learn during training) and forward transfer (did it learn something general it can apply to future years). The signal in any individual filing is known to be weak, but in aggregate a small, tradeable edge can be accumulated [ 37 ] (see also Figure 8 ). At baseline before training, Qwen3-8B sits around 50% and accuracy is noisy across years. Further details about the experimental setup including relevant hyperparameters are included in Appendix A.2.4\"\n },\n {\n \"type\": \"para\",\n \"text\": \"RL struggles when the reward signal is weak and noisy. GRPO shows a significant accuracy drop both on forward- and backward-looking sentiment; its success depends heavily on the reward ranking inside each group. If the reward is noisy, the algorithm may reinforce the wrong answer.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Stability-anchored methods show the strongest forward transfer with minimal past degradation. SDFT’s past-year accuracy stays flat around 0.50 0.50 while its future-year accuracy climbs to ≈ \\\\approx 0.62 0.62 by 2020: distilling against its own previous state avoids both drift and over-fitting to the current year. Cartridges is the most stable method we tested, with past and future hovering around 0.55 0.55 – 0.58 0.58 throughout the chain. In-place TTT is the locality reference: its accuracy stays in a tight 34.5–51.5% band across the chain (mean 46.9%) and never separates from baseline. The reset-per-input update geometry is incompatible with absorbing a slow-moving trend, but it pays no price for trying – the method neither acquires nor degrades.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prompting methods fail to extract useful knowledge for forward transfer. On this task, similar to RL, GEPA also struggles with learning meaningful information from noisy signals. ACE fits better on past-year data, but also completely fails to predict future years’ trend, which suggests that it does not accumulate meaningful knowledge.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Qualitatively, GEPA discovers generic financial-analysis heuristics (Figure 5 ). These edits fail to capture characteristics of an ideal adaptive analyst e.g. learn to discard irrelevant sections, normalize sentiment relative to sector or company baselines, or distinguish between newly informative elements and those that are already priced in by the market. While it is possible to construct a strategy with a positive Profit and Loss (PnL) with a rule as simple as focusing on relevant sections in the 10-K (see Figure 8 ), consistent with the observed drop in forward accuracy after optimization, the optimized prompt reduces even this forward PnL. We conclude that in this noisy, time-changing regime, GEPA overfits to transient textual heuristics while failing to preserve exposure to economically meaningful signals that remain predictive out-of-sample.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.4 Agentic Task\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The previous two experiments test the time axis under drift that comes from outside the model. Here we test a third sub-regime of time introduced in § 3 : the task family stays fixed (actions inside a web app), but the environment’s state drifts only because the agent itself acted on it, so state left behind by one step becomes part of the task the model faces at the next.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Data. We build the agentic suite on top of WebArena-Infinity [ 61 ] , seeding chain generation from each app’s description (which enumerates the available features) and its underlying state files (which fix the entities those features can act on). From these we generate sequential chains in which step i + 1 i{+}1 depends on a concrete piece of state that step i i leaves behind in the environment: in Gmail, create a label, rename it, then apply the renamed label to a specific email. Each step is paired with a programmatic verifier that reads the app’s state directly, so chain success is checked against the environment rather than the agent’s narration. Because the sequence arises from the agent’s own actions rather than an ordering we impose, we treat it as the most realistic instantiation of continual learning over time.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Setup. Beyond zero-shot evaluation, we ask whether experience compounds across a chain: whether learning from earlier trajectories improves later chain success. We add learning two ways on top of a browser-use agent driving a headless Chromium against the live app. First, an ACE playbook [ 59 ] on Qwen-32B, which accumulates reusable procedural guidance distilled from prior rollouts. Second, supervised fine-tuning of Qwen-8B on successful traces from a stronger teacher agent, distilling the larger agent’s behavior into the smaller model (per-level teachers are given in Appendix A.2.5 ). Within a chain we keep a single agent instance alive across all steps: the browser session is not reset, and each follow-up instruction is appended to the same conversation rather than starting a fresh context, so the model sees its own prior trajectory when acting on step i + 1 i{+}1 . Each step is given a fixed action budget and is scored independently by its verifier. Further details about the experimental setup including relevant hyperparameters are included in Appendix A.2.5 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Results. Both update mechanisms raise end-to-end chain success over their no-learning baseline at every horizon length (Figure 6 ). The ACE playbook lifts Qwen-32B over its zero-shot baseline at every length, most on the three-step (L3) chains ( 32.4 % 32.4\\\\% to 46.5 % 46.5\\\\% ). Without training, Qwen-8B collapses to zero past L1; supervised fine-tuning on the teacher’s traces recovers substantial success ( 40.0 % 40.0\\\\% at L3, 20.0 % 20.0\\\\% at L5, and 6.7 % 6.7\\\\% even at L10), and overtakes both Qwen-32B lines at L1 ( 60.0 % 60.0\\\\% ). Absolute success still decays with chain length for every configuration, as accumulated trajectory state grows and later steps must be grounded in increasingly distant environmental cues, converging near the floor by L10. This is the closest setting to deployed use, since the chain comes from the agent’s own actions rather than an ordering we picked, and future work could go deeper on this realistic agentic setting.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Discussion\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.1 Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We introduced a unified framework for evaluating continual learning in large language models, defining it as the problem of increasing competence as the world changes along two axes: space (domain shift) and time (temporal drift). By recasting widely used LLM benchmarks as sequential problems and evaluating eight methods under a common mechanism-agnostic protocol, we surface a consistent set of tradeoffs that span across method families. No single method handles all task regimes well: prompt-based methods fit quickly but overwrite prior capabilities; distillation-based methods accumulate knowledge stably but resist new updates; context compression improves efficiency without improving task acquisition; and online reinforcement learning adapts most effectively to factual change but degrades under noisy reward signals. Overall, our results suggest that continual learning is not a single capability: different patterns of environmental change require fundamentally different update behaviors. We hope that understanding where each method succeeds and fails will help guide the design of more capable continual learning systems.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.2 Limitations and Challenges\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To draw unified conclusions, we use a single model across all evaluations, Qwen3-8B; the relative behavior of the tested methods may change for larger models or models in reasoning mode. In addition, our benchmark suite captures only a subset of realistic environmental change: domain shifts, agentic sequences, noisy temporal drift, and discrete factual updates. While this is a step beyond fixed benchmarks and already reveals that different regimes require different update behaviors, deployed LLMs will face many more open-ended, changing, and uncertain environments and diverse use cases. The results should therefore be read as evidence about which update behaviors are required in different regimes, not as a definitive ranking of continual-learning algorithms.\"\n }\n]","isConversational":false}}},{"rowIdx":93,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.07707"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Co-LMLM: Continuous-Query Limited Memory Language Models"},"authors":{"kind":"list like","value":["Yair Feldman","Linxi Zhao","Nathan Godey","Dongyoung Go","Yilun Hua","Kilian Q. Weinberger","Jennifer J. Sun","Yoav Artzi"],"string":"[\n \"Yair Feldman\",\n \"Linxi Zhao\",\n \"Nathan Godey\",\n \"Dongyoung Go\",\n \"Yilun Hua\",\n \"Kilian Q. Weinberger\",\n \"Jennifer J. Sun\",\n \"Yoav Artzi\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.07707v1"},"published":{"kind":"timestamp","value":"2026-07-08T00:00:00","string":"2026-07-08T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Recently, there has been increasing interest in large language models (LLMs) that are trained to externalize knowledge (Ghosal et al., 2025 ; Zhao et al., 2026 ; Pouransari et al., 2026 ) . A particularly compelling approach is the limited memory language model (LMLM; Zhao et al., 2026 ) , where the LLM is pre-trained to externalize knowledge into a human-readable knowledge base (KB). This design offers several advantages. The KB is interpretable and easily editable, allowing for unlearning with no utility tradeoff, and enabling easy attribution of knowledge used to the source material."},{"type":"para","text":"Zhao et al. instantiate the LMLM approach with a relational KB, which we refer to as Rel-LMLM . Although representing a significant departure from conventional LLMs, Rel-LMLM proposes a training process that follows the common scalable next-token-prediction pre-training, with the key difference of pre-processing the data to simulate knowledge retrieval. Experiments demonstrate significantly lower perplexities compared to conventional LLMs and factuality scores similar to models several orders of magnitude larger, while retaining similar utility (i.e., NLU scores) to LLMs of the same size."},{"type":"para","text":"However, Rel-LMLM has several key limitations that constrain the scaling of the pre-training process and the knowledge retrieval expressivity. Pre-training relies on Wikipedia, where each article is centered on a specific entity, making it straightforward to automatically annotate relational queries at scale during data pre-processing. Although Wikipedia is sufficient for a proof-of-concept demonstration, it provides no avenue to scale much further. The relational representation itself, although human readable, introduces several limitations. It restricts the data that can be externalized to items that are the object of natural language relational tuples, where both the subject and relation are mentioned previously in the text. This same mechanism limits the expressivity of the retrieval itself to queries that can be expressed as simple subject-relation natural language queries, and generating these queries incurs the added cost of decoding multiple additional tokens during inference. Finally, synthesizing the exact queries in the data pre-processing step holds retrieval pathways as constant and pre-determined, significantly restricting the expressivity of the learned retrieval mechanism."},{"type":"para","text":"We propose continuous-query limited-memory language models ( Co-LMLM ), an approach that addresses these limitations, while retaining all the advantages that the LMLM design offers. The key modeling difference from Rel-LMLM is that queries are issued as single continuous vectors rather than structured relational tuples. The KB stores vector keys and string values instead of relational tuples. We jointly train the LLM Transformer for language modeling and retrieval using pre-training data pre-processed to simulate KB queries. The pre-processing specifies only when retrieval takes place and what string it returns, rather than dictating the explicit query representation. The query vector emitted by the LLM are optimized via a contrastive loss over synthesized positive and negative pairs. This allows Co-LMLM to retrieve knowledge at the cost of a single generation step, generate flexible queries, externalize a much broader set of knowledge without role or relational constraints, and generalize to pre-training data well beyond Wikipedia."},{"type":"para","text":"We evaluate Co-LMLM against Rel-LMLM models of comparable size and standard baselines trained on the same corpus without knowledge externalization. Across model scales, Co-LMLM achieves lower perplexity than standard models, substantially improves factual precision over both standard and Rel-LMLM , while preserving downstream NLU performance. At the 360M scale, Co-LMLM achieves lower perplexities than off-the-shelf HF/ SmolLM2-360M , which is pre-trained on 40 × \\times more data, and SimpleQA-verified (Haas et al., 2026 ) scores that are in line with gpt-4o-mini and higher than Claude Sonnet 4.5. At the same time, Co-LMLM retains the controllability benefits of LMLM : its external memory remains editable and supports direct unlearning through database operations. Finally, extending pre-training from Wikipedia to FineWeb-Edu further improves factual precision, showing Co-LMLM ’s benefit from scaling the pre-training data."},{"type":"para","text":"Code, data, and models are available at: https://lil-lab.github.io/co-lmlm-web ."},{"type":"heading","level":1,"text":"2 Related Work"},{"type":"heading","level":4,"text":"Parametric Knowledge and its Limitations"},{"type":"para","text":"Pre-trained language models store factual knowledge implicitly in their parameters and recall it at generation time (Petroni et al., 2019 ; Roberts et al., 2020 ) . This form of storage is inherently limited: factual recall scales with model size and exposure frequency, making long-tail knowledge difficult to capture (Kandpal et al., 2023 ; Mallen et al., 2023 ) . Moreover, factual associations are entangled with linguistic representations, making individual facts difficult to attribute, edit, or remove, and contributing to hallucinations and stale knowledge (Elhage et al., 2022 ; Dai et al., 2022 ; Huang et al., 2024 ) . They also make knowledge editing and unlearning challenging: removing or modifying specific facts typically requires additional training or specialized objectives, often trading off with general capabilities (Maini et al., 2024 ) . MemSinks isolates memorization-specific parameters, but still keeps knowledge within the model (Ghosal et al., 2025 ) . These challenges motivate approaches that move factual knowledge into separate, editable stores."},{"type":"heading","level":4,"text":"Externalizing Knowledge from Model Weights"},{"type":"para","text":"A growing line of work augments language models with non-parametric or semi-parametric memory, or aims to move factual knowledge more explicitly out of model parameters . kNN-LM retrieves nearest-neighbor examples from an external datastore and interpolates their probabilities with the model distribution (Khandelwal et al., 2020 ) , while RETRO retrieves neighboring chunks and conditions on them through cross-attention (Borgeaud et al., 2022 ) . Other approaches introduce sparse memory layers or learned key-value memories to increase model capacity and factuality (Lample et al., 2019 ; Berges et al., 2025 ; Wu et al., 2022 ; Peng et al., 2023 ; Yang et al., 2024 ; Cheng et al., 2026 ; Tseng and Sa, 2026 ) . These methods can improve perplexity or factuality, but the stored information remains not directly inspectable or editable."},{"type":"heading","level":4,"text":"Limited Memory Language Models"},{"type":"para","text":"Zhao et al. ( 2026 ) is the most related work to ours. They propose Limited Memory Language Models (LMLMs), which externalize factual knowledge into a textual relational KB during pretraining rather than encoding it in model parameters. We refer to this approach as Rel-LMLM to distinguish it from our continuous-query formulation. Rel-LMLM enables faster pre-training, interpretable retrieval, and direct editing, but its relational design limits the external memory coverage: facts must be expressible as Wikipedia-style subject-relation-object tuples, and retrieval depends on brittle textual queries and introduces additional generation token overhead. Co-LMLM preserves the benefits of externalized and editable KB while relaxing these constraints. It replaces relational tuples with free-form textual knowledge spans; and decoded relational queries with continuous queries emitted from the model’s hidden states. These queries can carry richer contextual information for retrieval, allowing Co-LMLM to scale beyond Wikipedia-style relational facts to general corpora (e.g., FineWeb-Edu), while reducing the token overhead of query generation."},{"type":"heading","level":4,"text":"Retrieval-Augmented Language Models"},{"type":"para","text":"Retrieval-Augmented Generation (RAG) conditions generation on retrieved passages during inference (Lewis et al., 2020 ; Izacard and Grave, 2021 ) , while retrieval-aware pretraining methods incorporate retrieval into model training (Guu et al., 2020 ; Borgeaud et al., 2022 ; Izacard et al., 2023 ) . Other methods let models decide when to retrieve and issue textual queries, as in Self-RAG, DRAGIN, and LMLM (Asai et al., 2024 ; Su et al., 2024 ; Zhao et al., 2026 ) . Our technique is related to methods that generate retrieval signals directly from internal representations rather than decoded queries (Muennighoff et al., 2025 ; Zhang et al., 2024a , 2025 ) . Unlike most retrieval-augmented methods, which add external context while leaving parametric knowledge largely intact, Co-LMLM uses retrieval to define the parametric/non-parametric boundary during pretraining: factual spans are externalized into a schema-free index, and the model learns when and how to retrieve them through a contrastive objective trained jointly with next-token prediction."},{"type":"heading","level":1,"text":"3 Method"},{"type":"para","text":"The Co-LMLM model is a decoder-only Transformer, which functions as both a language model and a dense retriever. Figure 2 provides a high-level overview of the method. The model generates query vectors similar to tool calling, and the returned values are encoded by the model, before decoding resumes ( Section 3.1 ). We jointly train the language modeling and retriever functions ( Section 3.2 ), using pre-training data that is pre-processed to simulate retrieval and synthetic questions that provide supervision for the contrastive retrieval objective ( Section 3.3 )."},{"type":"heading","level":2,"text":"3.1 Modeling and Inference"},{"type":"para","text":"Co-LMLM performs retrieval through continuous tool calls. The external memory is a key-value store: each key is a dense vector produced by the model, and each value is a factual text span. Unlike standard tool-calling methods (Schick et al., 2023 ) , the retrieval query is not represented as text. Instead, the query is the model’s hidden state at a special token."},{"type":"para","text":"At inference time, the model decodes tokens autoregressively. When it emits the special <FACT> token, we read the last-layer hidden state at that position as a retrieval query, query the index, and retrieve the top-1 fact text snippet. We splice the retrieved snippet into the context immediately after <FACT> , append a closing </FACT> special token, and resume autoregressive decoding from </FACT> onward. Each retrieval therefore costs one extra autoregressive step (the opening <FACT> token) and grows the context by the length of the retrieved span."},{"type":"heading","level":2,"text":"3.2 Training"},{"type":"para","text":"Our training data consists of raw text in which factual spans are enclosed by <FACT> and </FACT> . Each marked span is paired with a natural-language question whose answer is the span. Figure 2 (top) illustrates a single example. Section 3.3 describes how we pre-process raw pre-training data to create this data. Our objectives are to train the model to both predict the next token and externalize the information in the delimited spans, construct a KB from all such spans, and to emit retrieval queries to fetch spans from the KB as needed. The key to training the retrieval function is contrastive query pairs between documents and questions, allowing for both positive and negative pairs."},{"type":"para","text":"Let x x be an input document with factual spans bracketed by <FACT> and </FACT> . For each factual span, we have a question Q Q . The question is synthesized so its answer is the corresponding bracketed span in x x , and it is appended with a query marker token <FACT-q> . The retrieval vectors at the query markers appended to the questions are used as positive examples for the contrastive retrieval loss. The pre-training panel of Figure 2 (middle) visualized how x x and Q Q are used."},{"type":"para","text":"Let p ( x t | x < t ; θ ) p(x_{t}|x_{<t};\\theta) be the probability of the token x t x_{t} in the sequence document x x , parameterized by θ \\theta . In our model, p θ p_{\\theta} is computed by a Transformer. We denote ϕ t ( x < t ; θ ) \\phi_{t}(x_{<t};\\theta) as the features computed by the Transformer for index t t in sequence x x . 1 1 1 We L2-normalize the feature representations as they are used as retrieval queries (Chen et al., 2020 ) . . This is the same Transformer used for p θ p_{\\theta} ."},{"type":"para","text":"We jointly minimize next-token prediction (NTP) and bidirectional contrastive losses. The NTP loss is: ℒ NTP ( θ ) = − ∑ t ∉ M log p ( x t ∣ x < t ; θ ) \\mathcal{L}_{\\mathrm{NTP}}(\\theta)=-\\sum_{t\\notin M}\\log p(x_{t}\\mid x_{<t};\\theta) , where M M is the set of positions in x x that are strictly between <FACT> and </FACT> , inclusive of the closing </FACT> , but excluding the opening <FACT> . This NTP formulation means that we do not optimize our model to memorize the bracketed fact tokens, which represent the knowledge we aim to externalize. This is a critical design choice to externalize knowledge. We do optimize the loss of the opening <FACT> token, because the model must know to generate it in order to trigger retrieval. The closing </FACT> is appended mechanically, so does not require optimization."},{"type":"para","text":"The contrastive loss treats each fact query from the document x x and its corresponding question Q Q as a positive pair, and all other questions as negatives. For a fact that starts at index t t in a document x x , let f = ϕ t ( x < t ; θ ) f=\\phi_{t}(x_{<t};\\theta) be the query representation for retrieving this fact. Meaning, the feature representation for the <FACT> token opening this fact span. Correspondingly, for a question Q Q where the query marker <FACT-q> is at index t t , let q = ϕ t ( Q < t ) q=\\phi_{t}(Q_{<t}) be the query representation. That is, the feature representation computed for <FACT-q> . We use the InfoNCE loss (van den Oord et al., 2018 ) , compute on the current batch. Let ℬ = { ( f ( i ) , q ( i ) ) } i = 1 B \\mathcal{B}=\\{(f^{(i)},q^{(i)})\\}_{i=1}^{B} be the set of all paired fact document queries f ( i ) f^{(i)} and question queries q ( i ) q^{(i)} in the current batch. For each ( f ( i ) , q ( i ) ) ∈ B (f^{(i)},q^{(i)})\\in{B} , the loss terms are:"},{"type":"para","text":"where τ \\tau is a temperature hyperparameter. We average the two directions:"},{"type":"para","text":"We optimize the joint loss ℒ = ℒ NTP + λ ℒ CL \\mathcal{L}=\\mathcal{L}_{\\mathrm{NTP}}+\\lambda\\mathcal{L}_{\\mathrm{CL}} , with λ \\lambda a hyperparameter set to 0.25 0.25 by default."},{"type":"heading","level":4,"text":"Building the Index"},{"type":"para","text":"We build a dense retrieval index from our pre-training data. The index pairs vector keys with fact spans. We run the pre-trained language model over the pre-training corpus, extract the L2-normalized hidden state ϕ ( ⋅ ; θ ) \\phi(\\cdot;\\theta) at every <FACT> position, and store it as a key in a dense vector index whose value is the verbatim in-text span at that position. Indexing a new corpus therefore requires only pre-processing it ( Section 3.3 ) and a forward pass of Co-LMLM ."},{"type":"heading","level":2,"text":"3.3 Training Data Pre-processing"},{"type":"para","text":"Training requires documents annotated with fact spans and paired questions ( Section 3.2 ). We design an automated, scalable process to annotate raw pre-training data. The pre-processing pipeline has three stages: annotate a seed set with by prompting a state-of-the-art frontier LLM; fine-tune lightweight annotators on the seed data; and apply them to full pre-training corpus. Critically, we fine-tune smaller models because of the cost of annotating at scale with a large frontier model."},{"type":"heading","level":4,"text":"Annotation Specification"},{"type":"para","text":"We annotate each training document with a set of fact spans . A fact span is a sub-string of a pre-training document x x that encodes world knowledge we want our model to externalize – to retrieve rather than memorize. Each fact span is associated with two additional pieces of information: a context-dependent question that captures the retrieval intent at this position without itself carrying any factual content, and a snippet-form answer that answers the question in a form that need not match the surface form of the span. The question is used as the question Q Q during training. An annotated fact takes the form <FACT q={question} a={answer}>{span}</FACT> ."},{"type":"para","text":"The annotation reflects three guidelines: (a) spans are the narrowest self-contained sub-string that fully answers the question; (b) questions are written as standalone search queries that name their subject explicitly, and they avoid mentioning information that appears later in the document so that no future fact leaks through the question; and (c) answers vary the surface form of the span, so the model is exposed at training time to retrieved values that differ lexically from the surrounding context. We provide more details about the prompt used for the seed annotation in Appendix C ."},{"type":"heading","level":4,"text":"Annotation Pipeline"},{"type":"para","text":"Figure 3 visualizes the annotation pipeline. We first use Gemini-3.1-Pro to annotate a small seed of documents using the q=\"...\" / a=\"...\" format defined above. The seed set is then used to distill two efficient annotators for the entire pre-training corpus. The fact span annotator is a masked language model that detects fact spans on the original text via BIO tagging, 2 2 2 BIO stands for begin-inside-outside (Ramshaw and Marcus, 1995 ) . built on a ModernBERT encoder (Warner et al., 2025 ) . The question generator is a decoder-only generator that, conditioned on the document with span markers inserted, emits a question-answer pair for each tagged span; the document context is encoded once and reused across all facts."},{"type":"heading","level":4,"text":"Why Two Annotators?"},{"type":"para","text":"A single end-to-end generator that produces the full annotated text in one shot is the obvious alternative. We chose the split design for three reasons. First, for efficiency reasons: the question generator only emits questions and answers, not the document text, which lets us prefill the document context once and reuse it across all of its facts. Second, predicting spans as offsets into the original text makes the annotations faithful by construction : the in-text span is always a verbatim substring of the source document, while a generative annotator can drift away from the source (i.e., hallucinate), especially on long documents. 3 3 3 E.g. in preliminary experiments a Llama-3.2-1B (Grattafiori et al., 2024 ) model trained as the single generative annotator produced faithful annotations for only 33% of the documents. Finally, indexing new data is highly efficient as it only involves two forward passes (one for the span annotator and one for Co-LMLM )."},{"type":"heading","level":1,"text":"4 Experiments"},{"type":"heading","level":2,"text":"4.1 Experimental Setup"},{"type":"para","text":"We train Co-LMLM using the SmolLM2-135M and SmolLM2-360M architectures, with a standard language model trained at each size as the Standard baseline. We use a high-quality Wikipedia corpus ( ∼ \\sim 3B tokens) 4 4 4 https://huggingface.co/datasets/allenai/dolmino-mix-1124 , following Zhao et al. ( 2026 ) . We train all models for 100K steps with a context length of 4,096 tokens. For a fair comparison with Rel-LMLM , we train two Rel-LMLM models based on the same SmolLM2-135M and SmolLM2-360M architectures. We provide additional Rel-LMLM training details in Appendix A.5 . In addition, we evaluate the generalization of our process to general web text by training Co-LMLM and a corresponding Standard baseline on 90B tokens sampled from FineWeb-Edu (Penedo et al., 2024 ) ."},{"type":"para","text":"In all our retrieval experiments, we create the KB from the entire Wikipedia training corpus. This allows for a direct comparison with the Rel-LMLM KB that is constructed from this same corpus. It also guarantees the KB does not cover information not available to the baselines. The Co-LMLM KB includes ∼ 240 M \\sim 240\\text{M} items, compared to Rel-LMLM ’s ∼ 145 M \\sim 145\\text{M} , demonstrating the less restricted, and more aggressive externalization of Co-LMLM . Appendix A.1 provides additional details."},{"type":"heading","level":2,"text":"4.2 Language Modeling Perplexity"},{"type":"heading","level":4,"text":"Evaluation Setup"},{"type":"para","text":"We evaluate using perplexity on 1,000 randomly sampled Simple English Wikipedia articles. 5 5 5 https://simple.wikipedia.org/ These articles are not included in our pretraining corpus, but their underlying facts are likely covered by our index in different surface forms, because the information in Simple Wikipedia is largely a subset of Wikipedia. This setup avoids coverage issues, and helps distinguish retrieval or annotation errors in Co-LMLM from imperfect parametric memorization in Standard . The focus on Wikipedia knowledge makes the comparison to off-the-shelf models complex, because of the domain shift. The key evaluation is between the models we train, controlling for the training data. We also report numbers for off-the-shelf models, even though the comparison is not as clean."},{"type":"para","text":"We report two perplexity variants, following Zhao et al. ( 2026 ) . Static (Oracle) assumes perfect lookup behavior by supplying the correct retrieved spans as context, giving an optimistic estimate of language modeling quality. Dynamic performs actual lookups during decoding, so retrieval performance is reflected in the score. While dynamic follows the standard perplexity calculation, it introduces some complexities in our case. We discuss this in depth in Appendix A.6 and provide a worst-case analysis. Importantly, this worst-case analysis does not change our conclusions."},{"type":"heading","level":4,"text":"Results"},{"type":"para","text":"Figure 4 visualizes perplexity results. Co-LMLM consistently achieves lower perplexity than Standard across model sizes and perplexity variants. Under the more realistic Dynamic setting, Co-LMLM reduces perplexity from 14.4 to 11.8 for 135M and from 14.0 to 10.5 for 360M. These gains suggest that our annotated pretraining data and training objective successfully improve language modeling over factual text. The FineWeb (FW) results indicate that benefit of data scale, charting an initial trends for scaling. The comparison to off-the-shelf models (HF/ SmolLM2-135M and HF/ SmolLM2-360M , dashed line) provides context. The relative gap Co-LMLM -360M-FW shows is particularly promising given the orders of magnitude more training data of the HF models. However, this experiment has a likely domain confounder, so we consider this result qualified."},{"type":"para","text":"We also experiment with scaling up the index to include the entire FW and Wikipedia corpora used for training our models. This reflects a more realistic KB construction, both in source document diversity and size. This results in a much larger index of 2.2B entries, compared to the 240M entries in the Wiki-only index. 4(b) reports results on both the Simple English Wikipedia dataset, as well as a 895-document held-out test set from the FW dataset. We compute these results with higher quantization for an additional 2.5 × \\times memory reduction per vector, so we can fit all indices in memory. We present results for both a Wikipedia-only index and a combined FW and Wikipedia index."},{"type":"para","text":"We do not expect benefits from scaling up the index to include FW data for Simple-Wiki, because the corresponding knowledge is expected to be contained in the English Wikipedia dataset. Indeed, under the same quantization level, we only see a minor increase in perplexity even though the index is one order of magnitude larger, which speaks to the robustness of our learned retriever. For the FW test set, on the other hand, we observe a substantial improvement over the wiki-only index, producing a significantly lower perplexity than that of the HF/ SmolLM2-360M model, which is pretrained on those documents and overall much more data (4T tokens). The poor performance of the Wiki-only index on the FW test-set demonstrates Wikipedia’s limited coverage of knowledge and the need to scale up the factual KB. This result demonstrates effective scaling beyond Wikipedia data."},{"type":"heading","level":2,"text":"4.3 Factuality and General Benchmark Performance"},{"type":"heading","level":4,"text":"Evaluation Setup"},{"type":"para","text":"We evaluate factuality across five complementary settings that probe different forms of factual knowledge in pre-trained models. TriviaQA (Joshi et al., 2017 ) tests broad open-domain QA, while PopQA (Mallen et al., 2023 ) focuses on long-tail entities. SimpleQA Verified (hereafter SimpleQA; Wei et al., 2024 ; Haas et al., 2026 ) targets short-form, fact-seeking questions that are deliberately difficult and have a single, indisputable answer. T-REx (Petroni et al., 2021 ) evaluates controlled factual completion, where the model completes a factual statement with the correct object entity. FactScore (Min et al., 2023 ) evaluates long-form factual precision in biography generation by decomposing generations into atomic facts and verifying them against external evidence. We also evaluate Co-LMLM on standard NLU benchmarks to verify that factual offloading does not compromise language understanding."},{"type":"para","text":"We use greedy decoding for all factuality evaluations. For Rel-LMLM and Co-LMLM , retrieved factual spans are removed before scoring. The original Rel-LMLM experiments in Zhao et al. ( 2026 ) use partial KB as an approximation, due to the computational costs of indexing the entire KB. We use the complete KB, leading to slightly different numbers compared to the original paper. Appendix A.6 provides additional details."},{"type":"heading","level":4,"text":"Results"},{"type":"para","text":"Table 1 shows that Co-LMLM consistently improves over the corresponding Standard baselines, with especially large gains on short-form QA and FactScore, exceeding 18 18 points in each case. Similar to perplexity, we report numbers for off-the-shelf models for contextualization, but under similar caveats. Co-LMLM also outperforms Rel-LMLM , demonstrating that continuous queries are more robust than textual entity-relation queries. We attribute these gains to two factors: free-form factual spans provide richer external knowledge than relational triplets, and hidden-state queries can use the full generation context rather than a short and restricted decoded query. Training on FineWeb-Edu further improves short-form QA and knowledge completion, showing that our design remains effective beyond Wikipedia-style pretraining data. Co-LMLM -360M-FW achieves 21.7 on SimpleQA, in line with gpt-4o-mini and above Claude Sonnet 4.5 and Grok 2 according to the public leaderboard. 6 6 6 https://www.kaggle.com/benchmarks/deepmind/simpleqa-verified"},{"type":"para","text":"Co-LMLM achieves a better factuality-NLU trade-off that standard models, and benefits from the larger FineWeb-Edu pre-training data ( 5(a) ). Table 12 in Appendix B.1 details NLU performance across tasks, further illustrating that Co-LMLM retains similar NLU capabilities, while achieving much higher factuality."},{"type":"heading","level":4,"text":"Comparison with RAG"},{"type":"para","text":"RAG and Co-LMLM are complementary: RAG expands inference-time context through document retrieval, while Co-LMLM externalizes free-form factual knowledge during pretraining into an editable memory. We study the behavioral outcomes of Co-LMLM and a controlled RAG baseline, which uses BM25 to retrieve the top-4 100-word Wikipedia passages and prepends them to the input during generation ( Table 2 ). We also apply the same retrieval on top of Co-LMLM . RAG significantly improves factuality for standard LMs. Nevertheless, the superiority of Co-LMLM is mostly maintained in this setting as well, as RAG gains accumulate over the already-high factuality scores. This suggests that Co-LMLM +RAG is a viable path forward, where RAG provides broad document-level context, while Co-LMLM provides controllable, and editable factual memory."},{"type":"heading","level":2,"text":"4.4 Unlearning"},{"type":"para","text":"Externalizing knowledge simplifies unlearning: instead of retraining the model, we just remove the relevant entries from memory. We test whether Co-LMLM preserves this benefit while replacing relational queries with continuous queries. We evaluate this property with TOFU (Maini et al., 2024 ) , following the same setup as Rel-LMLM . TOFU tests whether a model can forget a designated forget set while preserving general utility. Its main metric, forget quality , tests whether the unlearned model is statistically indistinguishable from a retain-only model on the forgotten examples. Appendix A.7 provides additional details. 5(b) shows how simple memory operations achieve effective forgetting ( p p -value > 0.05 >0.05 ) while preserving model utility. In contrast, training-based unlearning methods such as NPO (Zhang et al., 2024b ) update model parameters and sacrifice utility due to entangled parametric knowledge. Co-LMLM retains the controllability benefit of Rel-LMLM while extending LMLMs to a more expressive and efficient continuous-query KB."},{"type":"heading","level":2,"text":"4.5 Knowledge Memorization"},{"type":"para","text":"We test whether Co-LMLM reduces factual memorization by evaluating it without the KB. Table 3 shows that factual precision drops substantially, forcing the model to rely on its internal parameters alone. This shows that Co-LMLM genuinely shifts factual knowledge from model weights to the external memory."},{"type":"heading","level":2,"text":"4.6 Continuous vs. Natural Language Queries"},{"type":"para","text":"We isolate the impact of using continuous queries by comparing Co-LMLM to a LMLM-Asker , a variant that generates free-form natural language questions as queries. LMLM-asker is trained on the same annotated corpus. It generates a <QUESTION> … </QUESTION> block before each factual span and retrieves using the decoded question, while Co-LMLM retrieves directly from the hidden state. Table 4 shows that Co-LMLM outperforms LMLM-Asker , demonstrating that advantage of a continuous queries. Intuitively, questions are constrained by the structure of language, whereas the continuous queries can learn arbitrary retrieval mechanism."},{"type":"heading","level":4,"text":"Inference Overhead"},{"type":"para","text":"Text queries also add generation cost: LMLM-Asker writes a question of arbitrary length before every lookup, while Rel-LMLM writes an entity-relation query. Co-LMLM instead produces the query from the hidden state, saving a cost that is linear in the length of the question. Empirically, forming a retrieval query costs Co-LMLM a single ∼ 2.2 {\\sim}2.2 ms hidden-state forward, versus ∼ 28 {\\sim}28 ms for LMLM-Asker , which is dominated by decoding the ∼ 13 {\\sim}13 -token question. We provide additional overhead comparisons in Appendix B.2 ."},{"type":"heading","level":2,"text":"4.7 Evaluating Query Timing"},{"type":"para","text":"We evaluate the ability of our model to form queries when needed by forcing query generation before answering short-form QA and knowledge-completion prompts. This setup differs from earlier results in that the model is forced to retrieve, while still using the query content generated by the model. Table 5 shows that forcing to query increase performance on both model scales across four benchmarks, illustrating that our retrieval performance is inhibited by the timing problem, and there remains room for improvement."},{"type":"heading","level":1,"text":"5 Discussion"},{"type":"para","text":"We present Co-LMLM , a scalable method to train LLMs for verifiable and controllable use of knowledge. This includes a large-scale annotation pipeline that can be applied to hundreds of billions of tokens at relatively modest costs. Critically, the Co-LMLM training process, beyond the pre-processing step, departs only modestly from the conventional LLM training process that has been shown to scale to many trillions of both tokens and model parameters."},{"type":"para","text":"Our method and results open an avenue to train LLMs with factuality, transparency, and controllability properties that are sorely lacking in current LLMs, which has significantly hampered their trustworthiness and utility. Co-LMLM provides a flexible free-form span substrate for studying how externalized knowledge scales across broader knowledge domains, memory sizes, and retrieval mechanisms. This establishes a foundation for future work to study knowledge scaling laws ."},{"type":"para","text":"Our current study has several limitations that outline important directions for future work. Although we experiment along model and data scaling axes to outline scaling trends, our experiments remain modest in model and data scale. We focus on probing the behavior of Co-LMLM in controlled pretraining settings, and leave larger-scale pretraining and downstream adaptation to future work. Our factual evaluations mainly target Wikipedia-style knowledge, a key factor in creating a clean experimental setup. Although, as shown, our pipeline can naturally extend to broader corpora such as FineWeb-Edu, there remains work to be done to systematically evaluate factuality across more diverse knowledge domains. Constructing a retrieval index at pretraining scale introduces a one-time computational cost. However, an important problem to address in the context of post-training and continual learning is how to adapt such an index once a model is fine-tuned."},{"type":"para","text":"Studying these and many other important problems will be enabled by the scalable design of our methods, and by our large-scale release of annotations of Wikipedia and 100B FineWeb tokens."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Recently, there has been increasing interest in large language models (LLMs) that are trained to externalize knowledge (Ghosal et al., 2025 ; Zhao et al., 2026 ; Pouransari et al., 2026 ) . A particularly compelling approach is the limited memory language model (LMLM; Zhao et al., 2026 ) , where the LLM is pre-trained to externalize knowledge into a human-readable knowledge base (KB). This design offers several advantages. The KB is interpretable and easily editable, allowing for unlearning with no utility tradeoff, and enabling easy attribution of knowledge used to the source material.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Zhao et al. instantiate the LMLM approach with a relational KB, which we refer to as Rel-LMLM . Although representing a significant departure from conventional LLMs, Rel-LMLM proposes a training process that follows the common scalable next-token-prediction pre-training, with the key difference of pre-processing the data to simulate knowledge retrieval. Experiments demonstrate significantly lower perplexities compared to conventional LLMs and factuality scores similar to models several orders of magnitude larger, while retaining similar utility (i.e., NLU scores) to LLMs of the same size.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"However, Rel-LMLM has several key limitations that constrain the scaling of the pre-training process and the knowledge retrieval expressivity. Pre-training relies on Wikipedia, where each article is centered on a specific entity, making it straightforward to automatically annotate relational queries at scale during data pre-processing. Although Wikipedia is sufficient for a proof-of-concept demonstration, it provides no avenue to scale much further. The relational representation itself, although human readable, introduces several limitations. It restricts the data that can be externalized to items that are the object of natural language relational tuples, where both the subject and relation are mentioned previously in the text. This same mechanism limits the expressivity of the retrieval itself to queries that can be expressed as simple subject-relation natural language queries, and generating these queries incurs the added cost of decoding multiple additional tokens during inference. Finally, synthesizing the exact queries in the data pre-processing step holds retrieval pathways as constant and pre-determined, significantly restricting the expressivity of the learned retrieval mechanism.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We propose continuous-query limited-memory language models ( Co-LMLM ), an approach that addresses these limitations, while retaining all the advantages that the LMLM design offers. The key modeling difference from Rel-LMLM is that queries are issued as single continuous vectors rather than structured relational tuples. The KB stores vector keys and string values instead of relational tuples. We jointly train the LLM Transformer for language modeling and retrieval using pre-training data pre-processed to simulate KB queries. The pre-processing specifies only when retrieval takes place and what string it returns, rather than dictating the explicit query representation. The query vector emitted by the LLM are optimized via a contrastive loss over synthesized positive and negative pairs. This allows Co-LMLM to retrieve knowledge at the cost of a single generation step, generate flexible queries, externalize a much broader set of knowledge without role or relational constraints, and generalize to pre-training data well beyond Wikipedia.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate Co-LMLM against Rel-LMLM models of comparable size and standard baselines trained on the same corpus without knowledge externalization. Across model scales, Co-LMLM achieves lower perplexity than standard models, substantially improves factual precision over both standard and Rel-LMLM , while preserving downstream NLU performance. At the 360M scale, Co-LMLM achieves lower perplexities than off-the-shelf HF/ SmolLM2-360M , which is pre-trained on 40 × \\\\times more data, and SimpleQA-verified (Haas et al., 2026 ) scores that are in line with gpt-4o-mini and higher than Claude Sonnet 4.5. At the same time, Co-LMLM retains the controllability benefits of LMLM : its external memory remains editable and supports direct unlearning through database operations. Finally, extending pre-training from Wikipedia to FineWeb-Edu further improves factual precision, showing Co-LMLM ’s benefit from scaling the pre-training data.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Code, data, and models are available at: https://lil-lab.github.io/co-lmlm-web .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Related Work\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Parametric Knowledge and its Limitations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Pre-trained language models store factual knowledge implicitly in their parameters and recall it at generation time (Petroni et al., 2019 ; Roberts et al., 2020 ) . This form of storage is inherently limited: factual recall scales with model size and exposure frequency, making long-tail knowledge difficult to capture (Kandpal et al., 2023 ; Mallen et al., 2023 ) . Moreover, factual associations are entangled with linguistic representations, making individual facts difficult to attribute, edit, or remove, and contributing to hallucinations and stale knowledge (Elhage et al., 2022 ; Dai et al., 2022 ; Huang et al., 2024 ) . They also make knowledge editing and unlearning challenging: removing or modifying specific facts typically requires additional training or specialized objectives, often trading off with general capabilities (Maini et al., 2024 ) . MemSinks isolates memorization-specific parameters, but still keeps knowledge within the model (Ghosal et al., 2025 ) . These challenges motivate approaches that move factual knowledge into separate, editable stores.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Externalizing Knowledge from Model Weights\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A growing line of work augments language models with non-parametric or semi-parametric memory, or aims to move factual knowledge more explicitly out of model parameters . kNN-LM retrieves nearest-neighbor examples from an external datastore and interpolates their probabilities with the model distribution (Khandelwal et al., 2020 ) , while RETRO retrieves neighboring chunks and conditions on them through cross-attention (Borgeaud et al., 2022 ) . Other approaches introduce sparse memory layers or learned key-value memories to increase model capacity and factuality (Lample et al., 2019 ; Berges et al., 2025 ; Wu et al., 2022 ; Peng et al., 2023 ; Yang et al., 2024 ; Cheng et al., 2026 ; Tseng and Sa, 2026 ) . These methods can improve perplexity or factuality, but the stored information remains not directly inspectable or editable.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Limited Memory Language Models\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Zhao et al. ( 2026 ) is the most related work to ours. They propose Limited Memory Language Models (LMLMs), which externalize factual knowledge into a textual relational KB during pretraining rather than encoding it in model parameters. We refer to this approach as Rel-LMLM to distinguish it from our continuous-query formulation. Rel-LMLM enables faster pre-training, interpretable retrieval, and direct editing, but its relational design limits the external memory coverage: facts must be expressible as Wikipedia-style subject-relation-object tuples, and retrieval depends on brittle textual queries and introduces additional generation token overhead. Co-LMLM preserves the benefits of externalized and editable KB while relaxing these constraints. It replaces relational tuples with free-form textual knowledge spans; and decoded relational queries with continuous queries emitted from the model’s hidden states. These queries can carry richer contextual information for retrieval, allowing Co-LMLM to scale beyond Wikipedia-style relational facts to general corpora (e.g., FineWeb-Edu), while reducing the token overhead of query generation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Retrieval-Augmented Language Models\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Retrieval-Augmented Generation (RAG) conditions generation on retrieved passages during inference (Lewis et al., 2020 ; Izacard and Grave, 2021 ) , while retrieval-aware pretraining methods incorporate retrieval into model training (Guu et al., 2020 ; Borgeaud et al., 2022 ; Izacard et al., 2023 ) . Other methods let models decide when to retrieve and issue textual queries, as in Self-RAG, DRAGIN, and LMLM (Asai et al., 2024 ; Su et al., 2024 ; Zhao et al., 2026 ) . Our technique is related to methods that generate retrieval signals directly from internal representations rather than decoded queries (Muennighoff et al., 2025 ; Zhang et al., 2024a , 2025 ) . Unlike most retrieval-augmented methods, which add external context while leaving parametric knowledge largely intact, Co-LMLM uses retrieval to define the parametric/non-parametric boundary during pretraining: factual spans are externalized into a schema-free index, and the model learns when and how to retrieve them through a contrastive objective trained jointly with next-token prediction.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Method\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The Co-LMLM model is a decoder-only Transformer, which functions as both a language model and a dense retriever. Figure 2 provides a high-level overview of the method. The model generates query vectors similar to tool calling, and the returned values are encoded by the model, before decoding resumes ( Section 3.1 ). We jointly train the language modeling and retriever functions ( Section 3.2 ), using pre-training data that is pre-processed to simulate retrieval and synthetic questions that provide supervision for the contrastive retrieval objective ( Section 3.3 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 Modeling and Inference\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Co-LMLM performs retrieval through continuous tool calls. The external memory is a key-value store: each key is a dense vector produced by the model, and each value is a factual text span. Unlike standard tool-calling methods (Schick et al., 2023 ) , the retrieval query is not represented as text. Instead, the query is the model’s hidden state at a special token.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"At inference time, the model decodes tokens autoregressively. When it emits the special <FACT> token, we read the last-layer hidden state at that position as a retrieval query, query the index, and retrieve the top-1 fact text snippet. We splice the retrieved snippet into the context immediately after <FACT> , append a closing </FACT> special token, and resume autoregressive decoding from </FACT> onward. Each retrieval therefore costs one extra autoregressive step (the opening <FACT> token) and grows the context by the length of the retrieved span.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2 Training\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our training data consists of raw text in which factual spans are enclosed by <FACT> and </FACT> . Each marked span is paired with a natural-language question whose answer is the span. Figure 2 (top) illustrates a single example. Section 3.3 describes how we pre-process raw pre-training data to create this data. Our objectives are to train the model to both predict the next token and externalize the information in the delimited spans, construct a KB from all such spans, and to emit retrieval queries to fetch spans from the KB as needed. The key to training the retrieval function is contrastive query pairs between documents and questions, allowing for both positive and negative pairs.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let x x be an input document with factual spans bracketed by <FACT> and </FACT> . For each factual span, we have a question Q Q . The question is synthesized so its answer is the corresponding bracketed span in x x , and it is appended with a query marker token <FACT-q> . The retrieval vectors at the query markers appended to the questions are used as positive examples for the contrastive retrieval loss. The pre-training panel of Figure 2 (middle) visualized how x x and Q Q are used.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let p ( x t | x < t ; θ ) p(x_{t}|x_{<t};\\\\theta) be the probability of the token x t x_{t} in the sequence document x x , parameterized by θ \\\\theta . In our model, p θ p_{\\\\theta} is computed by a Transformer. We denote ϕ t ( x < t ; θ ) \\\\phi_{t}(x_{<t};\\\\theta) as the features computed by the Transformer for index t t in sequence x x . 1 1 1 We L2-normalize the feature representations as they are used as retrieval queries (Chen et al., 2020 ) . . This is the same Transformer used for p θ p_{\\\\theta} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We jointly minimize next-token prediction (NTP) and bidirectional contrastive losses. The NTP loss is: ℒ NTP ( θ ) = − ∑ t ∉ M log p ( x t ∣ x < t ; θ ) \\\\mathcal{L}_{\\\\mathrm{NTP}}(\\\\theta)=-\\\\sum_{t\\\\notin M}\\\\log p(x_{t}\\\\mid x_{<t};\\\\theta) , where M M is the set of positions in x x that are strictly between <FACT> and </FACT> , inclusive of the closing </FACT> , but excluding the opening <FACT> . This NTP formulation means that we do not optimize our model to memorize the bracketed fact tokens, which represent the knowledge we aim to externalize. This is a critical design choice to externalize knowledge. We do optimize the loss of the opening <FACT> token, because the model must know to generate it in order to trigger retrieval. The closing </FACT> is appended mechanically, so does not require optimization.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The contrastive loss treats each fact query from the document x x and its corresponding question Q Q as a positive pair, and all other questions as negatives. For a fact that starts at index t t in a document x x , let f = ϕ t ( x < t ; θ ) f=\\\\phi_{t}(x_{<t};\\\\theta) be the query representation for retrieving this fact. Meaning, the feature representation for the <FACT> token opening this fact span. Correspondingly, for a question Q Q where the query marker <FACT-q> is at index t t , let q = ϕ t ( Q < t ) q=\\\\phi_{t}(Q_{<t}) be the query representation. That is, the feature representation computed for <FACT-q> . We use the InfoNCE loss (van den Oord et al., 2018 ) , compute on the current batch. Let ℬ = { ( f ( i ) , q ( i ) ) } i = 1 B \\\\mathcal{B}=\\\\{(f^{(i)},q^{(i)})\\\\}_{i=1}^{B} be the set of all paired fact document queries f ( i ) f^{(i)} and question queries q ( i ) q^{(i)} in the current batch. For each ( f ( i ) , q ( i ) ) ∈ B (f^{(i)},q^{(i)})\\\\in{B} , the loss terms are:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where τ \\\\tau is a temperature hyperparameter. We average the two directions:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We optimize the joint loss ℒ = ℒ NTP + λ ℒ CL \\\\mathcal{L}=\\\\mathcal{L}_{\\\\mathrm{NTP}}+\\\\lambda\\\\mathcal{L}_{\\\\mathrm{CL}} , with λ \\\\lambda a hyperparameter set to 0.25 0.25 by default.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Building the Index\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We build a dense retrieval index from our pre-training data. The index pairs vector keys with fact spans. We run the pre-trained language model over the pre-training corpus, extract the L2-normalized hidden state ϕ ( ⋅ ; θ ) \\\\phi(\\\\cdot;\\\\theta) at every <FACT> position, and store it as a key in a dense vector index whose value is the verbatim in-text span at that position. Indexing a new corpus therefore requires only pre-processing it ( Section 3.3 ) and a forward pass of Co-LMLM .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.3 Training Data Pre-processing\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Training requires documents annotated with fact spans and paired questions ( Section 3.2 ). We design an automated, scalable process to annotate raw pre-training data. The pre-processing pipeline has three stages: annotate a seed set with by prompting a state-of-the-art frontier LLM; fine-tune lightweight annotators on the seed data; and apply them to full pre-training corpus. Critically, we fine-tune smaller models because of the cost of annotating at scale with a large frontier model.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Annotation Specification\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We annotate each training document with a set of fact spans . A fact span is a sub-string of a pre-training document x x that encodes world knowledge we want our model to externalize – to retrieve rather than memorize. Each fact span is associated with two additional pieces of information: a context-dependent question that captures the retrieval intent at this position without itself carrying any factual content, and a snippet-form answer that answers the question in a form that need not match the surface form of the span. The question is used as the question Q Q during training. An annotated fact takes the form <FACT q={question} a={answer}>{span}</FACT> .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The annotation reflects three guidelines: (a) spans are the narrowest self-contained sub-string that fully answers the question; (b) questions are written as standalone search queries that name their subject explicitly, and they avoid mentioning information that appears later in the document so that no future fact leaks through the question; and (c) answers vary the surface form of the span, so the model is exposed at training time to retrieved values that differ lexically from the surrounding context. We provide more details about the prompt used for the seed annotation in Appendix C .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Annotation Pipeline\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 3 visualizes the annotation pipeline. We first use Gemini-3.1-Pro to annotate a small seed of documents using the q=\\\"...\\\" / a=\\\"...\\\" format defined above. The seed set is then used to distill two efficient annotators for the entire pre-training corpus. The fact span annotator is a masked language model that detects fact spans on the original text via BIO tagging, 2 2 2 BIO stands for begin-inside-outside (Ramshaw and Marcus, 1995 ) . built on a ModernBERT encoder (Warner et al., 2025 ) . The question generator is a decoder-only generator that, conditioned on the document with span markers inserted, emits a question-answer pair for each tagged span; the document context is encoded once and reused across all facts.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Why Two Annotators?\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A single end-to-end generator that produces the full annotated text in one shot is the obvious alternative. We chose the split design for three reasons. First, for efficiency reasons: the question generator only emits questions and answers, not the document text, which lets us prefill the document context once and reuse it across all of its facts. Second, predicting spans as offsets into the original text makes the annotations faithful by construction : the in-text span is always a verbatim substring of the source document, while a generative annotator can drift away from the source (i.e., hallucinate), especially on long documents. 3 3 3 E.g. in preliminary experiments a Llama-3.2-1B (Grattafiori et al., 2024 ) model trained as the single generative annotator produced faithful annotations for only 33% of the documents. Finally, indexing new data is highly efficient as it only involves two forward passes (one for the span annotator and one for Co-LMLM ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Experiments\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 Experimental Setup\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We train Co-LMLM using the SmolLM2-135M and SmolLM2-360M architectures, with a standard language model trained at each size as the Standard baseline. We use a high-quality Wikipedia corpus ( ∼ \\\\sim 3B tokens) 4 4 4 https://huggingface.co/datasets/allenai/dolmino-mix-1124 , following Zhao et al. ( 2026 ) . We train all models for 100K steps with a context length of 4,096 tokens. For a fair comparison with Rel-LMLM , we train two Rel-LMLM models based on the same SmolLM2-135M and SmolLM2-360M architectures. We provide additional Rel-LMLM training details in Appendix A.5 . In addition, we evaluate the generalization of our process to general web text by training Co-LMLM and a corresponding Standard baseline on 90B tokens sampled from FineWeb-Edu (Penedo et al., 2024 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In all our retrieval experiments, we create the KB from the entire Wikipedia training corpus. This allows for a direct comparison with the Rel-LMLM KB that is constructed from this same corpus. It also guarantees the KB does not cover information not available to the baselines. The Co-LMLM KB includes ∼ 240 M \\\\sim 240\\\\text{M} items, compared to Rel-LMLM ’s ∼ 145 M \\\\sim 145\\\\text{M} , demonstrating the less restricted, and more aggressive externalization of Co-LMLM . Appendix A.1 provides additional details.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 Language Modeling Perplexity\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Evaluation Setup\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate using perplexity on 1,000 randomly sampled Simple English Wikipedia articles. 5 5 5 https://simple.wikipedia.org/ These articles are not included in our pretraining corpus, but their underlying facts are likely covered by our index in different surface forms, because the information in Simple Wikipedia is largely a subset of Wikipedia. This setup avoids coverage issues, and helps distinguish retrieval or annotation errors in Co-LMLM from imperfect parametric memorization in Standard . The focus on Wikipedia knowledge makes the comparison to off-the-shelf models complex, because of the domain shift. The key evaluation is between the models we train, controlling for the training data. We also report numbers for off-the-shelf models, even though the comparison is not as clean.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We report two perplexity variants, following Zhao et al. ( 2026 ) . Static (Oracle) assumes perfect lookup behavior by supplying the correct retrieved spans as context, giving an optimistic estimate of language modeling quality. Dynamic performs actual lookups during decoding, so retrieval performance is reflected in the score. While dynamic follows the standard perplexity calculation, it introduces some complexities in our case. We discuss this in depth in Appendix A.6 and provide a worst-case analysis. Importantly, this worst-case analysis does not change our conclusions.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 4 visualizes perplexity results. Co-LMLM consistently achieves lower perplexity than Standard across model sizes and perplexity variants. Under the more realistic Dynamic setting, Co-LMLM reduces perplexity from 14.4 to 11.8 for 135M and from 14.0 to 10.5 for 360M. These gains suggest that our annotated pretraining data and training objective successfully improve language modeling over factual text. The FineWeb (FW) results indicate that benefit of data scale, charting an initial trends for scaling. The comparison to off-the-shelf models (HF/ SmolLM2-135M and HF/ SmolLM2-360M , dashed line) provides context. The relative gap Co-LMLM -360M-FW shows is particularly promising given the orders of magnitude more training data of the HF models. However, this experiment has a likely domain confounder, so we consider this result qualified.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We also experiment with scaling up the index to include the entire FW and Wikipedia corpora used for training our models. This reflects a more realistic KB construction, both in source document diversity and size. This results in a much larger index of 2.2B entries, compared to the 240M entries in the Wiki-only index. 4(b) reports results on both the Simple English Wikipedia dataset, as well as a 895-document held-out test set from the FW dataset. We compute these results with higher quantization for an additional 2.5 × \\\\times memory reduction per vector, so we can fit all indices in memory. We present results for both a Wikipedia-only index and a combined FW and Wikipedia index.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We do not expect benefits from scaling up the index to include FW data for Simple-Wiki, because the corresponding knowledge is expected to be contained in the English Wikipedia dataset. Indeed, under the same quantization level, we only see a minor increase in perplexity even though the index is one order of magnitude larger, which speaks to the robustness of our learned retriever. For the FW test set, on the other hand, we observe a substantial improvement over the wiki-only index, producing a significantly lower perplexity than that of the HF/ SmolLM2-360M model, which is pretrained on those documents and overall much more data (4T tokens). The poor performance of the Wiki-only index on the FW test-set demonstrates Wikipedia’s limited coverage of knowledge and the need to scale up the factual KB. This result demonstrates effective scaling beyond Wikipedia data.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3 Factuality and General Benchmark Performance\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Evaluation Setup\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate factuality across five complementary settings that probe different forms of factual knowledge in pre-trained models. TriviaQA (Joshi et al., 2017 ) tests broad open-domain QA, while PopQA (Mallen et al., 2023 ) focuses on long-tail entities. SimpleQA Verified (hereafter SimpleQA; Wei et al., 2024 ; Haas et al., 2026 ) targets short-form, fact-seeking questions that are deliberately difficult and have a single, indisputable answer. T-REx (Petroni et al., 2021 ) evaluates controlled factual completion, where the model completes a factual statement with the correct object entity. FactScore (Min et al., 2023 ) evaluates long-form factual precision in biography generation by decomposing generations into atomic facts and verifying them against external evidence. We also evaluate Co-LMLM on standard NLU benchmarks to verify that factual offloading does not compromise language understanding.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We use greedy decoding for all factuality evaluations. For Rel-LMLM and Co-LMLM , retrieved factual spans are removed before scoring. The original Rel-LMLM experiments in Zhao et al. ( 2026 ) use partial KB as an approximation, due to the computational costs of indexing the entire KB. We use the complete KB, leading to slightly different numbers compared to the original paper. Appendix A.6 provides additional details.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 1 shows that Co-LMLM consistently improves over the corresponding Standard baselines, with especially large gains on short-form QA and FactScore, exceeding 18 18 points in each case. Similar to perplexity, we report numbers for off-the-shelf models for contextualization, but under similar caveats. Co-LMLM also outperforms Rel-LMLM , demonstrating that continuous queries are more robust than textual entity-relation queries. We attribute these gains to two factors: free-form factual spans provide richer external knowledge than relational triplets, and hidden-state queries can use the full generation context rather than a short and restricted decoded query. Training on FineWeb-Edu further improves short-form QA and knowledge completion, showing that our design remains effective beyond Wikipedia-style pretraining data. Co-LMLM -360M-FW achieves 21.7 on SimpleQA, in line with gpt-4o-mini and above Claude Sonnet 4.5 and Grok 2 according to the public leaderboard. 6 6 6 https://www.kaggle.com/benchmarks/deepmind/simpleqa-verified\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Co-LMLM achieves a better factuality-NLU trade-off that standard models, and benefits from the larger FineWeb-Edu pre-training data ( 5(a) ). Table 12 in Appendix B.1 details NLU performance across tasks, further illustrating that Co-LMLM retains similar NLU capabilities, while achieving much higher factuality.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Comparison with RAG\"\n },\n {\n \"type\": \"para\",\n \"text\": \"RAG and Co-LMLM are complementary: RAG expands inference-time context through document retrieval, while Co-LMLM externalizes free-form factual knowledge during pretraining into an editable memory. We study the behavioral outcomes of Co-LMLM and a controlled RAG baseline, which uses BM25 to retrieve the top-4 100-word Wikipedia passages and prepends them to the input during generation ( Table 2 ). We also apply the same retrieval on top of Co-LMLM . RAG significantly improves factuality for standard LMs. Nevertheless, the superiority of Co-LMLM is mostly maintained in this setting as well, as RAG gains accumulate over the already-high factuality scores. This suggests that Co-LMLM +RAG is a viable path forward, where RAG provides broad document-level context, while Co-LMLM provides controllable, and editable factual memory.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.4 Unlearning\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Externalizing knowledge simplifies unlearning: instead of retraining the model, we just remove the relevant entries from memory. We test whether Co-LMLM preserves this benefit while replacing relational queries with continuous queries. We evaluate this property with TOFU (Maini et al., 2024 ) , following the same setup as Rel-LMLM . TOFU tests whether a model can forget a designated forget set while preserving general utility. Its main metric, forget quality , tests whether the unlearned model is statistically indistinguishable from a retain-only model on the forgotten examples. Appendix A.7 provides additional details. 5(b) shows how simple memory operations achieve effective forgetting ( p p -value > 0.05 >0.05 ) while preserving model utility. In contrast, training-based unlearning methods such as NPO (Zhang et al., 2024b ) update model parameters and sacrifice utility due to entangled parametric knowledge. Co-LMLM retains the controllability benefit of Rel-LMLM while extending LMLMs to a more expressive and efficient continuous-query KB.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.5 Knowledge Memorization\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We test whether Co-LMLM reduces factual memorization by evaluating it without the KB. Table 3 shows that factual precision drops substantially, forcing the model to rely on its internal parameters alone. This shows that Co-LMLM genuinely shifts factual knowledge from model weights to the external memory.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.6 Continuous vs. Natural Language Queries\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We isolate the impact of using continuous queries by comparing Co-LMLM to a LMLM-Asker , a variant that generates free-form natural language questions as queries. LMLM-asker is trained on the same annotated corpus. It generates a <QUESTION> … </QUESTION> block before each factual span and retrieves using the decoded question, while Co-LMLM retrieves directly from the hidden state. Table 4 shows that Co-LMLM outperforms LMLM-Asker , demonstrating that advantage of a continuous queries. Intuitively, questions are constrained by the structure of language, whereas the continuous queries can learn arbitrary retrieval mechanism.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Inference Overhead\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Text queries also add generation cost: LMLM-Asker writes a question of arbitrary length before every lookup, while Rel-LMLM writes an entity-relation query. Co-LMLM instead produces the query from the hidden state, saving a cost that is linear in the length of the question. Empirically, forming a retrieval query costs Co-LMLM a single ∼ 2.2 {\\\\sim}2.2 ms hidden-state forward, versus ∼ 28 {\\\\sim}28 ms for LMLM-Asker , which is dominated by decoding the ∼ 13 {\\\\sim}13 -token question. We provide additional overhead comparisons in Appendix B.2 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.7 Evaluating Query Timing\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate the ability of our model to form queries when needed by forcing query generation before answering short-form QA and knowledge-completion prompts. This setup differs from earlier results in that the model is forced to retrieve, while still using the query content generated by the model. Table 5 shows that forcing to query increase performance on both model scales across four benchmarks, illustrating that our retrieval performance is inhibited by the timing problem, and there remains room for improvement.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Discussion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We present Co-LMLM , a scalable method to train LLMs for verifiable and controllable use of knowledge. This includes a large-scale annotation pipeline that can be applied to hundreds of billions of tokens at relatively modest costs. Critically, the Co-LMLM training process, beyond the pre-processing step, departs only modestly from the conventional LLM training process that has been shown to scale to many trillions of both tokens and model parameters.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our method and results open an avenue to train LLMs with factuality, transparency, and controllability properties that are sorely lacking in current LLMs, which has significantly hampered their trustworthiness and utility. Co-LMLM provides a flexible free-form span substrate for studying how externalized knowledge scales across broader knowledge domains, memory sizes, and retrieval mechanisms. This establishes a foundation for future work to study knowledge scaling laws .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our current study has several limitations that outline important directions for future work. Although we experiment along model and data scaling axes to outline scaling trends, our experiments remain modest in model and data scale. We focus on probing the behavior of Co-LMLM in controlled pretraining settings, and leave larger-scale pretraining and downstream adaptation to future work. Our factual evaluations mainly target Wikipedia-style knowledge, a key factor in creating a clean experimental setup. Although, as shown, our pipeline can naturally extend to broader corpora such as FineWeb-Edu, there remains work to be done to systematically evaluate factuality across more diverse knowledge domains. Constructing a retrieval index at pretraining scale introduces a one-time computational cost. However, an important problem to address in the context of post-training and continual learning is how to adapt such an index once a model is fine-tuned.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Studying these and many other important problems will be enabled by the scalable design of our methods, and by our large-scale release of annotations of Wikipedia and 100B FineWeb tokens.\"\n }\n]","isConversational":false}}},{"rowIdx":94,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.07690"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Agon: Competitive Cross-Model RL with Implicit Rival Grading of Reasoning"},"authors":{"kind":"list like","value":["Vladislav Beliaev"],"string":"[\n \"Vladislav Beliaev\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.07690v1"},"published":{"kind":"timestamp","value":"2026-07-08T00:00:00","string":"2026-07-08T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Reinforcement learning from verifiable rewards has become the standard tool for sharpening the reasoning of large language models (LLMs) on tasks such as mathematics and code (Guo et al., 2025 ; Shao et al., 2024 ) . The dominant recipe is on-policy and outcome-based: sample a group of rollouts from the current policy, score each by whether its final answer is correct, and update with group-relative advantages, as in GRPO (Shao et al., 2024 ) . The reward is attached to the answer; the chain of thought that produced it is never graded. On problems the model can already partly solve this is benign, but on hard problems it creates a strong incentive to write more: a longer chain affords more chances to stumble onto the right answer, so additional text is the cheapest way to raise expected reward. The trace inflates with hedging and backtracking (“hmm,” “wait,” “let me reconsider”), and accuracy grows far slower than length (accuracy rises only modestly while length inflates by an order of magnitude (Aggarwal & Welleck, 2025 ) ). The model ends up producing more reasoning per problem without producing better reasoning per token."},{"type":"para","text":"The obvious fix, grading the reasoning directly, is intractable. There is no ground truth for “good thinking”: no label says which step was the insight and which was filler, and a learned process reward model is expensive, brittle, and itself unverifiable (Lightman et al., 2023 ) . So the trace stays unscored, and the length pathology persists. We ask instead: can a second model supply the missing signal? If a different policy attempts the same problem and we reward each model for out-reasoning the other , then the trace is graded implicitly by the rival, without any process labels. A step that leads to a win is reinforced; filler that the opponent exploits is punished."},{"type":"para","text":"The grader must be a different model, and the game must be competitive . Post-training RL is, by construction, self-improvement: the policy is optimized on signal it generates itself, which reinforces the very blind spots that created its errors: a model auditing its own work tends to plateau (Huang et al., 2024 ) . A second model with different failure modes breaks this closed loop. But merely having two models is not enough. Naively averaging or letting models agree, as in self-consistency and mixture-of-agents (Wang et al., 2023 , 2024 ) , tends to regress toward consensus and can dilute quality when the models differ in strength (Li et al., 2025 ) , while two near-identical models gain little from one another. The combination that works is a pair of comparable strength hypothesized to have different blind spots, placed under competitive pressure: each model must not only solve the problem but beat a peer that sees its work, so the objective rewards out-reasoning the peer rather than agreeing with it. Under this pressure the pair co-improves: each win raises the bar for the other on the next step, and both models end up past the ceiling where a single self-graded policy plateaus."},{"type":"para","text":"We instantiate this as Agon (Greek agōn , contest). Two distinct policies are trained head-to-head: a drafter produces candidate solutions; a challenger reads an opponent’s worked solution and is rewarded for solving correctly and beating it; roles rotate every step so both models learn both roles, and at inference the pair deploys the same way, one model drafting and the other answering after reading the draft. In principle the recipe applies to any pair of comparably strong models; for efficiency we realize the pair as two low-rank adapters over one frozen base ( ∼ 2 % {\\sim}2\\% memory overhead, not 2 × 2\\times ), kept divergent by different initialization and update streams. In this work the exchange happens in text: we first establish that the cross-model signal is useful before optimizing the channel (latent-space exchange; Section ˜ 6 ). We test two claims under a matched generation budget: that cross-model information exchange raises accuracy (the exchange claim ), and that adversarial pressure raises it further than cooperation (the competition claim ), using a 2 × 2 2\\times 2 design that crosses competition with exchange (one cell of which is provably inert under group-relative normalization; Section ˜ 4.4 ), plus a single-model self-refinement control, a zero-training orchestration control, and a budget-matched GRPO self-cascade control."},{"type":"heading","level":4,"text":"Contributions."},{"type":"para","text":"We reframe reasoning quality as an implicit reward that a competing peer model can supply, turning the unlabeled “good thinking” problem into a head-to-head game played during RL ( Section ˜ 4 )."},{"type":"para","text":"We propose Agon: a recipe for jointly training two distinct policies against each other with competitive GRPO, via draft-and-challenge rollouts, per-step role rotation, and a competitive reward (correctness plus a conversion bonus for out-solving the opponent). It runs on a stock GRPO trainer with no changes to the optimization core; a shared-base dual-adapter instantiation makes the pair nearly as cheap as a single model ( Section ˜ 4 )."},{"type":"para","text":"We lay out a 2 × 2 2\\times 2 matrix (competition × \\times information exchange) with self-refinement and zero-training controls ( Section ˜ 5 ). One cell, competition without exchange, is inert by construction under group-relative normalization ( Section ˜ 4.4 ), so the competition claim rests on the adversarial-vs-cooperative comparison; the fully isolated cell (per-rollout hidden drafts) is left for future work."},{"type":"para","text":"Under a matched generation budget on hard math, competition + + exchange beats cooperation, self-refinement, and vanilla GRPO on pass@1, with the largest gains on the smallest models. At the same doubled inference budget, the trained cascade nearly doubles the accuracy of an untrained Mixture-of-Agents pass over the same base, which itself barely improves on GRPO."},{"type":"heading","level":1,"text":"2 Related Work"},{"type":"heading","level":4,"text":"Outcome-only RL and the length pathology."},{"type":"para","text":"GRPO (Shao et al., 2024 ) and the R1 recipe (Guo et al., 2025 ) reward only the verified final answer. A well-documented side effect is uncontrolled growth of trace length (“overthinking”), with accuracy rising far more slowly than tokens (Aggarwal & Welleck, 2025 ; Chen et al., 2024a ) . Remedies act on the symptom: explicit length penalties or length-controlled objectives (Aggarwal & Welleck, 2025 ) , and dynamic sampling that discards degenerate all-correct/all-wrong groups (Yu et al., 2025 ) . None grades the content of the trace, because step-level labels do not exist and learned process reward models are costly and themselves unverifiable (Lightman et al., 2023 ) . Agon leaves the verifier untouched and instead derives a content signal from a competing peer."},{"type":"heading","level":4,"text":"Self-play and self-improvement."},{"type":"para","text":"A model can improve against versions of itself: SPIN (Chen et al., 2024b ) plays the current policy against its own past generations; Self-Rewarding LMs (Yuan et al., 2024 ) use the model as its own judge; and self-play reasoners such as Absolute Zero (Zhao et al., 2025 ) and R-Zero (Huang et al., 2025 ) co-evolve a proposer and a solver instantiated from one model. These methods are powerful but share a structural ceiling: a policy optimized on its own signal reinforces the blind spots that caused its errors, and self-correction without external feedback is unreliable (Huang et al., 2024 ) . Agon differs by training two distinct policies with different initialization and update streams, so the grader is a genuinely different agent rather than a copy of the graded. Multiagent finetuning (Subramaniam et al., 2025 ) also diversifies several models from one base via distinct data streams, but toward cooperative ensembling; Agon maintains divergence for adversarial grading , where the distinct rival is what supplies the reward."},{"type":"heading","level":4,"text":"Multi-model collaboration, debate, and ensembles."},{"type":"para","text":"Combining several models (self-consistency (Wang et al., 2023 ) , multi-agent debate (Du et al., 2023 ) , and Mixture-of-Agents (Wang et al., 2024 ) ) can improve robustness, but the aggregation is cooperative : it pushes toward consensus, which dilutes quality when members differ in strength (Li et al., 2025 ) , and yields little when they are near-identical. These are inference-time procedures over frozen models. Agon instead applies cross-model pressure during RL and makes it competitive: a model is rewarded for out-solving a peer that has seen its work, so the objective pays for beating the peer, not for matching it. We use a cooperative variant only as an ablation to isolate the value of competition, and include a two-agent MoA layer as a zero-training control; it recovers roughly one-eighth of Agon’s gain over GRPO ( Table ˜ 3 ). Closest in spirit is in-context self-refinement (Madaan et al., 2023 ) , where a model revises its own draft; Agon differs in that the draft comes from a competing, distinct peer and is selected by outcome under RL, not by self-feedback."},{"type":"heading","level":4,"text":"Adversarial and generator–verifier training."},{"type":"para","text":"Discriminator/verifier signals have long shaped generators, from GANs (Goodfellow et al., 2014 ) to prover–verifier games that improve the legibility of LLM solutions (Kirchner et al., 2024 ) and adversarial games over language (Cheng et al., 2024 ) . Debate (Irving et al., 2018 ) first framed a judged competition between two agents as a scalable proxy for supervising reasoning itself. Agon is symmetric and rotational rather than role-fixed: both models alternate between drafting and challenging, so neither degenerates into a pure critic, and both improve as solvers; the judge is a ground-truth verifier rather than a learned or human one. The eventual move from text to latent-space exchange connects to work on inter-model communication in representation space (Beliaev, 2025 ) , which we leave as future work."},{"type":"heading","level":1,"text":"3 Preliminaries: GRPO and the Unscored Trace"},{"type":"para","text":"For a problem x x with a verifier r ( x , y ) ∈ { 0 , 1 } r(x,y)\\in\\{0,1\\} that checks only the final answer of a completion y = ( trace , answer ) y=(\\text{trace},\\text{answer}) , GRPO (Shao et al., 2024 ) samples a group of G G rollouts y 1 , … , y G ∼ π θ ( ⋅ ∣ x ) y_{1},\\dots,y_{G}\\sim\\pi_{\\theta}(\\cdot\\mid x) and forms the group-relative advantage"},{"type":"para","text":"updating π θ \\pi_{\\theta} to raise the log-probability of above-average rollouts, with the usual KL regularization toward a reference (the base) omitted here for brevity; groups with zero reward variance (all-correct or all-wrong, where σ = 0 \\sigma=0 ) carry no signal and are dropped. The signal is entirely a function of the answer : two completions with the same answer receive the same advantage regardless of how they reasoned."},{"type":"heading","level":4,"text":"The unscored trace."},{"type":"para","text":"On hard problems the per-problem solve rate 𝔼 y ∼ π θ [ r ( x , y ) ] \\mathbb{E}_{y\\sim\\pi_{\\theta}}[r(x,y)] is small, and the policy raises expected reward most cheaply by increasing the number of distinct attempts inside one trace: each restart, case split, or “let me reconsider” is another chance at the answer. Because the trace is never scored, this padding is not penalized and is in fact reinforced through any correct rollout that contains it. The result is the empirically observed regime where length grows much faster than accuracy (Aggarwal & Welleck, 2025 ; Chen et al., 2024a ) ; in our own runs, GRPO training grows the mean trace from the zero-shot 6.1 6.1 k to 8.1 8.1 k tokens for + 7 +7 pp accuracy ( Table ˜ 3 ). The policy increases the quantity of reasoning while its density, the useful signal per token, stays flat. The bottleneck, then, is not quantity. The prevailing axes of scale (data, parameters, context, and sampled tokens) all add more reasoning; Agon instead targets a different axis, signal per token, by making low-density traces lose the competition."},{"type":"heading","level":4,"text":"The missing grader."},{"type":"para","text":"Closing this gap requires a reward that distinguishes a concise, sound trace from a long, lucky one, i.e. a signal on the reasoning itself. Verifiers do not provide it, and process reward models require labels we do not have (Lightman et al., 2023 ) . Agon obtains the signal relationally : a peer policy attempts the same x x , reads the other’s worked solution (the post-reasoning summary, not the raw trace; Section ˜ 4.3 ), and the reward depends on which model out-reasons the other . A trace whose flaws the opponent exploits (a sign error, a dead end it avoids) loses; a trace that leads to a win is reinforced. The grader is thus a second model, and, because both models are optimized, the bar rises over training. We formalize this next."},{"type":"heading","level":1,"text":"4 Method: Agon"},{"type":"para","text":"Agon trains two policies, A A and B B , head-to-head on the same problems. The design has four parts: a draft-and-challenge rollout that routes one model’s reasoning into the other’s context ( Section ˜ 4.3 ); a competitive reward that grades the challenger against the drafter ( Section ˜ 4.4 ); a compute-parity accounting so all comparisons are fair ( Section ˜ 4.5 ); and an efficient instantiation of the two policies as divergent adapters over a shared base ( Section ˜ 4.2 )."},{"type":"heading","level":2,"text":"4.1 Why a second model"},{"type":"para","text":"Post-training RL is self-improvement: the gradient in Equation ˜ 1 is computed from rollouts the policy itself produced, so it amplifies whatever the policy already does, including the systematic errors that define its blind spots. A model checking its own work inherits the bias that produced the mistake, which is why unaided self-correction often fails to add signal (Huang et al., 2024 ) ; we expect the same closed loop to cap self-play, however many rounds it runs (illustrated in Figure ˜ 3 , left). A different model is outside this loop: a peer hypothesized to have different failure modes audits what the policy cannot see in itself. In principle the peer could come from a different model family: the strongest models, open ones included, now cluster within a few points of one another on standard reasoning benchmarks (Stanford HAI, 2026 ) , so pairs of matched strength are easy to find; whether such pairs also differ enough in behavior is an empirical question we leave to future work. Our instantiation instead engineers the divergence structurally from one base ( Section ˜ 4.2 )."},{"type":"para","text":"Pairing helps only under two conditions. The models must be of comparable strength (otherwise, we expect, the weaker simply imitates the stronger and the game collapses into distillation; we do not ablate the strength gap), and they must have different failure modes , so that one can catch what the other misses (we make this concrete in Section ˜ 4.2 , Figure ˜ 4 ). We therefore keep the pair matched in capacity but divergent in behavior, and we make the interaction competitive so the objective rewards beating the peer rather than agreeing with it."},{"type":"heading","level":2,"text":"4.2 Instantiating the two policies"},{"type":"para","text":"Agon itself only requires two policies of matched strength and different blind spots; how they are parameterized is an implementation choice. We instantiate both as LoRA adapters (Hu et al., 2022 ) over a single frozen base model: π A = base + Δ A \\pi_{A}=\\mathrm{base}\\!+\\!\\Delta_{A} and π B = base + Δ B \\pi_{B}=\\mathrm{base}\\!+\\!\\Delta_{B} . The second policy costs one extra rank-16 adapter ( ∼ 10 {\\sim}10 M parameters, ∼ 2 % {\\sim}2\\% of the base, plus its optimizer state) rather than a full second model, and the shared base keeps the two policies in compatible representation spaces, useful for the eventual latent-space exchange ( Section ˜ 6 ). Two models alone are not enough, though: the useful regime is the one motivated in Section ˜ 4.1 , equal strength with hypothesized different blind spots ( Figure ˜ 4 ). We maintain divergence by (i) different initialization ( B B starts with small Gaussian noise around the zero matrix; A A uses the standard LoRA zero-init) and (ii) different update streams induced by role rotation ( Section ˜ 4.3 ): on any given problem one adapter drafts while the other challenges, so the two never receive the same gradient. All reported numbers use the same base checkpoint for both adapters. We stress that this instantiation is minimally divergent: at initialization the two policies differ only by adapter noise, so any complementarity must emerge from the diverging update streams during training rather than from pre-existing blind-spot differences. Different blind spots are therefore a hypothesis about the trained pair, not a measured property. The gap between cooperative exchange and single-adapter self-refinement (46 vs 32; Table ˜ 3 ) shows that training to read a peer’s draft beats training to read one’s own, but the comparison varies more than one factor (trainable adapter capacity, whose draft is read, the refiner prompt), so it does not isolate blind-spot divergence as the cause."},{"type":"heading","level":2,"text":"4.3 The draft-and-challenge step"},{"type":"para","text":"Each optimizer step ( Figure ˜ 2 , Algorithm ˜ 1 ) designates a drafter and a challenger . The drafter generates N N rollouts { a i } ∼ π draft ( ⋅ ∣ x ) \\{a_{i}\\}\\sim\\pi_{\\text{draft}}(\\cdot\\mid x) from the plain problem prompt and receives a vanilla GRPO update on them (correctness and format only, no opponent term); this standalone stream trains each adapter to solve from scratch, which is the first stage of deployment ( Section ˜ 4.5 ). The challenger then conditions on its paired opponent’s solution summary : the answer section the model writes after its private reasoning block, which recaps the derivation. The raw thinking trace is dropped, since the summary carries the distilled solution at a fraction of the tokens (cheaper to prefill, and free of the exploration noise that dominates long traces), and the final answer is withheld as a precaution against copying ( Figure ˜ 5 ), though a worked summary often implies its answer anyway. The summary is inserted as an attempt that “may or may not be correct,” and the challenger generates one rollout per pair, b i ∼ π chal ( ⋅ ∣ x , a i ) b_{i}\\sim\\pi_{\\text{chal}}(\\cdot\\mid x,a_{i}) , forming a group of N N . The challenger receives the competitive gradient; each stream’s GRPO advantage is computed within its own group of N N . Because each b i b_{i} sees a different opponent a i a_{i} , the group spans varied opponent contexts for the same x x ; this is what gives the conversion bonus within-group variance. Roles rotate every step : on even steps A A drafts and B B challenges, on odd steps the reverse. Without rotation, each adapter receives only one kind of gradient, one only ever solving from scratch and the other only ever solving against an opponent; the pair still trains, but ends up weaker (52 vs 61 pass@1; Table ˜ 7 ). With rotation, both adapters alternate between the two streams every step and co-improve. The challenger prompt instructs the model not to trust or paraphrase the opponent but to verify it and solve independently, winning by being correct when the opponent is wrong; the prompt is identical across the coop and adv reward stacks ( Section ˜ 4.4 ), so they differ only in the reward."},{"type":"heading","level":2,"text":"4.4 Reward: correctness plus a conversion bonus"},{"type":"para","text":"Let c ( y ) = r ( x , y ) ∈ { 0 , 1 } c(y)=r(x,y)\\in\\{0,1\\} be correctness. We report two reward stacks, applied to the challenger rollout b i b_{i} given its paired drafter solution a i a_{i} :"},{"type":"para","text":"where ϕ ( ⋅ ) \\phi(\\cdot) is a format term (well-formed <think> / <answer> structure) with weight λ = 0.5 \\lambda=0.5 (25% of the correctness term), identical across stacks so it introduces no bias between them. In coop the opponent appears only in the context , not in the reward, which isolates the value of information exchange (the exchange claim). In adv the conversion bonus c ( b i ) ( 1 − c ( a i ) ) c(b_{i})\\,(1-c(a_{i})) raises the effective correctness weight from 2 2 to 3 3 exactly on the paired opponent attempts that failed : solving where the opponent could not is worth more, which is the competitive pressure."},{"type":"para","text":"The form matters. A naïve margin c ( b i ) − c ( a i ) c(b_{i})-c(a_{i}) works poorly: since c ( a i ) c(a_{i}) is action-independent (fixed by the pairing a i a_{i} , not by the challenger’s rollout b i b_{i} ), its subtractive term contributes no expected policy gradient (exactly in the unnormalized objective; up to the nonlinearity of group standardization after it): it acts as a per-sample baseline, shifting each reward by the drafter’s outcome without adding any competitive direction. The tested margin stack, R = 2 c ( b i ) + ( c ( b i ) − c ( a i ) ) = 3 c ( b i ) − c ( a i ) R=2c(b_{i})+\\big(c(b_{i})-c(a_{i})\\big)=3c(b_{i})-c(a_{i}) , makes this concrete: the scalar 3 3 cancels under group-standard-deviation normalization and the − c ( a i ) -c(a_{i}) term is action-independent (zero expected policy gradient), so in expectation the margin points in the same direction as the cooperative gradient. The ablation is consistent with this prediction ( 49 49 vs coop 46 46 ), though the 3 3 pp difference lies within binomial sampling noise. The conversion bonus instead modulates the correctness weight by opponent difficulty : because it multiplies the action-dependent c ( b i ) c(b_{i}) , the reweighting shifts the gradient itself, and, given the within-group spread of opponent contexts { a i } \\{a_{i}\\} , tilts it toward the opponent attempts that failed. We confirm this empirically ( Table ˜ 7 ): the margin form barely improves on cooperation, whereas the conversion bonus clearly does. More generally, any reward that is affine in c ( b i ) c(b_{i}) with group-constant coefficients (e.g., a shared opponent draft for the entire challenger group, making c ( a ) c(a) identical for all N N rollouts) normalizes to exactly the advantage of the conversion-free reward in the same contexts under group-relative standardization; the conversion term then contributes zero expected policy gradient. Within-group variation in opponent difficulty is what makes the competitive term trainable. In the shared-opponent ablation we therefore fix one opponent draft per problem, hidden from the challenger’s context and used only for the reward (so the contexts are plain prompts, as in vanilla GRPO); the observed result (32) is consistent with the theoretical prediction that a group-constant conversion bonus adds no directional signal. This departs from the standard GRPO assumption that all rollouts in a group share the same prompt; here each challenger rollout conditions on its own opponent context. The resulting credit assignment is not merely noisier but systematically confounded: a rollout’s success probability depends on the quality of the draft it happened to draw (an action-independent draw), so group standardization mixes context difficulty with policy quality, and the conversion bonus, which pays exactly in the harder contexts, partially offsets that difficulty penalty. This admits an alternative reading of the adversarial gain: rather than “competitive pressure” in a game-theoretic sense, the bonus acts as difficulty-weighted reward shaping that upweights correct completions in hard contexts. Our experiments do not distinguish these two interpretations; we use the competitive framing for its constructive role in the design and leave the disentanglement to future work."},{"type":"para","text":"The resulting reward ladder (adv) is R − λ ϕ ∈ { 0 , 2 , 3 } R-\\lambda\\phi\\in\\{0,2,3\\} . The drafter’s standalone stream has no opponent, so Equation ˜ 2 applies as-is."},{"type":"para","text":"As an optional density lever , studied separately in Section ˜ 5.6 and not part of the headline recipe, we add a length tiebreak that fires only when both solutions are correct: R ( b i ) + = γ c ( b i ) c ( a i ) 1 [ | b i | < | a i | ] R(b_{i})\\mathrel{+{=}}\\gamma\\,c(b_{i})\\,c(a_{i})\\,\\mathbf{1}\\!\\left[\\,|b_{i}|<|a_{i}|\\,\\right] with weight γ \\gamma , rewarding the challenger for matching the opponent’s correctness in fewer tokens. Because the term requires c ( b i ) = 1 c(b_{i})=1 , it never directly penalizes correctness, only breaking ties among already-correct traces, so accuracy stays essentially flat while length drops, converting the emergent shortening ( Section ˜ 5.5 ) into a directly tunable target."},{"type":"heading","level":2,"text":"4.5 Compute parity"},{"type":"para","text":"Per step Agon generates N N drafter rollouts plus N N challenger rollouts ( 2 N 2N ), versus N N for vanilla GRPO. Both adapters are updated every step, the drafter on its standalone group and the challenger on its opponent-conditioned group; rotation swaps which adapter gets which stream. Our primary invariant is the generation budget : every method generates 2 N 2N rollouts per training problem (vanilla GRPO gets 2 N 2N instead of its default N N ) under the same completion-length cap and the same single pass over the training problems. Realized token counts then differ only through learned trace length, bounded for every method by the same cap; we do not audit total generated tokens (the drafter-stage length is not reported), but with the rollout count and length cap equalized, the gains are not attributable to a larger generation budget . The challenger additionally prefills the opponent summary, which is extra prefill compute we do not equalize (the summary is short relative to a full trace, is amortized cheaply against generation, and is absent at single-adapter inference). At inference Agon deploys exactly as it trains: a two-stage cascade in which one adapter drafts from the plain prompt and the other reads the draft’s summary and produces the final answer. This is two sequential generation passes, a doubled budget relative to a single-pass baseline, so Table ˜ 3 includes two-pass controls at the same budget (cross-refinement, self-refinement). Reported trace lengths refer to the final (challenger) stage. Both cascade directions ( A → B A{\\to}B and B → A B{\\to}A ) are evaluated and the better is reported, selected post hoc on the held-out set."},{"type":"heading","level":1,"text":"5 Experiments"},{"type":"heading","level":4,"text":"Setup."},{"type":"para","text":"We train Qwen3 models (Qwen Team, 2025 ) (primarily 0.6B; 1.7B/4B for scaling) on the hard split of DeepMath-103K (He et al., 2025 ) (difficulty 8, binary-answer questions removed), with a held-out set of 300 300 problems. We use group size G = N = 8 G=N=8 for both streams, LoRA rank 16 16 , learning rate 5 × 10 − 5 5\\times 10^{-5} (linear decay, 10 % 10\\% warmup), one training epoch over the 3 , 000 3{,}000 training problems, format weight λ = 0.5 \\lambda=0.5 , length tiebreak γ = 0.5 \\gamma=0.5 (used in Section ˜ 5.6 only), and a standard format reward (well-formed <think> / <answer> ) as ϕ \\phi . Generation uses vLLM (Kwon et al., 2023 ) and training uses TRL (von Werra et al., 2020 ) . The primary metric is pass@1 on the DeepMath held-out set; we additionally sanity-check on GSM8K (Cobbe et al., 2021 ) and MATH-500 (Hendrycks et al., 2021 ) ( Table ˜ 2 ). For the second-domain study ( Section ˜ 5.3 ) we train on the easy -difficulty subset of CodeContests (Li et al., 2022 ) with a unit-test verifier ( r = 1 r=1 iff all tests pass), under the same protocol and budget. Evaluation protocol: sampled decoding (temperature 0.6 0.6 , top- p p 0.95 0.95 ), one rollout per stage, generation budget matched to training ( 15 15 k completion tokens per stage), the answer parsed from the <answer> block; the 300 300 held-out problems are difficulty-8 only with binary-answer questions removed, split with a fixed seed before any training. Pair-trained methods are evaluated as the two-stage cascade ( Section ˜ 4.5 ) in both directions, reporting the better direction; mean trace length reports the final-stage completion. Reported pass@1 values include 95% Clopper–Pearson binomial confidence intervals over the 300-problem held-out set ( Table ˜ 4 )."},{"type":"heading","level":4,"text":"The 2 × 2 2\\times 2 design."},{"type":"para","text":"We cross two binary factors, competition and information exchange , giving four cells plus three controls ( Table ˜ 1 ): self-refinement, an untrained MoA at inference, and a budget-matched GRPO two-pass self-cascade. One caveat is structural: the “competition without exchange” cell uses a shared opponent draft, hidden from the challenger and used only for the reward, which by the analysis in Section ˜ 4.4 makes the conversion bonus group-constant and hence gradient-free, so this cell is theoretically equivalent to vanilla GRPO and serves as a consistency check rather than a true isolation of competition. Consequently, the exchange claim (exchange helps) is carried by the GRPO-vs-cooperative comparison, and the competition claim (competition helps beyond exchange) by the cooperative-vs-adversarial comparison, holding everything else fixed at a matched generation budget; a competition-without-visible-exchange cell with per-rollout hidden drafts is left for future work."},{"type":"heading","level":2,"text":"5.1 Main results"},{"type":"para","text":"Table ˜ 3 reports the main comparison on Qwen3-0.6B. All deltas below are read from the pass@1 column. The trained pair methods and the refinement controls are two-pass at inference; vanilla GRPO and zero-shot are single-pass (see the footnote of Table ˜ 3 ). Information exchange alone (cooperative exchange: 46 46 ) already improves over vanilla GRPO ( 30 30 ) and over self-refinement ( 32 32 ), supporting the exchange claim; adding competition (Agon: 61 61 ) improves further over cooperation, supporting the competition claim. The untrained MoA control is a two-agent Mixture-of-Agents layer at inference (Wang et al., 2024 ) : each model reads the other’s attempt and refines its own. It lifts GRPO by + 4 +4 pp ( 30 → 34 30\\!\\to\\!34 ); Agon lifts it by + 31 +31 pp, roughly eight times the gain of untrained aggregation. We note that the two levers contribute comparably: information exchange adds + 16 +16 pp (GRPO → \\to cooperative) and competition a further + 15 +15 pp (cooperative → \\to Agon). The dual-adapter architecture alone contributes little: the competitive-shared-opponent variant sits at 32 32 , barely above single-model GRPO at 30 30 , so the gain from exchange is attributable to the shared context, not to going from one model to two."},{"type":"para","text":"Each method is a single training run; the confidence intervals in Table ˜ 4 cover sampling variation over the 300 held-out problems ( ± ∼ 5.5 \\pm{\\sim}5.5 pp) but not run-to-run training variance. Both headline deltas (exchange: + 16 +16 pp; competition: + 15 +15 pp) exceed the width of these intervals. The Agon cascade reaches roughly twice vanilla GRPO’s held-out pass@1 while shortening final-stage traces; in gain terms, GRPO lifts the base by + 7 +7 pp ( 23 → 30 23\\!\\to\\!30 ) while Agon lifts it by + 31 +31 pp over GRPO ( 30 → 61 30\\!\\to\\!61 ), and + 26 +26 pp over a two-pass GRPO self-cascade at matched inference budget, nearly doubling the untrained MoA cascade."},{"type":"para","text":"To decompose the cascade gain, we evaluate the drafter adapter standalone after training (plain prompt, no opponent summary): it reaches 46 pass@1, a +16 pp lift over vanilla GRPO. This is notable given the budget asymmetry: each adapter drafts on only half the steps with group size N N , so it receives roughly a quarter of the standalone-rollout gradient volume of the vanilla GRPO baseline (which uses 2 N 2N on every problem), plus challenger-stream gradients on the alternating steps. Our working hypothesis is that the challenger-stream updates transfer to standalone solving (the adapter is trained to verify and re-derive, skills that a plain prompt also exercises), but we have not tested this with a control (e.g. the standalone accuracy of a cooperative-trained or self-refinement adapter), so the attribution of the drafter’s +16 pp to the competitive objective is a hypothesis, not an isolated measurement. The challenger then converts a net 15 of the drafter’s 54 failures into correct answers (with possible regressions among the drafter’s 46 successes), yielding the final 61. Under this reading, the drafter’s lift (+16 pp) and the challenger’s conversion (+15 pp) contribute comparable shares of the total gain, the former via the hypothesized transfer of challenger-stream skills to standalone solving."},{"type":"para","text":"To check that the gains are not an artifact of the training distribution, Table ˜ 2 reports the same checkpoints on GSM8K and MATH-500 without any further tuning: the ordering is preserved with smaller absolute margins than in-distribution ( + 13 +13 and + 19 +19 pp over zero-shot, vs + 38 +38 on DeepMath-hard), as expected for a method whose pressure targets the hard regime."},{"type":"heading","level":2,"text":"5.2 Scaling and generality"},{"type":"para","text":"We expect smaller models, which live deeper in the hard regime, to benefit most. Table ˜ 5 shows the gain over vanilla GRPO shrinking but remaining positive as the base grows, and the same pattern holding on two other model families, Qwen3.5 (Qwen Team, 2026 ) and Gemma 4 (Gemma Team, 2026 ) : the delta tracks zero-shot strength, not the family. The Agon cascade on the 0.6B base (61) exceeds not only zero-shot Qwen3-4B (52) but vanilla GRPO on the 4B base (59), a model ∼ 7 × {\\sim}7\\times larger, on the same held-out set. In every case the pair is two adapters over one base (our instantiation, not a requirement of the method); pairing across families is future work."},{"type":"heading","level":2,"text":"5.3 A second domain: competitive-programming code"},{"type":"para","text":"The recipe is not specific to math; any domain with a clean programmatic verifier qualifies. We repeat the core comparison on Qwen3-1.7B trained on the easy -difficulty subset of CodeContests (Li et al., 2022 ) , with r ( x , y ) = 1 r(x,y)=1 iff the generated program passes all unit tests, a held-out set of 300 300 problems, and the same protocol and generation budget. Table ˜ 6 shows the same ordering as Table ˜ 3 , exchange beats the baseline and competition beats cooperation, with traces again shortening. The absolute delta is smaller than on math: unit-test rewards are sparser at this scale, and code traces make the opponent prefill longer relative to generation. Across domains the claims should be read as directional rather than magnitude-matched."},{"type":"heading","level":2,"text":"5.4 Ablations"},{"type":"para","text":"Table ˜ 7 isolates each design choice on Qwen3-0.6B (held-out pass@1). The competition claim is carried by the competition row (adv vs coop). The information exchange row (Agon vs the shared-opponent variant) should be read with care: switching to a shared opponent removes the visible draft and , by the analysis in Section ˜ 4.4 , deactivates the competitive gradient, so this comparison changes two things at once and is a consistency check rather than an isolated exchange effect; the isolated exchange delta is GRPO vs cooperative in Table ˜ 3 . In the “shared opponent” ablation the challenger group is scored against one fixed opponent draft, hidden from its context (group-constant c ( a ) c(a) ), so the conversion bonus normalizes away ( Section ˜ 4.4 ); the result ( 32 32 vs 30 30 ) is consistent with theory and shows that within-group variance in opponent difficulty is required for the competitive term to contribute gradient. A per-rollout opponent pairing (distinct hidden drafts) is the honest ablation of competition without visible exchange; that variant is left for future work. Role rotation is ablated explicitly: with fixed roles each adapter receives only one gradient stream, yielding 52 pass@1 versus 61 with rotation."},{"type":"heading","level":2,"text":"5.5 Analysis: why traces shorten"},{"type":"para","text":"Empirically, Agon’s traces are shorter than GRPO’s ( Figure ˜ 7 a): the final (challenger) stage averages 3.5 3.5 k tokens against 8.1 8.1 k for a single GRPO pass. Note that this is a per-stage comparison: the cascade also runs a drafting stage, whose length we do not report, so total generated tokens per problem are not directly compared here and a full token-cost accounting is left for further analysis. The like-for-like comparison is against the two-stage untrained MoA control, where the same (final) stage is measured on both sides: Agon’s is half the length ( 3.5 3.5 k vs 6.9 6.9 k). The headline reward contains no length term. The observed shortening is consistent with the challenger having a candidate solution in context, so length-heavy exploration is no longer the only route to a correct answer. Challenger completions conditioned on a correct opponent summary are ∼ 35 % {\\sim}35\\% shorter than the challenger-stage average ( 2.3 2.3 k vs 3.5 3.5 k tokens on the final checkpoint), and Figure ˜ 6 shows the typical pattern: the challenger pinpoints the flawed step, re-derives it, and stops."},{"type":"heading","level":2,"text":"5.6 An optional density lever (auxiliary study)"},{"type":"para","text":"The shortening above is emergent: nothing in the headline reward asks for it. The same game can, however, also grade brevity: instead of taxing tokens with a penalty, we make being correct in fewer tokens than the rival a way to win. As a small side study, separate from the main recipe and from the two headline claims, we ask whether the shortening can thus be driven directly, using the length-tiebreak term from Section ˜ 4.4 (a bonus γ \\gamma when the challenger is correct and shorter than an equally-correct opponent). On Qwen3-0.6B, turning it on cuts mean trace length from 3.5 3.5 k to 2.6 2.6 k at essentially unchanged accuracy ( 61 → 60 61\\!\\to\\!60 pass@1), so the emergent shortening becomes directly controllable ( Table ˜ 8 ). Because the term only breaks ties among already-correct traces, it compresses without measurably trading away accuracy. We report this as an auxiliary knob; all headline numbers elsewhere use the length-free reward."},{"type":"heading","level":1,"text":"6 Conclusion"},{"type":"para","text":"Agon trains two models to reason over each other’s work, and the pair co-improves past where a single self-graded policy stalls. The competing peer supplies the signal outcome-only RL lacks: it reads the policy’s solution and grades it implicitly, with no process labels and no change to the verifier. Two distinct policies, trained head-to-head with rotating draft-and-challenge roles (instantiated cheaply as two adapters over one frozen base), let us test the idea. Under a matched generation budget, information exchange beats vanilla GRPO and self-refinement (the exchange claim), competition beats cooperation (the competition claim), and the resulting traces are shorter, a by-product of the competition rather than of an explicit penalty. At the same doubled inference budget, combining two copies of the base without training buys 4 4 points over GRPO, while training the pair to compete buys 31 31 ; the gain comes from the learned collaboration rather than from having two models."},{"type":"heading","level":4,"text":"Limitations."},{"type":"para","text":"Agon needs a verifier and reference problems, and we study math plus a smaller code study, both with clean programmatic rewards; noisier domains may behave differently. Gains depend on the pair being matched in strength but divergent in behavior (too large a gap collapses into distillation, too small a gap into self-play), and our divergence is maintained heuristically (init + + rotation) rather than guaranteed; complementarity is not quantified beyond the cascade win rate. Text exchange bounds the channel bandwidth. Inference is a two-stage cascade, i.e. two sequential generation passes and added latency; reported trace lengths cover the final stage only, and the better cascade direction is selected post hoc on the held-out set. Compute parity holds for generated tokens only: the challenger’s extra prefill of the opponent summary is not equalized ( Section ˜ 4.5 ). Groups with zero within-group variance (all-correct or all-wrong) yield undefined σ \\sigma in GRPO advantage; such groups are dropped. Every trained number in the paper is a single training run evaluated with a single sampled rollout per problem; the better cascade direction was selected post hoc on the held-out set. Run-to-run training variance is unquantified, and the reported deltas should be read with that in mind. The reported magnitudes come from a hard, decontaminated subset and should not be read as DeepMath-wide pass@1."},{"type":"heading","level":4,"text":"Future work."},{"type":"para","text":"This work establishes that the cross-model signal helps while exchanging it in text; the channel itself is the next axis. Exchanging hidden states directly, via KV-cache injection or gated latent bridges between the two adapters, removes the serialization bottleneck and lets models share information they cannot verbalize (Beliaev, 2025 ) . Beyond accuracy, our auxiliary density lever ( Section ˜ 5.6 ) is a first step; scaling it into a full multi-objective, compression traded explicitly against correctness, could make density a first-class, tunable target, and a diversity term over divergent solution strategies could make the pair complementary enough to ensemble."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Reinforcement learning from verifiable rewards has become the standard tool for sharpening the reasoning of large language models (LLMs) on tasks such as mathematics and code (Guo et al., 2025 ; Shao et al., 2024 ) . The dominant recipe is on-policy and outcome-based: sample a group of rollouts from the current policy, score each by whether its final answer is correct, and update with group-relative advantages, as in GRPO (Shao et al., 2024 ) . The reward is attached to the answer; the chain of thought that produced it is never graded. On problems the model can already partly solve this is benign, but on hard problems it creates a strong incentive to write more: a longer chain affords more chances to stumble onto the right answer, so additional text is the cheapest way to raise expected reward. The trace inflates with hedging and backtracking (“hmm,” “wait,” “let me reconsider”), and accuracy grows far slower than length (accuracy rises only modestly while length inflates by an order of magnitude (Aggarwal & Welleck, 2025 ) ). The model ends up producing more reasoning per problem without producing better reasoning per token.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The obvious fix, grading the reasoning directly, is intractable. There is no ground truth for “good thinking”: no label says which step was the insight and which was filler, and a learned process reward model is expensive, brittle, and itself unverifiable (Lightman et al., 2023 ) . So the trace stays unscored, and the length pathology persists. We ask instead: can a second model supply the missing signal? If a different policy attempts the same problem and we reward each model for out-reasoning the other , then the trace is graded implicitly by the rival, without any process labels. A step that leads to a win is reinforced; filler that the opponent exploits is punished.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The grader must be a different model, and the game must be competitive . Post-training RL is, by construction, self-improvement: the policy is optimized on signal it generates itself, which reinforces the very blind spots that created its errors: a model auditing its own work tends to plateau (Huang et al., 2024 ) . A second model with different failure modes breaks this closed loop. But merely having two models is not enough. Naively averaging or letting models agree, as in self-consistency and mixture-of-agents (Wang et al., 2023 , 2024 ) , tends to regress toward consensus and can dilute quality when the models differ in strength (Li et al., 2025 ) , while two near-identical models gain little from one another. The combination that works is a pair of comparable strength hypothesized to have different blind spots, placed under competitive pressure: each model must not only solve the problem but beat a peer that sees its work, so the objective rewards out-reasoning the peer rather than agreeing with it. Under this pressure the pair co-improves: each win raises the bar for the other on the next step, and both models end up past the ceiling where a single self-graded policy plateaus.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We instantiate this as Agon (Greek agōn , contest). Two distinct policies are trained head-to-head: a drafter produces candidate solutions; a challenger reads an opponent’s worked solution and is rewarded for solving correctly and beating it; roles rotate every step so both models learn both roles, and at inference the pair deploys the same way, one model drafting and the other answering after reading the draft. In principle the recipe applies to any pair of comparably strong models; for efficiency we realize the pair as two low-rank adapters over one frozen base ( ∼ 2 % {\\\\sim}2\\\\% memory overhead, not 2 × 2\\\\times ), kept divergent by different initialization and update streams. In this work the exchange happens in text: we first establish that the cross-model signal is useful before optimizing the channel (latent-space exchange; Section ˜ 6 ). We test two claims under a matched generation budget: that cross-model information exchange raises accuracy (the exchange claim ), and that adversarial pressure raises it further than cooperation (the competition claim ), using a 2 × 2 2\\\\times 2 design that crosses competition with exchange (one cell of which is provably inert under group-relative normalization; Section ˜ 4.4 ), plus a single-model self-refinement control, a zero-training orchestration control, and a budget-matched GRPO self-cascade control.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Contributions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We reframe reasoning quality as an implicit reward that a competing peer model can supply, turning the unlabeled “good thinking” problem into a head-to-head game played during RL ( Section ˜ 4 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We propose Agon: a recipe for jointly training two distinct policies against each other with competitive GRPO, via draft-and-challenge rollouts, per-step role rotation, and a competitive reward (correctness plus a conversion bonus for out-solving the opponent). It runs on a stock GRPO trainer with no changes to the optimization core; a shared-base dual-adapter instantiation makes the pair nearly as cheap as a single model ( Section ˜ 4 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We lay out a 2 × 2 2\\\\times 2 matrix (competition × \\\\times information exchange) with self-refinement and zero-training controls ( Section ˜ 5 ). One cell, competition without exchange, is inert by construction under group-relative normalization ( Section ˜ 4.4 ), so the competition claim rests on the adversarial-vs-cooperative comparison; the fully isolated cell (per-rollout hidden drafts) is left for future work.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Under a matched generation budget on hard math, competition + + exchange beats cooperation, self-refinement, and vanilla GRPO on pass@1, with the largest gains on the smallest models. At the same doubled inference budget, the trained cascade nearly doubles the accuracy of an untrained Mixture-of-Agents pass over the same base, which itself barely improves on GRPO.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Related Work\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Outcome-only RL and the length pathology.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"GRPO (Shao et al., 2024 ) and the R1 recipe (Guo et al., 2025 ) reward only the verified final answer. A well-documented side effect is uncontrolled growth of trace length (“overthinking”), with accuracy rising far more slowly than tokens (Aggarwal & Welleck, 2025 ; Chen et al., 2024a ) . Remedies act on the symptom: explicit length penalties or length-controlled objectives (Aggarwal & Welleck, 2025 ) , and dynamic sampling that discards degenerate all-correct/all-wrong groups (Yu et al., 2025 ) . None grades the content of the trace, because step-level labels do not exist and learned process reward models are costly and themselves unverifiable (Lightman et al., 2023 ) . Agon leaves the verifier untouched and instead derives a content signal from a competing peer.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Self-play and self-improvement.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A model can improve against versions of itself: SPIN (Chen et al., 2024b ) plays the current policy against its own past generations; Self-Rewarding LMs (Yuan et al., 2024 ) use the model as its own judge; and self-play reasoners such as Absolute Zero (Zhao et al., 2025 ) and R-Zero (Huang et al., 2025 ) co-evolve a proposer and a solver instantiated from one model. These methods are powerful but share a structural ceiling: a policy optimized on its own signal reinforces the blind spots that caused its errors, and self-correction without external feedback is unreliable (Huang et al., 2024 ) . Agon differs by training two distinct policies with different initialization and update streams, so the grader is a genuinely different agent rather than a copy of the graded. Multiagent finetuning (Subramaniam et al., 2025 ) also diversifies several models from one base via distinct data streams, but toward cooperative ensembling; Agon maintains divergence for adversarial grading , where the distinct rival is what supplies the reward.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Multi-model collaboration, debate, and ensembles.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Combining several models (self-consistency (Wang et al., 2023 ) , multi-agent debate (Du et al., 2023 ) , and Mixture-of-Agents (Wang et al., 2024 ) ) can improve robustness, but the aggregation is cooperative : it pushes toward consensus, which dilutes quality when members differ in strength (Li et al., 2025 ) , and yields little when they are near-identical. These are inference-time procedures over frozen models. Agon instead applies cross-model pressure during RL and makes it competitive: a model is rewarded for out-solving a peer that has seen its work, so the objective pays for beating the peer, not for matching it. We use a cooperative variant only as an ablation to isolate the value of competition, and include a two-agent MoA layer as a zero-training control; it recovers roughly one-eighth of Agon’s gain over GRPO ( Table ˜ 3 ). Closest in spirit is in-context self-refinement (Madaan et al., 2023 ) , where a model revises its own draft; Agon differs in that the draft comes from a competing, distinct peer and is selected by outcome under RL, not by self-feedback.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Adversarial and generator–verifier training.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Discriminator/verifier signals have long shaped generators, from GANs (Goodfellow et al., 2014 ) to prover–verifier games that improve the legibility of LLM solutions (Kirchner et al., 2024 ) and adversarial games over language (Cheng et al., 2024 ) . Debate (Irving et al., 2018 ) first framed a judged competition between two agents as a scalable proxy for supervising reasoning itself. Agon is symmetric and rotational rather than role-fixed: both models alternate between drafting and challenging, so neither degenerates into a pure critic, and both improve as solvers; the judge is a ground-truth verifier rather than a learned or human one. The eventual move from text to latent-space exchange connects to work on inter-model communication in representation space (Beliaev, 2025 ) , which we leave as future work.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Preliminaries: GRPO and the Unscored Trace\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For a problem x x with a verifier r ( x , y ) ∈ { 0 , 1 } r(x,y)\\\\in\\\\{0,1\\\\} that checks only the final answer of a completion y = ( trace , answer ) y=(\\\\text{trace},\\\\text{answer}) , GRPO (Shao et al., 2024 ) samples a group of G G rollouts y 1 , … , y G ∼ π θ ( ⋅ ∣ x ) y_{1},\\\\dots,y_{G}\\\\sim\\\\pi_{\\\\theta}(\\\\cdot\\\\mid x) and forms the group-relative advantage\"\n },\n {\n \"type\": \"para\",\n \"text\": \"updating π θ \\\\pi_{\\\\theta} to raise the log-probability of above-average rollouts, with the usual KL regularization toward a reference (the base) omitted here for brevity; groups with zero reward variance (all-correct or all-wrong, where σ = 0 \\\\sigma=0 ) carry no signal and are dropped. The signal is entirely a function of the answer : two completions with the same answer receive the same advantage regardless of how they reasoned.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"The unscored trace.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On hard problems the per-problem solve rate 𝔼 y ∼ π θ [ r ( x , y ) ] \\\\mathbb{E}_{y\\\\sim\\\\pi_{\\\\theta}}[r(x,y)] is small, and the policy raises expected reward most cheaply by increasing the number of distinct attempts inside one trace: each restart, case split, or “let me reconsider” is another chance at the answer. Because the trace is never scored, this padding is not penalized and is in fact reinforced through any correct rollout that contains it. The result is the empirically observed regime where length grows much faster than accuracy (Aggarwal & Welleck, 2025 ; Chen et al., 2024a ) ; in our own runs, GRPO training grows the mean trace from the zero-shot 6.1 6.1 k to 8.1 8.1 k tokens for + 7 +7 pp accuracy ( Table ˜ 3 ). The policy increases the quantity of reasoning while its density, the useful signal per token, stays flat. The bottleneck, then, is not quantity. The prevailing axes of scale (data, parameters, context, and sampled tokens) all add more reasoning; Agon instead targets a different axis, signal per token, by making low-density traces lose the competition.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"The missing grader.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Closing this gap requires a reward that distinguishes a concise, sound trace from a long, lucky one, i.e. a signal on the reasoning itself. Verifiers do not provide it, and process reward models require labels we do not have (Lightman et al., 2023 ) . Agon obtains the signal relationally : a peer policy attempts the same x x , reads the other’s worked solution (the post-reasoning summary, not the raw trace; Section ˜ 4.3 ), and the reward depends on which model out-reasons the other . A trace whose flaws the opponent exploits (a sign error, a dead end it avoids) loses; a trace that leads to a win is reinforced. The grader is thus a second model, and, because both models are optimized, the bar rises over training. We formalize this next.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Method: Agon\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Agon trains two policies, A A and B B , head-to-head on the same problems. The design has four parts: a draft-and-challenge rollout that routes one model’s reasoning into the other’s context ( Section ˜ 4.3 ); a competitive reward that grades the challenger against the drafter ( Section ˜ 4.4 ); a compute-parity accounting so all comparisons are fair ( Section ˜ 4.5 ); and an efficient instantiation of the two policies as divergent adapters over a shared base ( Section ˜ 4.2 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 Why a second model\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Post-training RL is self-improvement: the gradient in Equation ˜ 1 is computed from rollouts the policy itself produced, so it amplifies whatever the policy already does, including the systematic errors that define its blind spots. A model checking its own work inherits the bias that produced the mistake, which is why unaided self-correction often fails to add signal (Huang et al., 2024 ) ; we expect the same closed loop to cap self-play, however many rounds it runs (illustrated in Figure ˜ 3 , left). A different model is outside this loop: a peer hypothesized to have different failure modes audits what the policy cannot see in itself. In principle the peer could come from a different model family: the strongest models, open ones included, now cluster within a few points of one another on standard reasoning benchmarks (Stanford HAI, 2026 ) , so pairs of matched strength are easy to find; whether such pairs also differ enough in behavior is an empirical question we leave to future work. Our instantiation instead engineers the divergence structurally from one base ( Section ˜ 4.2 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Pairing helps only under two conditions. The models must be of comparable strength (otherwise, we expect, the weaker simply imitates the stronger and the game collapses into distillation; we do not ablate the strength gap), and they must have different failure modes , so that one can catch what the other misses (we make this concrete in Section ˜ 4.2 , Figure ˜ 4 ). We therefore keep the pair matched in capacity but divergent in behavior, and we make the interaction competitive so the objective rewards beating the peer rather than agreeing with it.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 Instantiating the two policies\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Agon itself only requires two policies of matched strength and different blind spots; how they are parameterized is an implementation choice. We instantiate both as LoRA adapters (Hu et al., 2022 ) over a single frozen base model: π A = base + Δ A \\\\pi_{A}=\\\\mathrm{base}\\\\!+\\\\!\\\\Delta_{A} and π B = base + Δ B \\\\pi_{B}=\\\\mathrm{base}\\\\!+\\\\!\\\\Delta_{B} . The second policy costs one extra rank-16 adapter ( ∼ 10 {\\\\sim}10 M parameters, ∼ 2 % {\\\\sim}2\\\\% of the base, plus its optimizer state) rather than a full second model, and the shared base keeps the two policies in compatible representation spaces, useful for the eventual latent-space exchange ( Section ˜ 6 ). Two models alone are not enough, though: the useful regime is the one motivated in Section ˜ 4.1 , equal strength with hypothesized different blind spots ( Figure ˜ 4 ). We maintain divergence by (i) different initialization ( B B starts with small Gaussian noise around the zero matrix; A A uses the standard LoRA zero-init) and (ii) different update streams induced by role rotation ( Section ˜ 4.3 ): on any given problem one adapter drafts while the other challenges, so the two never receive the same gradient. All reported numbers use the same base checkpoint for both adapters. We stress that this instantiation is minimally divergent: at initialization the two policies differ only by adapter noise, so any complementarity must emerge from the diverging update streams during training rather than from pre-existing blind-spot differences. Different blind spots are therefore a hypothesis about the trained pair, not a measured property. The gap between cooperative exchange and single-adapter self-refinement (46 vs 32; Table ˜ 3 ) shows that training to read a peer’s draft beats training to read one’s own, but the comparison varies more than one factor (trainable adapter capacity, whose draft is read, the refiner prompt), so it does not isolate blind-spot divergence as the cause.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3 The draft-and-challenge step\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each optimizer step ( Figure ˜ 2 , Algorithm ˜ 1 ) designates a drafter and a challenger . The drafter generates N N rollouts { a i } ∼ π draft ( ⋅ ∣ x ) \\\\{a_{i}\\\\}\\\\sim\\\\pi_{\\\\text{draft}}(\\\\cdot\\\\mid x) from the plain problem prompt and receives a vanilla GRPO update on them (correctness and format only, no opponent term); this standalone stream trains each adapter to solve from scratch, which is the first stage of deployment ( Section ˜ 4.5 ). The challenger then conditions on its paired opponent’s solution summary : the answer section the model writes after its private reasoning block, which recaps the derivation. The raw thinking trace is dropped, since the summary carries the distilled solution at a fraction of the tokens (cheaper to prefill, and free of the exploration noise that dominates long traces), and the final answer is withheld as a precaution against copying ( Figure ˜ 5 ), though a worked summary often implies its answer anyway. The summary is inserted as an attempt that “may or may not be correct,” and the challenger generates one rollout per pair, b i ∼ π chal ( ⋅ ∣ x , a i ) b_{i}\\\\sim\\\\pi_{\\\\text{chal}}(\\\\cdot\\\\mid x,a_{i}) , forming a group of N N . The challenger receives the competitive gradient; each stream’s GRPO advantage is computed within its own group of N N . Because each b i b_{i} sees a different opponent a i a_{i} , the group spans varied opponent contexts for the same x x ; this is what gives the conversion bonus within-group variance. Roles rotate every step : on even steps A A drafts and B B challenges, on odd steps the reverse. Without rotation, each adapter receives only one kind of gradient, one only ever solving from scratch and the other only ever solving against an opponent; the pair still trains, but ends up weaker (52 vs 61 pass@1; Table ˜ 7 ). With rotation, both adapters alternate between the two streams every step and co-improve. The challenger prompt instructs the model not to trust or paraphrase the opponent but to verify it and solve independently, winning by being correct when the opponent is wrong; the prompt is identical across the coop and adv reward stacks ( Section ˜ 4.4 ), so they differ only in the reward.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.4 Reward: correctness plus a conversion bonus\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let c ( y ) = r ( x , y ) ∈ { 0 , 1 } c(y)=r(x,y)\\\\in\\\\{0,1\\\\} be correctness. We report two reward stacks, applied to the challenger rollout b i b_{i} given its paired drafter solution a i a_{i} :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where ϕ ( ⋅ ) \\\\phi(\\\\cdot) is a format term (well-formed <think> / <answer> structure) with weight λ = 0.5 \\\\lambda=0.5 (25% of the correctness term), identical across stacks so it introduces no bias between them. In coop the opponent appears only in the context , not in the reward, which isolates the value of information exchange (the exchange claim). In adv the conversion bonus c ( b i ) ( 1 − c ( a i ) ) c(b_{i})\\\\,(1-c(a_{i})) raises the effective correctness weight from 2 2 to 3 3 exactly on the paired opponent attempts that failed : solving where the opponent could not is worth more, which is the competitive pressure.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The form matters. A naïve margin c ( b i ) − c ( a i ) c(b_{i})-c(a_{i}) works poorly: since c ( a i ) c(a_{i}) is action-independent (fixed by the pairing a i a_{i} , not by the challenger’s rollout b i b_{i} ), its subtractive term contributes no expected policy gradient (exactly in the unnormalized objective; up to the nonlinearity of group standardization after it): it acts as a per-sample baseline, shifting each reward by the drafter’s outcome without adding any competitive direction. The tested margin stack, R = 2 c ( b i ) + ( c ( b i ) − c ( a i ) ) = 3 c ( b i ) − c ( a i ) R=2c(b_{i})+\\\\big(c(b_{i})-c(a_{i})\\\\big)=3c(b_{i})-c(a_{i}) , makes this concrete: the scalar 3 3 cancels under group-standard-deviation normalization and the − c ( a i ) -c(a_{i}) term is action-independent (zero expected policy gradient), so in expectation the margin points in the same direction as the cooperative gradient. The ablation is consistent with this prediction ( 49 49 vs coop 46 46 ), though the 3 3 pp difference lies within binomial sampling noise. The conversion bonus instead modulates the correctness weight by opponent difficulty : because it multiplies the action-dependent c ( b i ) c(b_{i}) , the reweighting shifts the gradient itself, and, given the within-group spread of opponent contexts { a i } \\\\{a_{i}\\\\} , tilts it toward the opponent attempts that failed. We confirm this empirically ( Table ˜ 7 ): the margin form barely improves on cooperation, whereas the conversion bonus clearly does. More generally, any reward that is affine in c ( b i ) c(b_{i}) with group-constant coefficients (e.g., a shared opponent draft for the entire challenger group, making c ( a ) c(a) identical for all N N rollouts) normalizes to exactly the advantage of the conversion-free reward in the same contexts under group-relative standardization; the conversion term then contributes zero expected policy gradient. Within-group variation in opponent difficulty is what makes the competitive term trainable. In the shared-opponent ablation we therefore fix one opponent draft per problem, hidden from the challenger’s context and used only for the reward (so the contexts are plain prompts, as in vanilla GRPO); the observed result (32) is consistent with the theoretical prediction that a group-constant conversion bonus adds no directional signal. This departs from the standard GRPO assumption that all rollouts in a group share the same prompt; here each challenger rollout conditions on its own opponent context. The resulting credit assignment is not merely noisier but systematically confounded: a rollout’s success probability depends on the quality of the draft it happened to draw (an action-independent draw), so group standardization mixes context difficulty with policy quality, and the conversion bonus, which pays exactly in the harder contexts, partially offsets that difficulty penalty. This admits an alternative reading of the adversarial gain: rather than “competitive pressure” in a game-theoretic sense, the bonus acts as difficulty-weighted reward shaping that upweights correct completions in hard contexts. Our experiments do not distinguish these two interpretations; we use the competitive framing for its constructive role in the design and leave the disentanglement to future work.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The resulting reward ladder (adv) is R − λ ϕ ∈ { 0 , 2 , 3 } R-\\\\lambda\\\\phi\\\\in\\\\{0,2,3\\\\} . The drafter’s standalone stream has no opponent, so Equation ˜ 2 applies as-is.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As an optional density lever , studied separately in Section ˜ 5.6 and not part of the headline recipe, we add a length tiebreak that fires only when both solutions are correct: R ( b i ) + = γ c ( b i ) c ( a i ) 1 [ | b i | < | a i | ] R(b_{i})\\\\mathrel{+{=}}\\\\gamma\\\\,c(b_{i})\\\\,c(a_{i})\\\\,\\\\mathbf{1}\\\\!\\\\left[\\\\,|b_{i}|<|a_{i}|\\\\,\\\\right] with weight γ \\\\gamma , rewarding the challenger for matching the opponent’s correctness in fewer tokens. Because the term requires c ( b i ) = 1 c(b_{i})=1 , it never directly penalizes correctness, only breaking ties among already-correct traces, so accuracy stays essentially flat while length drops, converting the emergent shortening ( Section ˜ 5.5 ) into a directly tunable target.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.5 Compute parity\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Per step Agon generates N N drafter rollouts plus N N challenger rollouts ( 2 N 2N ), versus N N for vanilla GRPO. Both adapters are updated every step, the drafter on its standalone group and the challenger on its opponent-conditioned group; rotation swaps which adapter gets which stream. Our primary invariant is the generation budget : every method generates 2 N 2N rollouts per training problem (vanilla GRPO gets 2 N 2N instead of its default N N ) under the same completion-length cap and the same single pass over the training problems. Realized token counts then differ only through learned trace length, bounded for every method by the same cap; we do not audit total generated tokens (the drafter-stage length is not reported), but with the rollout count and length cap equalized, the gains are not attributable to a larger generation budget . The challenger additionally prefills the opponent summary, which is extra prefill compute we do not equalize (the summary is short relative to a full trace, is amortized cheaply against generation, and is absent at single-adapter inference). At inference Agon deploys exactly as it trains: a two-stage cascade in which one adapter drafts from the plain prompt and the other reads the draft’s summary and produces the final answer. This is two sequential generation passes, a doubled budget relative to a single-pass baseline, so Table ˜ 3 includes two-pass controls at the same budget (cross-refinement, self-refinement). Reported trace lengths refer to the final (challenger) stage. Both cascade directions ( A → B A{\\\\to}B and B → A B{\\\\to}A ) are evaluated and the better is reported, selected post hoc on the held-out set.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Experiments\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Setup.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We train Qwen3 models (Qwen Team, 2025 ) (primarily 0.6B; 1.7B/4B for scaling) on the hard split of DeepMath-103K (He et al., 2025 ) (difficulty 8, binary-answer questions removed), with a held-out set of 300 300 problems. We use group size G = N = 8 G=N=8 for both streams, LoRA rank 16 16 , learning rate 5 × 10 − 5 5\\\\times 10^{-5} (linear decay, 10 % 10\\\\% warmup), one training epoch over the 3 , 000 3{,}000 training problems, format weight λ = 0.5 \\\\lambda=0.5 , length tiebreak γ = 0.5 \\\\gamma=0.5 (used in Section ˜ 5.6 only), and a standard format reward (well-formed <think> / <answer> ) as ϕ \\\\phi . Generation uses vLLM (Kwon et al., 2023 ) and training uses TRL (von Werra et al., 2020 ) . The primary metric is pass@1 on the DeepMath held-out set; we additionally sanity-check on GSM8K (Cobbe et al., 2021 ) and MATH-500 (Hendrycks et al., 2021 ) ( Table ˜ 2 ). For the second-domain study ( Section ˜ 5.3 ) we train on the easy -difficulty subset of CodeContests (Li et al., 2022 ) with a unit-test verifier ( r = 1 r=1 iff all tests pass), under the same protocol and budget. Evaluation protocol: sampled decoding (temperature 0.6 0.6 , top- p p 0.95 0.95 ), one rollout per stage, generation budget matched to training ( 15 15 k completion tokens per stage), the answer parsed from the <answer> block; the 300 300 held-out problems are difficulty-8 only with binary-answer questions removed, split with a fixed seed before any training. Pair-trained methods are evaluated as the two-stage cascade ( Section ˜ 4.5 ) in both directions, reporting the better direction; mean trace length reports the final-stage completion. Reported pass@1 values include 95% Clopper–Pearson binomial confidence intervals over the 300-problem held-out set ( Table ˜ 4 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"The 2 × 2 2\\\\times 2 design.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We cross two binary factors, competition and information exchange , giving four cells plus three controls ( Table ˜ 1 ): self-refinement, an untrained MoA at inference, and a budget-matched GRPO two-pass self-cascade. One caveat is structural: the “competition without exchange” cell uses a shared opponent draft, hidden from the challenger and used only for the reward, which by the analysis in Section ˜ 4.4 makes the conversion bonus group-constant and hence gradient-free, so this cell is theoretically equivalent to vanilla GRPO and serves as a consistency check rather than a true isolation of competition. Consequently, the exchange claim (exchange helps) is carried by the GRPO-vs-cooperative comparison, and the competition claim (competition helps beyond exchange) by the cooperative-vs-adversarial comparison, holding everything else fixed at a matched generation budget; a competition-without-visible-exchange cell with per-rollout hidden drafts is left for future work.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.1 Main results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table ˜ 3 reports the main comparison on Qwen3-0.6B. All deltas below are read from the pass@1 column. The trained pair methods and the refinement controls are two-pass at inference; vanilla GRPO and zero-shot are single-pass (see the footnote of Table ˜ 3 ). Information exchange alone (cooperative exchange: 46 46 ) already improves over vanilla GRPO ( 30 30 ) and over self-refinement ( 32 32 ), supporting the exchange claim; adding competition (Agon: 61 61 ) improves further over cooperation, supporting the competition claim. The untrained MoA control is a two-agent Mixture-of-Agents layer at inference (Wang et al., 2024 ) : each model reads the other’s attempt and refines its own. It lifts GRPO by + 4 +4 pp ( 30 → 34 30\\\\!\\\\to\\\\!34 ); Agon lifts it by + 31 +31 pp, roughly eight times the gain of untrained aggregation. We note that the two levers contribute comparably: information exchange adds + 16 +16 pp (GRPO → \\\\to cooperative) and competition a further + 15 +15 pp (cooperative → \\\\to Agon). The dual-adapter architecture alone contributes little: the competitive-shared-opponent variant sits at 32 32 , barely above single-model GRPO at 30 30 , so the gain from exchange is attributable to the shared context, not to going from one model to two.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Each method is a single training run; the confidence intervals in Table ˜ 4 cover sampling variation over the 300 held-out problems ( ± ∼ 5.5 \\\\pm{\\\\sim}5.5 pp) but not run-to-run training variance. Both headline deltas (exchange: + 16 +16 pp; competition: + 15 +15 pp) exceed the width of these intervals. The Agon cascade reaches roughly twice vanilla GRPO’s held-out pass@1 while shortening final-stage traces; in gain terms, GRPO lifts the base by + 7 +7 pp ( 23 → 30 23\\\\!\\\\to\\\\!30 ) while Agon lifts it by + 31 +31 pp over GRPO ( 30 → 61 30\\\\!\\\\to\\\\!61 ), and + 26 +26 pp over a two-pass GRPO self-cascade at matched inference budget, nearly doubling the untrained MoA cascade.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To decompose the cascade gain, we evaluate the drafter adapter standalone after training (plain prompt, no opponent summary): it reaches 46 pass@1, a +16 pp lift over vanilla GRPO. This is notable given the budget asymmetry: each adapter drafts on only half the steps with group size N N , so it receives roughly a quarter of the standalone-rollout gradient volume of the vanilla GRPO baseline (which uses 2 N 2N on every problem), plus challenger-stream gradients on the alternating steps. Our working hypothesis is that the challenger-stream updates transfer to standalone solving (the adapter is trained to verify and re-derive, skills that a plain prompt also exercises), but we have not tested this with a control (e.g. the standalone accuracy of a cooperative-trained or self-refinement adapter), so the attribution of the drafter’s +16 pp to the competitive objective is a hypothesis, not an isolated measurement. The challenger then converts a net 15 of the drafter’s 54 failures into correct answers (with possible regressions among the drafter’s 46 successes), yielding the final 61. Under this reading, the drafter’s lift (+16 pp) and the challenger’s conversion (+15 pp) contribute comparable shares of the total gain, the former via the hypothesized transfer of challenger-stream skills to standalone solving.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To check that the gains are not an artifact of the training distribution, Table ˜ 2 reports the same checkpoints on GSM8K and MATH-500 without any further tuning: the ordering is preserved with smaller absolute margins than in-distribution ( + 13 +13 and + 19 +19 pp over zero-shot, vs + 38 +38 on DeepMath-hard), as expected for a method whose pressure targets the hard regime.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.2 Scaling and generality\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We expect smaller models, which live deeper in the hard regime, to benefit most. Table ˜ 5 shows the gain over vanilla GRPO shrinking but remaining positive as the base grows, and the same pattern holding on two other model families, Qwen3.5 (Qwen Team, 2026 ) and Gemma 4 (Gemma Team, 2026 ) : the delta tracks zero-shot strength, not the family. The Agon cascade on the 0.6B base (61) exceeds not only zero-shot Qwen3-4B (52) but vanilla GRPO on the 4B base (59), a model ∼ 7 × {\\\\sim}7\\\\times larger, on the same held-out set. In every case the pair is two adapters over one base (our instantiation, not a requirement of the method); pairing across families is future work.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.3 A second domain: competitive-programming code\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The recipe is not specific to math; any domain with a clean programmatic verifier qualifies. We repeat the core comparison on Qwen3-1.7B trained on the easy -difficulty subset of CodeContests (Li et al., 2022 ) , with r ( x , y ) = 1 r(x,y)=1 iff the generated program passes all unit tests, a held-out set of 300 300 problems, and the same protocol and generation budget. Table ˜ 6 shows the same ordering as Table ˜ 3 , exchange beats the baseline and competition beats cooperation, with traces again shortening. The absolute delta is smaller than on math: unit-test rewards are sparser at this scale, and code traces make the opponent prefill longer relative to generation. Across domains the claims should be read as directional rather than magnitude-matched.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.4 Ablations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table ˜ 7 isolates each design choice on Qwen3-0.6B (held-out pass@1). The competition claim is carried by the competition row (adv vs coop). The information exchange row (Agon vs the shared-opponent variant) should be read with care: switching to a shared opponent removes the visible draft and , by the analysis in Section ˜ 4.4 , deactivates the competitive gradient, so this comparison changes two things at once and is a consistency check rather than an isolated exchange effect; the isolated exchange delta is GRPO vs cooperative in Table ˜ 3 . In the “shared opponent” ablation the challenger group is scored against one fixed opponent draft, hidden from its context (group-constant c ( a ) c(a) ), so the conversion bonus normalizes away ( Section ˜ 4.4 ); the result ( 32 32 vs 30 30 ) is consistent with theory and shows that within-group variance in opponent difficulty is required for the competitive term to contribute gradient. A per-rollout opponent pairing (distinct hidden drafts) is the honest ablation of competition without visible exchange; that variant is left for future work. Role rotation is ablated explicitly: with fixed roles each adapter receives only one gradient stream, yielding 52 pass@1 versus 61 with rotation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.5 Analysis: why traces shorten\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Empirically, Agon’s traces are shorter than GRPO’s ( Figure ˜ 7 a): the final (challenger) stage averages 3.5 3.5 k tokens against 8.1 8.1 k for a single GRPO pass. Note that this is a per-stage comparison: the cascade also runs a drafting stage, whose length we do not report, so total generated tokens per problem are not directly compared here and a full token-cost accounting is left for further analysis. The like-for-like comparison is against the two-stage untrained MoA control, where the same (final) stage is measured on both sides: Agon’s is half the length ( 3.5 3.5 k vs 6.9 6.9 k). The headline reward contains no length term. The observed shortening is consistent with the challenger having a candidate solution in context, so length-heavy exploration is no longer the only route to a correct answer. Challenger completions conditioned on a correct opponent summary are ∼ 35 % {\\\\sim}35\\\\% shorter than the challenger-stage average ( 2.3 2.3 k vs 3.5 3.5 k tokens on the final checkpoint), and Figure ˜ 6 shows the typical pattern: the challenger pinpoints the flawed step, re-derives it, and stops.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.6 An optional density lever (auxiliary study)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The shortening above is emergent: nothing in the headline reward asks for it. The same game can, however, also grade brevity: instead of taxing tokens with a penalty, we make being correct in fewer tokens than the rival a way to win. As a small side study, separate from the main recipe and from the two headline claims, we ask whether the shortening can thus be driven directly, using the length-tiebreak term from Section ˜ 4.4 (a bonus γ \\\\gamma when the challenger is correct and shorter than an equally-correct opponent). On Qwen3-0.6B, turning it on cuts mean trace length from 3.5 3.5 k to 2.6 2.6 k at essentially unchanged accuracy ( 61 → 60 61\\\\!\\\\to\\\\!60 pass@1), so the emergent shortening becomes directly controllable ( Table ˜ 8 ). Because the term only breaks ties among already-correct traces, it compresses without measurably trading away accuracy. We report this as an auxiliary knob; all headline numbers elsewhere use the length-free reward.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6 Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Agon trains two models to reason over each other’s work, and the pair co-improves past where a single self-graded policy stalls. The competing peer supplies the signal outcome-only RL lacks: it reads the policy’s solution and grades it implicitly, with no process labels and no change to the verifier. Two distinct policies, trained head-to-head with rotating draft-and-challenge roles (instantiated cheaply as two adapters over one frozen base), let us test the idea. Under a matched generation budget, information exchange beats vanilla GRPO and self-refinement (the exchange claim), competition beats cooperation (the competition claim), and the resulting traces are shorter, a by-product of the competition rather than of an explicit penalty. At the same doubled inference budget, combining two copies of the base without training buys 4 4 points over GRPO, while training the pair to compete buys 31 31 ; the gain comes from the learned collaboration rather than from having two models.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Limitations.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Agon needs a verifier and reference problems, and we study math plus a smaller code study, both with clean programmatic rewards; noisier domains may behave differently. Gains depend on the pair being matched in strength but divergent in behavior (too large a gap collapses into distillation, too small a gap into self-play), and our divergence is maintained heuristically (init + + rotation) rather than guaranteed; complementarity is not quantified beyond the cascade win rate. Text exchange bounds the channel bandwidth. Inference is a two-stage cascade, i.e. two sequential generation passes and added latency; reported trace lengths cover the final stage only, and the better cascade direction is selected post hoc on the held-out set. Compute parity holds for generated tokens only: the challenger’s extra prefill of the opponent summary is not equalized ( Section ˜ 4.5 ). Groups with zero within-group variance (all-correct or all-wrong) yield undefined σ \\\\sigma in GRPO advantage; such groups are dropped. Every trained number in the paper is a single training run evaluated with a single sampled rollout per problem; the better cascade direction was selected post hoc on the held-out set. Run-to-run training variance is unquantified, and the reported deltas should be read with that in mind. The reported magnitudes come from a hard, decontaminated subset and should not be read as DeepMath-wide pass@1.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Future work.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This work establishes that the cross-model signal helps while exchanging it in text; the channel itself is the next axis. Exchanging hidden states directly, via KV-cache injection or gated latent bridges between the two adapters, removes the serialization bottleneck and lets models share information they cannot verbalize (Beliaev, 2025 ) . Beyond accuracy, our auxiliary density lever ( Section ˜ 5.6 ) is a first step; scaling it into a full multi-objective, compression traded explicitly against correctness, could make density a first-class, tunable target, and a diversity term over divergent solution strategies could make the pair complementary enough to ensemble.\"\n }\n]","isConversational":false}}},{"rowIdx":95,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.07674"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Max Out GRPO Signal: Adaptive Trace Prefix Control for Hard Reasoning Problems"},"authors":{"kind":"list like","value":["Vladislav Beliaev"],"string":"[\n \"Vladislav Beliaev\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.07674v1"},"published":{"kind":"timestamp","value":"2026-07-08T00:00:00","string":"2026-07-08T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Reinforcement learning from verifiable rewards is now the standard tool for improving the reasoning of large language models (LLMs) on tasks such as mathematics and code (Guo et al., 2025 ; Lambert et al., 2024 ) . Building on policy-gradient methods (Schulman et al., 2017 ; Ahmadian et al., 2024 ) and RLHF (Ouyang et al., 2022 ) , the dominant recipe is on-policy: sample a group of G G rollouts from the current policy, score each by an outcome reward, and update using group-relative advantages, as in GRPO (Shao et al., 2024 ) . This recipe has a structural blind spot. On a hard problem, one the model almost never solves, all G G rollouts fail, every reward is identical, the group-relative advantage is exactly zero, and the gradient contribution of that problem is zero (Yu et al., 2025 ; Zhang et al., 2025 ) . The model burns sampling compute on the problem and learns nothing from it. Worse, these are frequently the problems we most want to learn: the ones at the frontier of the model’s ability."},{"type":"para","text":"Common remedies work around this rather than fix it. Oversampling and dynamic filtering, as in DAPO (Yu et al., 2025 ) , discard degenerate all-correct/all-wrong groups but cannot manufacture a correct rollout where the model has none. Warm-starting RL with supervised fine-tuning on reference traces (or rejection-sampled ones, STaR/RAFT-style (Zelikman et al., 2022 ; Dong et al., 2023 ) ) injects competence but induces entropy collapse that hurts subsequent exploration (Chu et al., 2025 ; Cui et al., 2025 ) . A natural idea is instead to make hard problems temporarily easier so they yield a signal, then remove the assistance. If we prepend a correct prefix of a solution and ask the model to complete it, the conditional success rate rises (empirically near-monotonically) with prefix length: with no prefix the model solves almost nothing, with a long enough prefix it solves almost everything. Prefix length is therefore a continuous dial on difficulty , and for most reachable success rates k / G k/G there is a prefix length that achieves it. This observation underlies a line of concurrent work (Setlur et al., 2026 ; Qu et al., 2026 ; Huang et al., 2025 ) ."},{"type":"para","text":"The central question this paper addresses is what success rate to aim for, and how to hit it , i.e. how to max out the GRPO signal on hard problems. Concurrent methods fix a small set of prefix lengths per problem, chosen once from the base model so that the conditioned base accuracy is “reasonable” (Setlur et al., 2026 ) . We argue this leaves most of the available signal on the table for two reasons. First, the GRPO signal is not maximized merely by being non-zero; prior work establishes an intermediate optimal k / G k/G (near 50 % 50\\% ) for RL learning signal (Bae et al., 2025 ; Gao et al., 2025 ) (Section 4.1 ). Second, the prefix length that achieves a given k / G k/G is per-sample and non-stationary: as the policy improves during training, the same prefix yields a higher k / G k/G , drifting the problem out of the productive regime. A schedule fixed at the start cannot track this."},{"type":"para","text":"We introduce AdaPrefix-GRPO , which treats prefix length as a closed-loop controller . During training we measure the realized batch-mean k / G k/G and adjust a single global base prefix length toward a target k / G k/G using a root-finding update (secant or binary search); each sample is offset from the base by a static difficulty-dependent shift, and all prefixes are annealed to zero so that the deployed policy never sees a prefix at test time. The method requires only minimal changes to the RL trainer: prefixes are injected as an assistant message in the data pipeline, gradients are masked on the prefix, and the reward is computed on the full completion. We hold the total training compute (FLOPs) fixed across all methods, so improvements are not attributable to extra training compute; the one-time calibration is a per-model preprocessing step, amortized across all runs on the same model and data, and excluded from the training budgets."},{"type":"heading","level":4,"text":"Contributions."},{"type":"para","text":"We frame prefix length as a closed-loop controller of GRPO’s success rate k / G k/G , converting an uncontrolled observable into a tracked set-point (a global closed loop with per-sample difficulty offsets; Section 4 )."},{"type":"para","text":"We adopt the established intermediate ( ≈ 50 % \\approx\\!50\\% ) solve-rate set-point for RL signal (Bae et al., 2025 ; Gao et al., 2025 ) and make it reachable across the batch through prefix length (Section 4.1 )."},{"type":"para","text":"We propose AdaPrefix-GRPO: cold-start initialization, global closed-loop adaptation of a base prefix with per-sample difficulty offsets, and annealing to zero, implemented purely in data preparation on top of a stock GRPO trainer (Section 4.2 )."},{"type":"para","text":"At matched training FLOPs on hard math, AdaPrefix-GRPO improves pass@1 over vanilla GRPO and a fixed-prefix baseline without collapsing pass@16, and the gain grows as the model shrinks (Section 5 )."},{"type":"heading","level":1,"text":"2 Related Work"},{"type":"heading","level":4,"text":"The dead zone in group-relative RL."},{"type":"para","text":"GRPO (Shao et al., 2024 ) and its variants estimate advantages by centering rewards within a group of rollouts. When a problem is solved by none (or all) of the group, the centered advantages collapse to zero and the problem produces no gradient, the “learning cliff” (Zhang et al., 2025 ) that DAPO’s dynamic filtering works around (Yu et al., 2025 ) . Classical exploration remedies (entropy bonuses, looser clipping, pass@ k k objectives) do not resolve this and can destabilize optimization (Qu et al., 2026 ) ."},{"type":"heading","level":4,"text":"Prefix- and hint-guided RL."},{"type":"para","text":"Closest is PrefixRL (Setlur et al., 2026 ) : it conditions on-policy RL on prefixes of off-policy correct traces (rejection-sampled to “reuse” prior compute), masks gradients on the prefix, and reports back-generalization from prefixed to unprefixed problems, but it fixes a handful of prefix lengths per problem, chosen once from base accuracy. POPE (Qu et al., 2026 ) uses the minimal prefix that yields a non-zero reward; Prefix-RFT (Huang et al., 2025 ) anneals a single global prefix on a cosine schedule; Scaf-/Hint-GRPO (Zhang et al., 2025 ; Huang et al., 2025 ) give graded in-prompt hints; R3 (Xi et al., 2024 ) slides the RL start state backward along a demonstration, a reverse curriculum over prefix position. These treat assistance as a fixed or global heuristic. We additionally show it is closed-loop control, not prefixing per se, that carries the gain (Section 5.3 ). Off-policy methods such as LUFFY (Yan et al., 2025 ) and ReLIFT (Ma et al., 2025 ) use demonstrations as training targets; like PrefixRL we stay on-policy and use solutions only to seed exploration."},{"type":"heading","level":4,"text":"Difficulty curricula."},{"type":"para","text":"Ordering data by difficulty is classic (Bengio et al., 2009 ; Kumar et al., 2010 ) , and recent RL work curates toward an intermediate success rate (Bae et al., 2025 ; Gao et al., 2025 ) . Instead of selecting problems that already sit at the right difficulty, we move every problem toward the target and track it over time."},{"type":"heading","level":1,"text":"3 Preliminaries: GRPO and the Dead Zone"},{"type":"para","text":"For a problem x x with reference solution and a verifier r ( x , y ) ∈ { 0 , 1 } r(x,y)\\in\\{0,1\\} , GRPO samples a group of G G rollouts y 1 , … , y G ∼ π θ ( ⋅ ∣ x ) y_{1},\\dots,y_{G}\\sim\\pi_{\\theta}(\\cdot\\mid x) and forms the group-relative advantage"},{"type":"para","text":"We use this un-normalized centered advantage (no division by the group std). The choice matters for Section 4.1 : std-normalization does not move the set-point (the aggregate positive mass still peaks at k / G = 0.5 k/G{}\\!=\\!0.5 ), but it inflates the per-sample advantage of a rare correct rollout, which scales as ( 1 − p ) / p \\sqrt{(1-p)/p} and diverges as p → 0 p\\!\\to\\!0 . The un-normalized form keeps a well-behaved p ( 1 − p ) p(1-p) scaling without that instability. Un-normalized advantages are also competitive with the std-normalized variant in recent practice (Liu et al., 2025 ) , so this departure does not weaken the baseline. Let k = ∑ i r ( x , y i ) k=\\sum_{i}r(x,y_{i}) be the number of correct rollouts, so the empirical success rate is k / G k/G . If k = 0 k=0 (or k = G k=G ), all A i = 0 A_{i}=0 and the gradient contribution of x x vanishes. We call the set of problems with pass @ G ≈ 0 \\mathrm{pass}@G\\approx 0 the dead zone : at exactly zero success probability no amount of sampling extracts signal, and under finite sampling near-zero rates yield negligible signal. On our hard split this is the common case: vanilla GRPO solves only ∼ \\sim 35% of problems at pass@16, leaving a substantial dead zone on the training portion; controlled prefixing converts a substantial fraction of these into learnable problems."},{"type":"heading","level":4,"text":"Prefix as a difficulty dial."},{"type":"para","text":"Given a reference solution z = ( z 1 , … , z L ) z=(z_{1},\\dots,z_{L}) for x x , a prefix of ratio ρ ∈ [ 0 , 1 ] \\rho\\in[0,1] is z 1 : ⌈ ρ L ⌉ z_{1:\\lceil\\rho L\\rceil} , prepended to the prompt as a partial answer the model must continue. The conditional success rate κ ( ρ ) = 𝔼 [ r ( x , y ) ∣ y ∼ π ( ⋅ ∣ x , z 1 : ρ L ) ] \\kappa(\\rho)=\\mathbb{E}[r(x,y)\\mid y\\sim\\pi(\\cdot\\mid x,z_{1:\\rho L})] is empirically near-monotone in ρ \\rho ( Figure ˜ 2 a), with κ ( 0 ) \\kappa(0) the unconditioned rate and κ ( 1 ) → 1 \\kappa(1)\\!\\to\\!1 ; longer, partly off-distribution prefixes occasionally fail to help, a violation rate we report (Section 5.4 ). We do not require strict monotonicity: the bracketing (binary-search) update only needs κ \\kappa to cross τ \\tau once inside [ ρ − , ρ + ] [\\rho^{-},\\rho^{+}] and tolerates local non-monotonicity, and it is the fallback whenever the secant step is ill-conditioned. For any reachable target τ ∈ ( κ ( 0 ) , 1 ) \\tau\\in(\\kappa(0),1) there is a ρ ⋆ \\rho^{\\star} with κ ( ρ ⋆ ) = τ \\kappa(\\rho^{\\star})=\\tau ; for easy problems κ ( 0 ) > τ \\kappa(0)\\!>\\!\\tau already, so the controller simply sets ρ = 0 \\rho=0 . Gradients are masked on the prefix tokens; the reward is computed on the full (prefix + + generated) sequence; evaluation always uses ρ = 0 \\rho=0 ."},{"type":"heading","level":1,"text":"4 Method: AdaPrefix-GRPO"},{"type":"heading","level":2,"text":"4.1 The intermediate set-point"},{"type":"para","text":"Escaping the dead zone only requires k ≥ 1 k\\geq 1 , but the magnitude of the learning signal depends on where in ( 0 , 1 ) (0,1) the group success rate p = k / G p=k/G{} sits. Work on RL data selection finds, empirically and through analysis of the optimization dynamics, that an intermediate solve rate, commonly near 50 % 50\\% , maximizes useful signal per update, while groups that are almost always or almost never solved contribute little (Bae et al., 2025 ; Gao et al., 2025 ) ; classical RL curricula targeted the same intermediate-success band long before LLMs (Florensa et al., 2018 ) ."},{"type":"para","text":"Two intuitions match this empirical optimum, both stated for the un-normalized advantages we use (Section 3 ). First, the aggregate positive gradient in a group scales with the reward spread p ( 1 − p ) p(1-p) , which peaks at p = 0.5 p=0.5 . Second, the number of contrastive (correct, incorrect) pairs the group baseline can separate is k ( G − k ) k(G-k) , maximal at k = G / 2 k=G/2 and zero at p ∈ { 0 , 1 } p\\!\\in\\!\\{0,1\\} , recovering the dead zone as the degenerate case. Neither argument is a proof. The set-point itself is robust to advantage normalization (Section 3 ), so the choice of un-normalized advantages does not affect where the optimum sits. The empirical target sweep in Section 5.3 is the actual evidence; reward noise and many valid solution paths can shift the optimum somewhat (Gao et al., 2025 ) , so we treat τ = 0.5 \\tau=0.5 as a default. Our contribution is the mechanism that reaches this set-point: prefix length lets us hold the batch near τ \\tau throughout training."},{"type":"heading","level":2,"text":"4.2 Algorithm"},{"type":"para","text":"The set-point fixes where we want each group to sit ( k / G ≈ τ k/G{}\\!\\approx\\!\\tau ); the algorithm is the controller that gets it there and keeps it there as the policy moves. AdaPrefix-GRPO has three stages (Algorithm 1 ): a cold start that places the population roughly in range, a global closed loop with per-sample difficulty offsets that tracks τ \\tau throughout training, and an anneal to ρ = 0 \\rho=0 so deployment needs no prefix."},{"type":"heading","level":4,"text":"Cold start."},{"type":"para","text":"Before the first step we set the initial base prefix ratio ρ b ( 0 ) \\rho_{b}^{(0)} , shared by the whole population. We support two estimators: a conservative constant (e.g. 0.8 0.8 , which over-prefixes but makes k ≥ 1 k\\!\\geq\\!1 overwhelmingly likely from step one), and a one-shot calibration sweep : 4 rollouts per problem on a random 512-problem subset, over the grid ρ ∈ { 0 , 0.2 , 0.4 , 0.6 , 0.8 } \\rho\\in\\{0,0.2,0.4,0.6,0.8\\} , give the population-mean κ ( ρ ) \\kappa(\\rho) curve, which we invert at τ \\tau to set ρ b ( 0 ) \\rho_{b}^{(0)} . Independently of which estimator sets the base, a difficulty probe always runs: four rollouts per problem at ρ b ( 0 ) \\rho_{b}^{(0)} give every sample a difficulty estimate d x d_{x} (its measured pass rate) that the offsets below consume; the conservative variant thus skips only the grid sweep, not the probe. Sweep and probe are one-time preprocessing, amortized across all runs on the same model and data (seeds, hyperparameter sweeps, ablations), and excluded from the training budgets. The cold start only needs to land inside ( κ ( 0 ) , 1 ) (\\kappa(0),1) ; the closed loop removes the residual error. The sweep costs a single pre-training inference pass and lands near the optimum, so our main runs use it; the hyperparameter-free conservative constant trails it by only ∼ \\sim 1.8 pp (Table 3 ) and is a reasonable fallback when even that overhead is unwanted."},{"type":"heading","level":4,"text":"Closed-loop adaptation with per-sample offsets."},{"type":"para","text":"The controller maintains a single base prefix ratio ρ b \\rho_{b} , shared by the whole population. Every N N optimizer steps we read the realized k / G k/G{} , pooled over all rollouts in the window and smoothed with an EMA ( β = 0.7 \\beta{=}0.7 ), and move ρ b \\rho_{b} toward τ \\tau . Because κ ( ρ ) \\kappa(\\rho) is monotone, this is a one-dimensional root-find. Binary search keeps bracketing bounds [ ρ − , ρ + ] [\\rho^{-},\\rho^{+}] and halves toward τ \\tau (robust, one bit per update); secant interpolates from the two most recent ( ρ b , k / G ) (\\rho_{b},k/G{}) pairs,"},{"type":"para","text":"which converges in fewer updates when κ \\kappa is locally smooth. We guard the denominator (falling back to a bisection step when k / G ≈ t k / G t − 1 k/G{}_{t}\\!\\approx\\!k/G{}_{t-1} ; in our runs this fires on ∼ \\sim 17% of updates, mostly early when k / G k/G estimates are noisiest), clip the per-update step in ρ b \\rho_{b} to 0.05 0.05 , and keep ρ ∈ [ 0 , ρ max ] \\rho\\in[0,\\rho_{\\max}] . Pooling over the window makes the loop feasible: a single sample yields one heavily quantized Bernoulli estimate ( k / G ∈ { 0 , 1 G , … , 1 } k/G{}\\in\\{0,\\tfrac{1}{G},\\dots,1\\} ) per visit, and visits are ∼ \\sim 100 steps apart, far too sparse to track a moving target, whereas the window aggregates thousands of rollouts."},{"type":"para","text":"Holding the population mean at τ \\tau is not enough: the same mean can hide a wide spread, with half the samples saturated ( k = G k\\!=\\!G ) and half dead ( k = 0 k\\!=\\!0 ), so most rollouts still carry no signal. To compress the spread, each sample is offset from the base by a static difficulty shift, ρ x = clip ( ρ b + δ ( d x ) , 0 , ρ max ) \\rho_{x}=\\mathrm{clip}\\big(\\rho_{b}+\\delta(d_{x}),\\,0,\\,\\rho_{\\max}\\big) , where d x d_{x} is the sample’s cold-start difficulty and δ \\delta is a small monotone (linear) map with a hand-tuned slope, spanning ± 0.15 \\pm 0.15 around the base across the difficulty range in our runs: harder samples get slightly longer prefixes than the base, easier ones shorter. The offset is open-loop, its constants fixed across training; all tracking of the policy’s improvement happens through the single global loop. On easy problems the negative offset saturates at the clip, ρ x = 0 \\rho_{x}\\!=\\!0 , so prefixing is reduced on easy problems (we verify no regression on the GSM8K slice included in every method’s training mixture; Section 5 )."},{"type":"heading","level":4,"text":"Why closed-loop, and why offsets."},{"type":"para","text":"A single global schedule (Prefix-RFT) or a fixed per-problem length (PrefixRL) can only be right on average at one moment: problems differ widely in κ ( ⋅ ) \\kappa(\\cdot) , and as the policy improves every κ \\kappa curve shifts left, so any static or open-loop choice drifts off τ \\tau , typically toward k = G k\\!=\\!G , back into a zero-variance regime ( Figure ˜ 2 b). The global loop absorbs this non-stationarity across time. The per-sample offset addresses the orthogonal problem, heterogeneity across problems: mean-only control can leave the batch bimodal, many samples saturated or dead around a healthy average, so most rollouts are wasted; the offset narrows the spread of k / G k/G around the tracked mean, keeping the whole batch, not just its mean, near the signal-maximizing set-point ( Figure ˜ 5 ). The granularity ablation (Section 5.3 ) isolates how much of the gain the offset adds on top of base-only control. Two caveats. We use “controller” loosely: the update is a smoothed, quantized root-finder with no formal stability guarantee, and the annealing envelope (below) overrides the loop in the final phase. The offset is a static heuristic: it corrects difficulty ordering across problems but cannot adapt to a sample whose difficulty was mis-estimated at cold start."},{"type":"heading","level":4,"text":"Annealing."},{"type":"para","text":"To remove the train/test mismatch we impose a shrinking upper envelope and run the controller inside it: ρ x ← min ( ρ ¯ t , ρ b + δ ( d x ) ) \\rho_{x}\\leftarrow\\min\\!\\big(\\bar{\\rho}_{t},\\ \\rho_{b}+\\delta(d_{x})\\big) . The envelope is held near ρ max \\rho_{\\max} for the first fraction w w of training and then decayed to zero, ρ ¯ t = ρ max ⋅ clip ( T − t ( 1 − w ) T , 0 , 1 ) \\bar{\\rho}_{t}=\\rho_{\\max}\\cdot\\mathrm{clip}\\!\\big(\\tfrac{T-t}{(1-w)T},0,1\\big) (we use w ≈ 0.8 w\\!\\approx\\!0.8 ). Early on the envelope is slack and the loop is free to hold τ \\tau ; late in training the envelope binds and forces every ρ x → 0 \\rho_{x}\\!\\to\\!0 , trading some signal for on-distribution (no-prefix) practice. The final policy is thus trained to solve unaided, and all evaluation is at ρ = 0 \\rho=0 . As the envelope binds, the controller can no longer hold τ \\tau and the measured k / G k/G dips modestly below it ( Figure ˜ 2 b); but the no-prefix held-out curve shows no sustained regression ( Figure ˜ 1 ), so this phase consolidates what was learned. To test this, we report the final ρ = 0 \\rho=0 pass rate on the full originally-dead set (Section 5.4 ); a non-trivial unaided pass rate on problems that started at κ ( 0 ) ≈ 0 \\kappa(0)\\!\\approx\\!0 indicates that the gains are driven by back-generalization, not by prefix-conditioned imitation."},{"type":"heading","level":4,"text":"Prefix source."},{"type":"para","text":"Prefixes can be cut from reference solutions (always available) or from model-generated correct traces (rejection-sampled, on-distribution, as in PrefixRL). A reference path may be off-distribution and harder to continue, so the source matters; we default to reference solutions for availability and show in Section 5.3 that the controller is orthogonal to it: swapping in off-policy prefixes retains most of the gain."},{"type":"heading","level":4,"text":"Implementation."},{"type":"para","text":"AdaPrefix-GRPO touches the trainer only through the prefix loss mask; everything else lives in data preparation. Prefixes are appended as an assistant message that the model is asked to continue; the trainer masks loss on prefix tokens and scores the full sequence. Prefix cuts snap to sentence boundaries to avoid mid-token artifacts."},{"type":"heading","level":4,"text":"Compute accounting."},{"type":"para","text":"We compare methods at matched training compute , not matched steps or episodes. Following the accounting of Setlur et al. ( 2026 ); Snell et al. ( 2024 ) , a run’s cost is FLOPs = 2 N D samp + 6 N D upd \\mathrm{FLOPs}=2ND_{\\mathrm{samp}}+6ND_{\\mathrm{upd}} , where N N is the number of model parameters (we train with LoRA, but sampling and backpropagation traverse the full model, so we use the full parameter count), D samp D_{\\mathrm{samp}} counts all tokens processed at rollout time (including the prefill of supplied prefixes , so prefixing is not free), and D upd D_{\\mathrm{upd}} counts tokens in gradient updates. Every method runs to the same cumulative budget; AdaPrefix-GRPO’s one-time calibration sweep and difficulty probe are treated as amortized per-model preprocessing and excluded from the training budget. Because a prefixed rollout decodes only the continuation, AdaPrefix-GRPO completes more, shorter episodes within the budget, roughly two passes over the data where vanilla GRPO completes one; Figure ˜ 3 separates this revisit effect from the controller itself by reporting both compute-matched and iteration-matched views, and by including a vanilla variant that spends the same budget on more samples per problem. Prefix prefill stays under ∼ \\sim 15% of total training tokens because prefixes are capped and annealed. Table 1 reports generated length at evaluation ( ρ = 0 \\rho=0 ), where no prefix is present, so the trace-length saving reflects the deployed policy rather than the mechanical effect of starting mid-solution."},{"type":"heading","level":1,"text":"5 Experiments"},{"type":"heading","level":4,"text":"Setup."},{"type":"para","text":"We train on three Qwen3 sizes (0.6B, 1.7B, 4B) (Yang et al., 2025 ) and additionally validate cross-family on Qwen3.5-2B (Qwen Team, 2026 ) and Gemma 4 E4B (Gemma Team, 2026 ) , to confirm the controller is not Qwen-specific. Training data is the DeepMath-103K (He et al., 2025 ) hard split (difficulty 8, binary-answer questions removed): ∼ \\sim 6.8K problems with verified reference solutions (R1-generated traces shipped with the dataset). All methods train on the identical mixture: this hard split plus a 5% slice of GSM8K training problems (included so that self-disabling of prefixing on easy data can be verified; the GSM8K test split is disjoint). We use G = 8 G=8 rollouts, a batch of 64 prompts per optimizer step, and LoRA (Hu et al., 2022 ) . All methods run to the same training-FLOPs budget ( ≈ \\approx 6 × \\times 10 18 for the 1.7B model) under the accounting of Section 4 . For vanilla GRPO this budget corresponds to a single pass over the mixture ( ∼ \\sim 112 optimizer steps, 8 full-length rollouts per problem); because prefixed rollouts decode only the continuation, AdaPrefix-GRPO completes ∼ \\sim 2.1 passes ( ∼ \\sim 240 steps) of shorter episodes within the same budget. The controller updates every N = 10 N{=}10 steps. The main comparison rows (vanilla GRPO, PrefixRL, AdaPrefix-GRPO) are averaged over 3 seeds (per-seed spread ≤ \\leq 1.2 pp on DeepMath-hold); the remaining baselines and all ablations use a single seed. Figure ˜ 3 additionally reports the iteration-matched view, in which baselines run the same number of optimizer steps and are therefore granted more FLOPs. Training is built on TRL (von Werra et al., 2020 ) with vLLM (Kwon et al., 2023 ) generation. Evaluation spans GSM8K (Cobbe et al., 2021 ) as an easy-task sanity check, MATH-500 (Hendrycks et al., 2021 ) , AIME 2024/2025 (averaged), AMC 2023 (40 problems), and a DeepMath held-out split (1,000 problems held out from the same difficulty-8 hard split, disjoint from training). We report pass@1 and pass@16 estimated from 8 and 16 samples per problem respectively (avg@8 for the small AIME/AMC sets) with 95% CIs from a problem-level bootstrap (resampling problems together with their 8 samples, which accounts for per-problem clustering), and mean trace length. Evaluation uses temperature 1.0 with a strict verifier and no system prompt or thinking mode ; official Qwen3 reports use thinking mode and larger sampling budgets, so absolute zero-shot numbers are lower here. We additionally report held-out pass@16 to verify that gains do not come from collapsing the policy’s exploration onto a few modes."},{"type":"heading","level":4,"text":"Baselines."},{"type":"para","text":"Zero-shot : base model, no RL."},{"type":"para","text":"Vanilla GRPO : standard on-policy GRPO, no prefix."},{"type":"para","text":"Vanilla GRPO ( G = 16 G{=}16 ) : doubles the group size within the same FLOPs budget, which halves the number of optimizer steps; controls for the possibility that simply re-allocating compute toward more attempts per problem explains the gain."},{"type":"para","text":"SFT-warmup → \\to GRPO : mid-train on reference solutions, then GRPO (the “why not just SFT” control). The SFT phase is charged to the method’s FLOPs budget ( ∼ \\sim 8%, one epoch over the reference traces), shortening its RL phase accordingly."},{"type":"para","text":"Difficulty-filter GRPO : drop dead-zone problems, GRPO on the rest."},{"type":"para","text":"PrefixRL (Setlur et al., 2026 ) : a fixed set of prefix lengths per problem chosen once from base accuracy, static across training; the primary baseline. To keep the comparison fair, it uses the same reference solutions and prefix source as AdaPrefix-GRPO, the same FLOPs budget, and its per-problem length set is tuned with a matched length sweep; the only difference is that lengths are fixed rather than tracked. We additionally report a PrefixRL + anneal variant to separate the effect of annealing from the effect of closing the loop."},{"type":"para","text":"Prefix-RFT (Huang et al., 2025 ) : uniform decay, a single global prefix ratio annealed 0.8 → 0 0.8\\to 0 ."},{"type":"para","text":"AdaPrefix-GRPO (ours) : global secant control of the base prefix toward τ = 0.5 \\tau=0.5 , with per-sample difficulty offsets, annealed to 0 ."},{"type":"heading","level":2,"text":"5.1 Main results"},{"type":"para","text":"Table ˜ 1 reports the FLOPs-matched comparison on Qwen3-1.7B; Figure ˜ 3 shows the full compute curves and the iteration-matched control. AdaPrefix-GRPO improves pass@1 across benchmarks. The extra passes over the data that AdaPrefix-GRPO affords within the budget do not by themselves explain the gain: giving vanilla GRPO the same budget as a larger group ( G = 16 G{=}16 ) recovers only a small fraction of it. In absolute terms it adds +6.4 pp over GRPO on MATH-500 ( +17.0 pp on the DeepMath held-out split at budget end) versus +1.7/+6.1 pp for the PrefixRL baseline, i.e. 3–4 × \\times its gain on both. Head-to-head, AdaPrefix-GRPO beats the PrefixRL baseline on every benchmark (+4.7 MATH-500, +3.5 AIME, +3.4 AMC, +12.0 DeepMath-hold); under a paired problem-level bootstrap (both methods are evaluated on the same problems, so we resample per-problem score differences) the DeepMath gap reaches significance while the remaining three are positive and directionally consistent; all this at half the trace length (4.4k vs. 8.0k) and with no GSM8K regression; the gain over fixed prefixing thus does not hinge on the held-out split. Because every number here is measured at ρ = 0 \\rho=0 (no prefix), the ∼ \\sim 2 × \\times shorter outputs come from the deployed policy itself, not from counting only the completion of a prefixed problem. Among the baselines, Prefix-RFT (uniform decay) is the strongest simple alternative, edging out fixed prefixing on MATH-500 and DeepMath with shorter traces, which confirms that annealing helps; but without the closed loop it still trails AdaPrefix-GRPO by +11.4 pp on DeepMath-hold (significant under the paired bootstrap) and by +2.7 on MATH-500. The gain also does not come from narrowing the policy: pass@16 on the DeepMath held-out split improves alongside pass@1 (base 32.8 / GRPO 35.1 / ours 56.3%), so the distribution sharpens without collapsing. On easy GSM8K all methods are near ceiling and AdaPrefix-GRPO is best or tied (92.4%), confirming the controller does no harm where prefixing is unneeded, whereas fixed prefixing slips slightly (90.4%) by over-prefixing easy problems. AIME columns average the 2024 and 2025 sets. Beyond the benchmark numbers, the solvable set itself expands: ∼ \\sim 17% of originally-dead problems (zero pass@16 under the base model at ρ = 0 \\rho=0 before training) are solved at pass@1 after annealing."},{"type":"heading","level":2,"text":"5.2 Scaling and cross-family generalization"},{"type":"para","text":"We expect smaller models to live deeper in the dead zone and therefore benefit more. Each model size runs both methods to the same per-size FLOPs budget, set by the same rule as in the main setup, one vanilla-GRPO pass over the mixture at that size (so budgets scale with N N and trace length); within every row of Table ˜ 2 the two columns are therefore directly comparable, while absolute budgets differ across rows. Table ˜ 2 shows the pass@1 gain over vanilla GRPO on the DeepMath held-out split shrinking but staying positive as the Qwen3 base grows, and the same controller transferring to a different architecture family (Qwen3.5-2B, Gemma 4), ruling out a Qwen-specific effect. Notably, the 0.6B model trained with AdaPrefix-GRPO surpasses the 3 × 3\\times larger Qwen3-1.7B trained with vanilla GRPO (41.8 vs. 30.1) despite its smaller absolute budget: at this scale the training recipe buys more than the next model size."},{"type":"heading","level":2,"text":"5.3 Ablations"},{"type":"para","text":"Table ˜ 3 isolates each design choice on Qwen3-1.7B. The target sweep is consistent with the intermediate-set-point prediction (Section 4.1 ): extreme targets clearly hurt ( τ = 0.125 \\tau{=}0.125 : 41.9), while 0.375–0.625 sit within CI of one another; we take τ = 0.5 \\tau=0.5 as the default from prior work and do not claim a sharp empirical optimum. Control granularity shows the gain comes from closed-loop control, not prefixing per se (PrefixRL and global-decay trail by a wide margin; the bulk of the improvement is from closing the loop). Adding our annealing envelope to the PrefixRL baseline (the PrefixRL + anneal row) recovers only a small part of that gap, confirming annealing alone is not the driver. The per-sample difficulty offset adds a smaller increment over base-only control (+2.9 pp, at the edge of the single-run CI), consistent with the view that compressing the across-problem spread of k / G k/G matters beyond centering its mean. The advantage-norm. row addresses a confound: running AdaPrefix-GRPO with std-normalized advantages (which leave the τ \\tau set-point unchanged but over-weight rare successes; Section 3 ) still helps but by less, indicating the gain is not purely a global gradient-norm effect. The annealing row shows the delayed envelope modestly beats no annealing (48.2 vs 44.8), i.e. forced consolidation helps, but the method works even with the envelope removed entirely: back-generalization carries most of the gain, and Section 5.4 discusses why the train/test distribution shift one might expect fails to materialize. The controller is also robust to the prefix source : swapping reference for rejection-sampled off-policy prefixes (PrefixRL’s regime) retains most of the gain. Finally, the group size G G trades off estimator quality against update count: at a fixed FLOPs budget, doubling G G halves the number of optimizer steps, so the G = 16 G\\!=\\!16 row buys a sharper k / G k/G estimate for the controller with half the updates. Accuracy improves from G = 4 G\\!=\\!4 to G = 8 G\\!=\\!8 and is essentially flat at G = 16 G\\!=\\!16 (48.7 vs 48.2, within CI), i.e. the better estimate roughly pays for the lost updates but no more, making G = 8 G\\!=\\!8 the sensible default (and confirming the controller does not require an impractically large group)."},{"type":"heading","level":2,"text":"5.4 Analysis"},{"type":"heading","level":4,"text":"The controller tracks the set-point."},{"type":"para","text":"The closed loop holds the batch-mean k / G k/G at τ = 0.5 \\tau\\!=\\!0.5 within a few update intervals and keeps it there as the policy improves ( Figure ˜ 2 b) and keeps the whole distribution of per-sample k / G k/G concentrated near τ \\tau , whereas fixed and global prefixes drift toward saturation, piling mass at k / G = 1 k/G{}\\!=\\!1 while a hard core stays at 0 ( Figure ˜ 5 ). Tracking difficulty cohorts, their ρ x \\rho_{x} and realized k / G k/G are driven toward the tracked mean by the global controller plus static offsets and anneal cleanly to ρ = 0 \\rho=0 ( Figure ˜ 4 ). This is the mechanism that DAPO approximates with dynamic sampling, oversampling to discard all-correct/all-wrong groups (Yu et al., 2025 ) , except that prefix control creates signal on problems no amount of resampling can rescue."},{"type":"heading","level":4,"text":"Why prefixing transfers to no prefix."},{"type":"para","text":"Training on a partial solution still requires the model to generate the remaining reasoning and be graded on the full trace, so it practices the same sub-skills it must later chain unaided; as prefixes anneal away, those skills are exercised from progressively earlier states. This matches the back-generalization PrefixRL reports (Setlur et al., 2026 ) : held-out (no-prefix) pass@1 rises throughout training ( Figure ˜ 1 ), not only once prefixes vanish. Three mechanisms plausibly explain why the train/test distribution shift one might expect does not materialize. First, the loss is masked on prefix tokens: the model never imitates the (possibly off-distribution) reference text and is only reinforced on its own successful continuations, so every token that receives gradient is on-policy and no imitation drift accumulates (conditioning still shifts the state distribution; the final ρ = 0 \\rho=0 pass rates below are the direct check that this shift does not persist). Second, a prefix changes only the start state of the episode, not the policy being learned: as in reverse-curriculum RL (Xi et al., 2024 ) , the sub-skills reinforced on the remainder (algebraic manipulation, case analysis, self-verification) are the same ones a from-scratch solution must chain, and they transfer across starting positions. Third, the controller itself performs a soft anneal: as the policy improves, every κ \\kappa curve shifts left and the loop lowers ρ x \\rho_{x} to keep k / G k/G at τ \\tau ( Figure ˜ 4 b), so the distribution of training start states drifts toward unprefixed problems on its own ; the envelope merely forces the final step to exactly ρ = 0 \\rho=0 . As the direct test of consolidation, the final ρ = 0 \\rho=0 policy solves ∼ \\sim 17% of the full originally-dead set at pass@1 ( ∼ \\sim 22% at pass@16), evidence that the gains persist without prefixes."},{"type":"heading","level":4,"text":"Monotonicity in practice."},{"type":"para","text":"Across problems the response κ ( ρ ) \\kappa(\\rho) violates monotonicity on ∼ \\sim 6% of measured ( ρ , κ ) (\\rho,\\kappa) pairs (mostly long, off-distribution reference prefixes), and on those the controller falls back to bracketing without diverging; the mid-training snapshot in Figure ˜ 5 shows the resulting concentration of per-sample k / G k/G near τ \\tau on raw (un-smoothed) quantized rates."},{"type":"heading","level":4,"text":"Why traces shorten."},{"type":"para","text":"Because each sample is held near a moderate k / G k/G , positive advantages reward efficient correct completions rather than long, lucky explorations; combined with annealing, the policy converges to shorter unaided ( ρ = 0 \\rho=0 ) solutions than vanilla GRPO, which only finds reward through length-heavy exploration. (We treat this as a hypothesis to verify with length-conditioned analyses at ρ = 0 \\rho=0 .)"},{"type":"heading","level":1,"text":"6 Conclusion"},{"type":"para","text":"GRPO discards its hardest problems because zero within-group reward variance yields zero gradient. Prefix length is a continuous, per-sample dial that can place any problem at almost any reachable success rate. Conceptually this removes GRPO’s sampling barrier, the dead zone in which hard problems yield no signal under a finite rollout budget, and shifts the binding constraint to the model’s own capacity : with a controllable signal, whether a problem is learnable turns on whether the model can complete it given enough scaffolding, not on whether it can already solve it unaided. Prefixing cannot exceed that capacity ceiling or rescue a reference path the model is unable to follow, but within it, difficulty is no longer a hard cutoff on trainability. The contribution of AdaPrefix-GRPO is to control that dial in closed loop toward the intermediate, signal-maximizing target k / G k/G (Bae et al., 2025 ; Gao et al., 2025 ) and anneal it away, rather than fixing it in advance. At matched training compute this extracts signal where GRPO got none, beats a fixed-prefix baseline, and yields shorter solutions, with the largest gains on the smallest models."},{"type":"heading","level":4,"text":"Limitations."},{"type":"para","text":"AdaPrefix-GRPO needs reference (or off-policy) solutions and a reliable, near-binary verifier; we evaluate only on math reasoning and do not test noisy or non-verifiable domains, where behavior may differ. We adopt the intermediate ( ≈ 0.5 \\approx\\!0.5 ) set-point from prior work (Bae et al., 2025 ; Gao et al., 2025 ) and locate it empirically; we do not derive the exact per-sample optimum under reward noise. A subtler caveat: a 50 % 50\\% success rate on the short remainder of a heavily prefixed problem exercises credit assignment only over that remainder, which may be an easier learning signal than 50 % 50\\% on a from-scratch trajectory; annealing is what forces the harder, full-length signal back in, and disentangling the two is future work. The offset map δ \\delta is a hand-tuned static heuristic: a sample whose cold-start difficulty is mis-estimated keeps a biased prefix that the global loop cannot correct individually, and upgrading the offset to a genuine per-sample second loop would require denser per-sample k / G k/G estimates than G = 8 G{=}8 with sparse revisits provides. The cold-start calibration and difficulty probe, while one-time and amortized across runs, are not free; cheap or training-free per-sample difficulty estimation is an interesting direction for future work. The prefix source also matters (reference paths can be off-distribution), though our ablation suggests the controller is largely orthogonal to it. Small benchmarks (AIME, AMC) carry wide intervals, so we rely on several benchmarks and avg@8 sampling rather than any single number. Underlying per-step data and training code are not released."},{"type":"para","text":"Quantities quoted in the main text are measured as follows. Monotonicity violation rate ( ∼ \\sim 6%, Section 5.4 ): the fraction of adjacent ( ρ , κ ( ρ ) ) (\\rho,\\kappa(\\rho)) pairs with Δ κ < 0 \\Delta\\kappa<0 , pooled over the calibration sweep (512 problems × \\times 4 adjacent grid intervals = = 2,048 pairs) and all subsequent controller reads. Secant fallback rate ( ∼ \\sim 17%, Section 4.2 ): 4 of the 24 controller updates ( ∼ \\sim 240 steps, N = 10 N{=}10 ) triggered the bisection fallback, 3 of them in the first third of training. Prefill overhead ( < < 15%, Section 4 ): prefix tokens account for 14.6% of all training-time tokens (prefill + + generated), measured over the full AdaPrefix-GRPO run."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Reinforcement learning from verifiable rewards is now the standard tool for improving the reasoning of large language models (LLMs) on tasks such as mathematics and code (Guo et al., 2025 ; Lambert et al., 2024 ) . Building on policy-gradient methods (Schulman et al., 2017 ; Ahmadian et al., 2024 ) and RLHF (Ouyang et al., 2022 ) , the dominant recipe is on-policy: sample a group of G G rollouts from the current policy, score each by an outcome reward, and update using group-relative advantages, as in GRPO (Shao et al., 2024 ) . This recipe has a structural blind spot. On a hard problem, one the model almost never solves, all G G rollouts fail, every reward is identical, the group-relative advantage is exactly zero, and the gradient contribution of that problem is zero (Yu et al., 2025 ; Zhang et al., 2025 ) . The model burns sampling compute on the problem and learns nothing from it. Worse, these are frequently the problems we most want to learn: the ones at the frontier of the model’s ability.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Common remedies work around this rather than fix it. Oversampling and dynamic filtering, as in DAPO (Yu et al., 2025 ) , discard degenerate all-correct/all-wrong groups but cannot manufacture a correct rollout where the model has none. Warm-starting RL with supervised fine-tuning on reference traces (or rejection-sampled ones, STaR/RAFT-style (Zelikman et al., 2022 ; Dong et al., 2023 ) ) injects competence but induces entropy collapse that hurts subsequent exploration (Chu et al., 2025 ; Cui et al., 2025 ) . A natural idea is instead to make hard problems temporarily easier so they yield a signal, then remove the assistance. If we prepend a correct prefix of a solution and ask the model to complete it, the conditional success rate rises (empirically near-monotonically) with prefix length: with no prefix the model solves almost nothing, with a long enough prefix it solves almost everything. Prefix length is therefore a continuous dial on difficulty , and for most reachable success rates k / G k/G there is a prefix length that achieves it. This observation underlies a line of concurrent work (Setlur et al., 2026 ; Qu et al., 2026 ; Huang et al., 2025 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The central question this paper addresses is what success rate to aim for, and how to hit it , i.e. how to max out the GRPO signal on hard problems. Concurrent methods fix a small set of prefix lengths per problem, chosen once from the base model so that the conditioned base accuracy is “reasonable” (Setlur et al., 2026 ) . We argue this leaves most of the available signal on the table for two reasons. First, the GRPO signal is not maximized merely by being non-zero; prior work establishes an intermediate optimal k / G k/G (near 50 % 50\\\\% ) for RL learning signal (Bae et al., 2025 ; Gao et al., 2025 ) (Section 4.1 ). Second, the prefix length that achieves a given k / G k/G is per-sample and non-stationary: as the policy improves during training, the same prefix yields a higher k / G k/G , drifting the problem out of the productive regime. A schedule fixed at the start cannot track this.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We introduce AdaPrefix-GRPO , which treats prefix length as a closed-loop controller . During training we measure the realized batch-mean k / G k/G and adjust a single global base prefix length toward a target k / G k/G using a root-finding update (secant or binary search); each sample is offset from the base by a static difficulty-dependent shift, and all prefixes are annealed to zero so that the deployed policy never sees a prefix at test time. The method requires only minimal changes to the RL trainer: prefixes are injected as an assistant message in the data pipeline, gradients are masked on the prefix, and the reward is computed on the full completion. We hold the total training compute (FLOPs) fixed across all methods, so improvements are not attributable to extra training compute; the one-time calibration is a per-model preprocessing step, amortized across all runs on the same model and data, and excluded from the training budgets.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Contributions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We frame prefix length as a closed-loop controller of GRPO’s success rate k / G k/G , converting an uncontrolled observable into a tracked set-point (a global closed loop with per-sample difficulty offsets; Section 4 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We adopt the established intermediate ( ≈ 50 % \\\\approx\\\\!50\\\\% ) solve-rate set-point for RL signal (Bae et al., 2025 ; Gao et al., 2025 ) and make it reachable across the batch through prefix length (Section 4.1 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We propose AdaPrefix-GRPO: cold-start initialization, global closed-loop adaptation of a base prefix with per-sample difficulty offsets, and annealing to zero, implemented purely in data preparation on top of a stock GRPO trainer (Section 4.2 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"At matched training FLOPs on hard math, AdaPrefix-GRPO improves pass@1 over vanilla GRPO and a fixed-prefix baseline without collapsing pass@16, and the gain grows as the model shrinks (Section 5 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Related Work\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"The dead zone in group-relative RL.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"GRPO (Shao et al., 2024 ) and its variants estimate advantages by centering rewards within a group of rollouts. When a problem is solved by none (or all) of the group, the centered advantages collapse to zero and the problem produces no gradient, the “learning cliff” (Zhang et al., 2025 ) that DAPO’s dynamic filtering works around (Yu et al., 2025 ) . Classical exploration remedies (entropy bonuses, looser clipping, pass@ k k objectives) do not resolve this and can destabilize optimization (Qu et al., 2026 ) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Prefix- and hint-guided RL.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Closest is PrefixRL (Setlur et al., 2026 ) : it conditions on-policy RL on prefixes of off-policy correct traces (rejection-sampled to “reuse” prior compute), masks gradients on the prefix, and reports back-generalization from prefixed to unprefixed problems, but it fixes a handful of prefix lengths per problem, chosen once from base accuracy. POPE (Qu et al., 2026 ) uses the minimal prefix that yields a non-zero reward; Prefix-RFT (Huang et al., 2025 ) anneals a single global prefix on a cosine schedule; Scaf-/Hint-GRPO (Zhang et al., 2025 ; Huang et al., 2025 ) give graded in-prompt hints; R3 (Xi et al., 2024 ) slides the RL start state backward along a demonstration, a reverse curriculum over prefix position. These treat assistance as a fixed or global heuristic. We additionally show it is closed-loop control, not prefixing per se, that carries the gain (Section 5.3 ). Off-policy methods such as LUFFY (Yan et al., 2025 ) and ReLIFT (Ma et al., 2025 ) use demonstrations as training targets; like PrefixRL we stay on-policy and use solutions only to seed exploration.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Difficulty curricula.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Ordering data by difficulty is classic (Bengio et al., 2009 ; Kumar et al., 2010 ) , and recent RL work curates toward an intermediate success rate (Bae et al., 2025 ; Gao et al., 2025 ) . Instead of selecting problems that already sit at the right difficulty, we move every problem toward the target and track it over time.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Preliminaries: GRPO and the Dead Zone\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For a problem x x with reference solution and a verifier r ( x , y ) ∈ { 0 , 1 } r(x,y)\\\\in\\\\{0,1\\\\} , GRPO samples a group of G G rollouts y 1 , … , y G ∼ π θ ( ⋅ ∣ x ) y_{1},\\\\dots,y_{G}\\\\sim\\\\pi_{\\\\theta}(\\\\cdot\\\\mid x) and forms the group-relative advantage\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We use this un-normalized centered advantage (no division by the group std). The choice matters for Section 4.1 : std-normalization does not move the set-point (the aggregate positive mass still peaks at k / G = 0.5 k/G{}\\\\!=\\\\!0.5 ), but it inflates the per-sample advantage of a rare correct rollout, which scales as ( 1 − p ) / p \\\\sqrt{(1-p)/p} and diverges as p → 0 p\\\\!\\\\to\\\\!0 . The un-normalized form keeps a well-behaved p ( 1 − p ) p(1-p) scaling without that instability. Un-normalized advantages are also competitive with the std-normalized variant in recent practice (Liu et al., 2025 ) , so this departure does not weaken the baseline. Let k = ∑ i r ( x , y i ) k=\\\\sum_{i}r(x,y_{i}) be the number of correct rollouts, so the empirical success rate is k / G k/G . If k = 0 k=0 (or k = G k=G ), all A i = 0 A_{i}=0 and the gradient contribution of x x vanishes. We call the set of problems with pass @ G ≈ 0 \\\\mathrm{pass}@G\\\\approx 0 the dead zone : at exactly zero success probability no amount of sampling extracts signal, and under finite sampling near-zero rates yield negligible signal. On our hard split this is the common case: vanilla GRPO solves only ∼ \\\\sim 35% of problems at pass@16, leaving a substantial dead zone on the training portion; controlled prefixing converts a substantial fraction of these into learnable problems.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Prefix as a difficulty dial.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given a reference solution z = ( z 1 , … , z L ) z=(z_{1},\\\\dots,z_{L}) for x x , a prefix of ratio ρ ∈ [ 0 , 1 ] \\\\rho\\\\in[0,1] is z 1 : ⌈ ρ L ⌉ z_{1:\\\\lceil\\\\rho L\\\\rceil} , prepended to the prompt as a partial answer the model must continue. The conditional success rate κ ( ρ ) = 𝔼 [ r ( x , y ) ∣ y ∼ π ( ⋅ ∣ x , z 1 : ρ L ) ] \\\\kappa(\\\\rho)=\\\\mathbb{E}[r(x,y)\\\\mid y\\\\sim\\\\pi(\\\\cdot\\\\mid x,z_{1:\\\\rho L})] is empirically near-monotone in ρ \\\\rho ( Figure ˜ 2 a), with κ ( 0 ) \\\\kappa(0) the unconditioned rate and κ ( 1 ) → 1 \\\\kappa(1)\\\\!\\\\to\\\\!1 ; longer, partly off-distribution prefixes occasionally fail to help, a violation rate we report (Section 5.4 ). We do not require strict monotonicity: the bracketing (binary-search) update only needs κ \\\\kappa to cross τ \\\\tau once inside [ ρ − , ρ + ] [\\\\rho^{-},\\\\rho^{+}] and tolerates local non-monotonicity, and it is the fallback whenever the secant step is ill-conditioned. For any reachable target τ ∈ ( κ ( 0 ) , 1 ) \\\\tau\\\\in(\\\\kappa(0),1) there is a ρ ⋆ \\\\rho^{\\\\star} with κ ( ρ ⋆ ) = τ \\\\kappa(\\\\rho^{\\\\star})=\\\\tau ; for easy problems κ ( 0 ) > τ \\\\kappa(0)\\\\!>\\\\!\\\\tau already, so the controller simply sets ρ = 0 \\\\rho=0 . Gradients are masked on the prefix tokens; the reward is computed on the full (prefix + + generated) sequence; evaluation always uses ρ = 0 \\\\rho=0 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Method: AdaPrefix-GRPO\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 The intermediate set-point\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Escaping the dead zone only requires k ≥ 1 k\\\\geq 1 , but the magnitude of the learning signal depends on where in ( 0 , 1 ) (0,1) the group success rate p = k / G p=k/G{} sits. Work on RL data selection finds, empirically and through analysis of the optimization dynamics, that an intermediate solve rate, commonly near 50 % 50\\\\% , maximizes useful signal per update, while groups that are almost always or almost never solved contribute little (Bae et al., 2025 ; Gao et al., 2025 ) ; classical RL curricula targeted the same intermediate-success band long before LLMs (Florensa et al., 2018 ) .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Two intuitions match this empirical optimum, both stated for the un-normalized advantages we use (Section 3 ). First, the aggregate positive gradient in a group scales with the reward spread p ( 1 − p ) p(1-p) , which peaks at p = 0.5 p=0.5 . Second, the number of contrastive (correct, incorrect) pairs the group baseline can separate is k ( G − k ) k(G-k) , maximal at k = G / 2 k=G/2 and zero at p ∈ { 0 , 1 } p\\\\!\\\\in\\\\!\\\\{0,1\\\\} , recovering the dead zone as the degenerate case. Neither argument is a proof. The set-point itself is robust to advantage normalization (Section 3 ), so the choice of un-normalized advantages does not affect where the optimum sits. The empirical target sweep in Section 5.3 is the actual evidence; reward noise and many valid solution paths can shift the optimum somewhat (Gao et al., 2025 ) , so we treat τ = 0.5 \\\\tau=0.5 as a default. Our contribution is the mechanism that reaches this set-point: prefix length lets us hold the batch near τ \\\\tau throughout training.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 Algorithm\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The set-point fixes where we want each group to sit ( k / G ≈ τ k/G{}\\\\!\\\\approx\\\\!\\\\tau ); the algorithm is the controller that gets it there and keeps it there as the policy moves. AdaPrefix-GRPO has three stages (Algorithm 1 ): a cold start that places the population roughly in range, a global closed loop with per-sample difficulty offsets that tracks τ \\\\tau throughout training, and an anneal to ρ = 0 \\\\rho=0 so deployment needs no prefix.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Cold start.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Before the first step we set the initial base prefix ratio ρ b ( 0 ) \\\\rho_{b}^{(0)} , shared by the whole population. We support two estimators: a conservative constant (e.g. 0.8 0.8 , which over-prefixes but makes k ≥ 1 k\\\\!\\\\geq\\\\!1 overwhelmingly likely from step one), and a one-shot calibration sweep : 4 rollouts per problem on a random 512-problem subset, over the grid ρ ∈ { 0 , 0.2 , 0.4 , 0.6 , 0.8 } \\\\rho\\\\in\\\\{0,0.2,0.4,0.6,0.8\\\\} , give the population-mean κ ( ρ ) \\\\kappa(\\\\rho) curve, which we invert at τ \\\\tau to set ρ b ( 0 ) \\\\rho_{b}^{(0)} . Independently of which estimator sets the base, a difficulty probe always runs: four rollouts per problem at ρ b ( 0 ) \\\\rho_{b}^{(0)} give every sample a difficulty estimate d x d_{x} (its measured pass rate) that the offsets below consume; the conservative variant thus skips only the grid sweep, not the probe. Sweep and probe are one-time preprocessing, amortized across all runs on the same model and data (seeds, hyperparameter sweeps, ablations), and excluded from the training budgets. The cold start only needs to land inside ( κ ( 0 ) , 1 ) (\\\\kappa(0),1) ; the closed loop removes the residual error. The sweep costs a single pre-training inference pass and lands near the optimum, so our main runs use it; the hyperparameter-free conservative constant trails it by only ∼ \\\\sim 1.8 pp (Table 3 ) and is a reasonable fallback when even that overhead is unwanted.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Closed-loop adaptation with per-sample offsets.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The controller maintains a single base prefix ratio ρ b \\\\rho_{b} , shared by the whole population. Every N N optimizer steps we read the realized k / G k/G{} , pooled over all rollouts in the window and smoothed with an EMA ( β = 0.7 \\\\beta{=}0.7 ), and move ρ b \\\\rho_{b} toward τ \\\\tau . Because κ ( ρ ) \\\\kappa(\\\\rho) is monotone, this is a one-dimensional root-find. Binary search keeps bracketing bounds [ ρ − , ρ + ] [\\\\rho^{-},\\\\rho^{+}] and halves toward τ \\\\tau (robust, one bit per update); secant interpolates from the two most recent ( ρ b , k / G ) (\\\\rho_{b},k/G{}) pairs,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"which converges in fewer updates when κ \\\\kappa is locally smooth. We guard the denominator (falling back to a bisection step when k / G ≈ t k / G t − 1 k/G{}_{t}\\\\!\\\\approx\\\\!k/G{}_{t-1} ; in our runs this fires on ∼ \\\\sim 17% of updates, mostly early when k / G k/G estimates are noisiest), clip the per-update step in ρ b \\\\rho_{b} to 0.05 0.05 , and keep ρ ∈ [ 0 , ρ max ] \\\\rho\\\\in[0,\\\\rho_{\\\\max}] . Pooling over the window makes the loop feasible: a single sample yields one heavily quantized Bernoulli estimate ( k / G ∈ { 0 , 1 G , … , 1 } k/G{}\\\\in\\\\{0,\\\\tfrac{1}{G},\\\\dots,1\\\\} ) per visit, and visits are ∼ \\\\sim 100 steps apart, far too sparse to track a moving target, whereas the window aggregates thousands of rollouts.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Holding the population mean at τ \\\\tau is not enough: the same mean can hide a wide spread, with half the samples saturated ( k = G k\\\\!=\\\\!G ) and half dead ( k = 0 k\\\\!=\\\\!0 ), so most rollouts still carry no signal. To compress the spread, each sample is offset from the base by a static difficulty shift, ρ x = clip ( ρ b + δ ( d x ) , 0 , ρ max ) \\\\rho_{x}=\\\\mathrm{clip}\\\\big(\\\\rho_{b}+\\\\delta(d_{x}),\\\\,0,\\\\,\\\\rho_{\\\\max}\\\\big) , where d x d_{x} is the sample’s cold-start difficulty and δ \\\\delta is a small monotone (linear) map with a hand-tuned slope, spanning ± 0.15 \\\\pm 0.15 around the base across the difficulty range in our runs: harder samples get slightly longer prefixes than the base, easier ones shorter. The offset is open-loop, its constants fixed across training; all tracking of the policy’s improvement happens through the single global loop. On easy problems the negative offset saturates at the clip, ρ x = 0 \\\\rho_{x}\\\\!=\\\\!0 , so prefixing is reduced on easy problems (we verify no regression on the GSM8K slice included in every method’s training mixture; Section 5 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Why closed-loop, and why offsets.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A single global schedule (Prefix-RFT) or a fixed per-problem length (PrefixRL) can only be right on average at one moment: problems differ widely in κ ( ⋅ ) \\\\kappa(\\\\cdot) , and as the policy improves every κ \\\\kappa curve shifts left, so any static or open-loop choice drifts off τ \\\\tau , typically toward k = G k\\\\!=\\\\!G , back into a zero-variance regime ( Figure ˜ 2 b). The global loop absorbs this non-stationarity across time. The per-sample offset addresses the orthogonal problem, heterogeneity across problems: mean-only control can leave the batch bimodal, many samples saturated or dead around a healthy average, so most rollouts are wasted; the offset narrows the spread of k / G k/G around the tracked mean, keeping the whole batch, not just its mean, near the signal-maximizing set-point ( Figure ˜ 5 ). The granularity ablation (Section 5.3 ) isolates how much of the gain the offset adds on top of base-only control. Two caveats. We use “controller” loosely: the update is a smoothed, quantized root-finder with no formal stability guarantee, and the annealing envelope (below) overrides the loop in the final phase. The offset is a static heuristic: it corrects difficulty ordering across problems but cannot adapt to a sample whose difficulty was mis-estimated at cold start.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Annealing.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To remove the train/test mismatch we impose a shrinking upper envelope and run the controller inside it: ρ x ← min ( ρ ¯ t , ρ b + δ ( d x ) ) \\\\rho_{x}\\\\leftarrow\\\\min\\\\!\\\\big(\\\\bar{\\\\rho}_{t},\\\\ \\\\rho_{b}+\\\\delta(d_{x})\\\\big) . The envelope is held near ρ max \\\\rho_{\\\\max} for the first fraction w w of training and then decayed to zero, ρ ¯ t = ρ max ⋅ clip ( T − t ( 1 − w ) T , 0 , 1 ) \\\\bar{\\\\rho}_{t}=\\\\rho_{\\\\max}\\\\cdot\\\\mathrm{clip}\\\\!\\\\big(\\\\tfrac{T-t}{(1-w)T},0,1\\\\big) (we use w ≈ 0.8 w\\\\!\\\\approx\\\\!0.8 ). Early on the envelope is slack and the loop is free to hold τ \\\\tau ; late in training the envelope binds and forces every ρ x → 0 \\\\rho_{x}\\\\!\\\\to\\\\!0 , trading some signal for on-distribution (no-prefix) practice. The final policy is thus trained to solve unaided, and all evaluation is at ρ = 0 \\\\rho=0 . As the envelope binds, the controller can no longer hold τ \\\\tau and the measured k / G k/G dips modestly below it ( Figure ˜ 2 b); but the no-prefix held-out curve shows no sustained regression ( Figure ˜ 1 ), so this phase consolidates what was learned. To test this, we report the final ρ = 0 \\\\rho=0 pass rate on the full originally-dead set (Section 5.4 ); a non-trivial unaided pass rate on problems that started at κ ( 0 ) ≈ 0 \\\\kappa(0)\\\\!\\\\approx\\\\!0 indicates that the gains are driven by back-generalization, not by prefix-conditioned imitation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Prefix source.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prefixes can be cut from reference solutions (always available) or from model-generated correct traces (rejection-sampled, on-distribution, as in PrefixRL). A reference path may be off-distribution and harder to continue, so the source matters; we default to reference solutions for availability and show in Section 5.3 that the controller is orthogonal to it: swapping in off-policy prefixes retains most of the gain.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Implementation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"AdaPrefix-GRPO touches the trainer only through the prefix loss mask; everything else lives in data preparation. Prefixes are appended as an assistant message that the model is asked to continue; the trainer masks loss on prefix tokens and scores the full sequence. Prefix cuts snap to sentence boundaries to avoid mid-token artifacts.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Compute accounting.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We compare methods at matched training compute , not matched steps or episodes. Following the accounting of Setlur et al. ( 2026 ); Snell et al. ( 2024 ) , a run’s cost is FLOPs = 2 N D samp + 6 N D upd \\\\mathrm{FLOPs}=2ND_{\\\\mathrm{samp}}+6ND_{\\\\mathrm{upd}} , where N N is the number of model parameters (we train with LoRA, but sampling and backpropagation traverse the full model, so we use the full parameter count), D samp D_{\\\\mathrm{samp}} counts all tokens processed at rollout time (including the prefill of supplied prefixes , so prefixing is not free), and D upd D_{\\\\mathrm{upd}} counts tokens in gradient updates. Every method runs to the same cumulative budget; AdaPrefix-GRPO’s one-time calibration sweep and difficulty probe are treated as amortized per-model preprocessing and excluded from the training budget. Because a prefixed rollout decodes only the continuation, AdaPrefix-GRPO completes more, shorter episodes within the budget, roughly two passes over the data where vanilla GRPO completes one; Figure ˜ 3 separates this revisit effect from the controller itself by reporting both compute-matched and iteration-matched views, and by including a vanilla variant that spends the same budget on more samples per problem. Prefix prefill stays under ∼ \\\\sim 15% of total training tokens because prefixes are capped and annealed. Table 1 reports generated length at evaluation ( ρ = 0 \\\\rho=0 ), where no prefix is present, so the trace-length saving reflects the deployed policy rather than the mechanical effect of starting mid-solution.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Experiments\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Setup.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We train on three Qwen3 sizes (0.6B, 1.7B, 4B) (Yang et al., 2025 ) and additionally validate cross-family on Qwen3.5-2B (Qwen Team, 2026 ) and Gemma 4 E4B (Gemma Team, 2026 ) , to confirm the controller is not Qwen-specific. Training data is the DeepMath-103K (He et al., 2025 ) hard split (difficulty 8, binary-answer questions removed): ∼ \\\\sim 6.8K problems with verified reference solutions (R1-generated traces shipped with the dataset). All methods train on the identical mixture: this hard split plus a 5% slice of GSM8K training problems (included so that self-disabling of prefixing on easy data can be verified; the GSM8K test split is disjoint). We use G = 8 G=8 rollouts, a batch of 64 prompts per optimizer step, and LoRA (Hu et al., 2022 ) . All methods run to the same training-FLOPs budget ( ≈ \\\\approx 6 × \\\\times 10 18 for the 1.7B model) under the accounting of Section 4 . For vanilla GRPO this budget corresponds to a single pass over the mixture ( ∼ \\\\sim 112 optimizer steps, 8 full-length rollouts per problem); because prefixed rollouts decode only the continuation, AdaPrefix-GRPO completes ∼ \\\\sim 2.1 passes ( ∼ \\\\sim 240 steps) of shorter episodes within the same budget. The controller updates every N = 10 N{=}10 steps. The main comparison rows (vanilla GRPO, PrefixRL, AdaPrefix-GRPO) are averaged over 3 seeds (per-seed spread ≤ \\\\leq 1.2 pp on DeepMath-hold); the remaining baselines and all ablations use a single seed. Figure ˜ 3 additionally reports the iteration-matched view, in which baselines run the same number of optimizer steps and are therefore granted more FLOPs. Training is built on TRL (von Werra et al., 2020 ) with vLLM (Kwon et al., 2023 ) generation. Evaluation spans GSM8K (Cobbe et al., 2021 ) as an easy-task sanity check, MATH-500 (Hendrycks et al., 2021 ) , AIME 2024/2025 (averaged), AMC 2023 (40 problems), and a DeepMath held-out split (1,000 problems held out from the same difficulty-8 hard split, disjoint from training). We report pass@1 and pass@16 estimated from 8 and 16 samples per problem respectively (avg@8 for the small AIME/AMC sets) with 95% CIs from a problem-level bootstrap (resampling problems together with their 8 samples, which accounts for per-problem clustering), and mean trace length. Evaluation uses temperature 1.0 with a strict verifier and no system prompt or thinking mode ; official Qwen3 reports use thinking mode and larger sampling budgets, so absolute zero-shot numbers are lower here. We additionally report held-out pass@16 to verify that gains do not come from collapsing the policy’s exploration onto a few modes.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Baselines.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Zero-shot : base model, no RL.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Vanilla GRPO : standard on-policy GRPO, no prefix.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Vanilla GRPO ( G = 16 G{=}16 ) : doubles the group size within the same FLOPs budget, which halves the number of optimizer steps; controls for the possibility that simply re-allocating compute toward more attempts per problem explains the gain.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"SFT-warmup → \\\\to GRPO : mid-train on reference solutions, then GRPO (the “why not just SFT” control). The SFT phase is charged to the method’s FLOPs budget ( ∼ \\\\sim 8%, one epoch over the reference traces), shortening its RL phase accordingly.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Difficulty-filter GRPO : drop dead-zone problems, GRPO on the rest.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"PrefixRL (Setlur et al., 2026 ) : a fixed set of prefix lengths per problem chosen once from base accuracy, static across training; the primary baseline. To keep the comparison fair, it uses the same reference solutions and prefix source as AdaPrefix-GRPO, the same FLOPs budget, and its per-problem length set is tuned with a matched length sweep; the only difference is that lengths are fixed rather than tracked. We additionally report a PrefixRL + anneal variant to separate the effect of annealing from the effect of closing the loop.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Prefix-RFT (Huang et al., 2025 ) : uniform decay, a single global prefix ratio annealed 0.8 → 0 0.8\\\\to 0 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"AdaPrefix-GRPO (ours) : global secant control of the base prefix toward τ = 0.5 \\\\tau=0.5 , with per-sample difficulty offsets, annealed to 0 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.1 Main results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table ˜ 1 reports the FLOPs-matched comparison on Qwen3-1.7B; Figure ˜ 3 shows the full compute curves and the iteration-matched control. AdaPrefix-GRPO improves pass@1 across benchmarks. The extra passes over the data that AdaPrefix-GRPO affords within the budget do not by themselves explain the gain: giving vanilla GRPO the same budget as a larger group ( G = 16 G{=}16 ) recovers only a small fraction of it. In absolute terms it adds +6.4 pp over GRPO on MATH-500 ( +17.0 pp on the DeepMath held-out split at budget end) versus +1.7/+6.1 pp for the PrefixRL baseline, i.e. 3–4 × \\\\times its gain on both. Head-to-head, AdaPrefix-GRPO beats the PrefixRL baseline on every benchmark (+4.7 MATH-500, +3.5 AIME, +3.4 AMC, +12.0 DeepMath-hold); under a paired problem-level bootstrap (both methods are evaluated on the same problems, so we resample per-problem score differences) the DeepMath gap reaches significance while the remaining three are positive and directionally consistent; all this at half the trace length (4.4k vs. 8.0k) and with no GSM8K regression; the gain over fixed prefixing thus does not hinge on the held-out split. Because every number here is measured at ρ = 0 \\\\rho=0 (no prefix), the ∼ \\\\sim 2 × \\\\times shorter outputs come from the deployed policy itself, not from counting only the completion of a prefixed problem. Among the baselines, Prefix-RFT (uniform decay) is the strongest simple alternative, edging out fixed prefixing on MATH-500 and DeepMath with shorter traces, which confirms that annealing helps; but without the closed loop it still trails AdaPrefix-GRPO by +11.4 pp on DeepMath-hold (significant under the paired bootstrap) and by +2.7 on MATH-500. The gain also does not come from narrowing the policy: pass@16 on the DeepMath held-out split improves alongside pass@1 (base 32.8 / GRPO 35.1 / ours 56.3%), so the distribution sharpens without collapsing. On easy GSM8K all methods are near ceiling and AdaPrefix-GRPO is best or tied (92.4%), confirming the controller does no harm where prefixing is unneeded, whereas fixed prefixing slips slightly (90.4%) by over-prefixing easy problems. AIME columns average the 2024 and 2025 sets. Beyond the benchmark numbers, the solvable set itself expands: ∼ \\\\sim 17% of originally-dead problems (zero pass@16 under the base model at ρ = 0 \\\\rho=0 before training) are solved at pass@1 after annealing.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.2 Scaling and cross-family generalization\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We expect smaller models to live deeper in the dead zone and therefore benefit more. Each model size runs both methods to the same per-size FLOPs budget, set by the same rule as in the main setup, one vanilla-GRPO pass over the mixture at that size (so budgets scale with N N and trace length); within every row of Table ˜ 2 the two columns are therefore directly comparable, while absolute budgets differ across rows. Table ˜ 2 shows the pass@1 gain over vanilla GRPO on the DeepMath held-out split shrinking but staying positive as the Qwen3 base grows, and the same controller transferring to a different architecture family (Qwen3.5-2B, Gemma 4), ruling out a Qwen-specific effect. Notably, the 0.6B model trained with AdaPrefix-GRPO surpasses the 3 × 3\\\\times larger Qwen3-1.7B trained with vanilla GRPO (41.8 vs. 30.1) despite its smaller absolute budget: at this scale the training recipe buys more than the next model size.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.3 Ablations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table ˜ 3 isolates each design choice on Qwen3-1.7B. The target sweep is consistent with the intermediate-set-point prediction (Section 4.1 ): extreme targets clearly hurt ( τ = 0.125 \\\\tau{=}0.125 : 41.9), while 0.375–0.625 sit within CI of one another; we take τ = 0.5 \\\\tau=0.5 as the default from prior work and do not claim a sharp empirical optimum. Control granularity shows the gain comes from closed-loop control, not prefixing per se (PrefixRL and global-decay trail by a wide margin; the bulk of the improvement is from closing the loop). Adding our annealing envelope to the PrefixRL baseline (the PrefixRL + anneal row) recovers only a small part of that gap, confirming annealing alone is not the driver. The per-sample difficulty offset adds a smaller increment over base-only control (+2.9 pp, at the edge of the single-run CI), consistent with the view that compressing the across-problem spread of k / G k/G matters beyond centering its mean. The advantage-norm. row addresses a confound: running AdaPrefix-GRPO with std-normalized advantages (which leave the τ \\\\tau set-point unchanged but over-weight rare successes; Section 3 ) still helps but by less, indicating the gain is not purely a global gradient-norm effect. The annealing row shows the delayed envelope modestly beats no annealing (48.2 vs 44.8), i.e. forced consolidation helps, but the method works even with the envelope removed entirely: back-generalization carries most of the gain, and Section 5.4 discusses why the train/test distribution shift one might expect fails to materialize. The controller is also robust to the prefix source : swapping reference for rejection-sampled off-policy prefixes (PrefixRL’s regime) retains most of the gain. Finally, the group size G G trades off estimator quality against update count: at a fixed FLOPs budget, doubling G G halves the number of optimizer steps, so the G = 16 G\\\\!=\\\\!16 row buys a sharper k / G k/G estimate for the controller with half the updates. Accuracy improves from G = 4 G\\\\!=\\\\!4 to G = 8 G\\\\!=\\\\!8 and is essentially flat at G = 16 G\\\\!=\\\\!16 (48.7 vs 48.2, within CI), i.e. the better estimate roughly pays for the lost updates but no more, making G = 8 G\\\\!=\\\\!8 the sensible default (and confirming the controller does not require an impractically large group).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.4 Analysis\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"The controller tracks the set-point.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The closed loop holds the batch-mean k / G k/G at τ = 0.5 \\\\tau\\\\!=\\\\!0.5 within a few update intervals and keeps it there as the policy improves ( Figure ˜ 2 b) and keeps the whole distribution of per-sample k / G k/G concentrated near τ \\\\tau , whereas fixed and global prefixes drift toward saturation, piling mass at k / G = 1 k/G{}\\\\!=\\\\!1 while a hard core stays at 0 ( Figure ˜ 5 ). Tracking difficulty cohorts, their ρ x \\\\rho_{x} and realized k / G k/G are driven toward the tracked mean by the global controller plus static offsets and anneal cleanly to ρ = 0 \\\\rho=0 ( Figure ˜ 4 ). This is the mechanism that DAPO approximates with dynamic sampling, oversampling to discard all-correct/all-wrong groups (Yu et al., 2025 ) , except that prefix control creates signal on problems no amount of resampling can rescue.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Why prefixing transfers to no prefix.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Training on a partial solution still requires the model to generate the remaining reasoning and be graded on the full trace, so it practices the same sub-skills it must later chain unaided; as prefixes anneal away, those skills are exercised from progressively earlier states. This matches the back-generalization PrefixRL reports (Setlur et al., 2026 ) : held-out (no-prefix) pass@1 rises throughout training ( Figure ˜ 1 ), not only once prefixes vanish. Three mechanisms plausibly explain why the train/test distribution shift one might expect does not materialize. First, the loss is masked on prefix tokens: the model never imitates the (possibly off-distribution) reference text and is only reinforced on its own successful continuations, so every token that receives gradient is on-policy and no imitation drift accumulates (conditioning still shifts the state distribution; the final ρ = 0 \\\\rho=0 pass rates below are the direct check that this shift does not persist). Second, a prefix changes only the start state of the episode, not the policy being learned: as in reverse-curriculum RL (Xi et al., 2024 ) , the sub-skills reinforced on the remainder (algebraic manipulation, case analysis, self-verification) are the same ones a from-scratch solution must chain, and they transfer across starting positions. Third, the controller itself performs a soft anneal: as the policy improves, every κ \\\\kappa curve shifts left and the loop lowers ρ x \\\\rho_{x} to keep k / G k/G at τ \\\\tau ( Figure ˜ 4 b), so the distribution of training start states drifts toward unprefixed problems on its own ; the envelope merely forces the final step to exactly ρ = 0 \\\\rho=0 . As the direct test of consolidation, the final ρ = 0 \\\\rho=0 policy solves ∼ \\\\sim 17% of the full originally-dead set at pass@1 ( ∼ \\\\sim 22% at pass@16), evidence that the gains persist without prefixes.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Monotonicity in practice.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Across problems the response κ ( ρ ) \\\\kappa(\\\\rho) violates monotonicity on ∼ \\\\sim 6% of measured ( ρ , κ ) (\\\\rho,\\\\kappa) pairs (mostly long, off-distribution reference prefixes), and on those the controller falls back to bracketing without diverging; the mid-training snapshot in Figure ˜ 5 shows the resulting concentration of per-sample k / G k/G near τ \\\\tau on raw (un-smoothed) quantized rates.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Why traces shorten.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Because each sample is held near a moderate k / G k/G , positive advantages reward efficient correct completions rather than long, lucky explorations; combined with annealing, the policy converges to shorter unaided ( ρ = 0 \\\\rho=0 ) solutions than vanilla GRPO, which only finds reward through length-heavy exploration. (We treat this as a hypothesis to verify with length-conditioned analyses at ρ = 0 \\\\rho=0 .)\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6 Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"GRPO discards its hardest problems because zero within-group reward variance yields zero gradient. Prefix length is a continuous, per-sample dial that can place any problem at almost any reachable success rate. Conceptually this removes GRPO’s sampling barrier, the dead zone in which hard problems yield no signal under a finite rollout budget, and shifts the binding constraint to the model’s own capacity : with a controllable signal, whether a problem is learnable turns on whether the model can complete it given enough scaffolding, not on whether it can already solve it unaided. Prefixing cannot exceed that capacity ceiling or rescue a reference path the model is unable to follow, but within it, difficulty is no longer a hard cutoff on trainability. The contribution of AdaPrefix-GRPO is to control that dial in closed loop toward the intermediate, signal-maximizing target k / G k/G (Bae et al., 2025 ; Gao et al., 2025 ) and anneal it away, rather than fixing it in advance. At matched training compute this extracts signal where GRPO got none, beats a fixed-prefix baseline, and yields shorter solutions, with the largest gains on the smallest models.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Limitations.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"AdaPrefix-GRPO needs reference (or off-policy) solutions and a reliable, near-binary verifier; we evaluate only on math reasoning and do not test noisy or non-verifiable domains, where behavior may differ. We adopt the intermediate ( ≈ 0.5 \\\\approx\\\\!0.5 ) set-point from prior work (Bae et al., 2025 ; Gao et al., 2025 ) and locate it empirically; we do not derive the exact per-sample optimum under reward noise. A subtler caveat: a 50 % 50\\\\% success rate on the short remainder of a heavily prefixed problem exercises credit assignment only over that remainder, which may be an easier learning signal than 50 % 50\\\\% on a from-scratch trajectory; annealing is what forces the harder, full-length signal back in, and disentangling the two is future work. The offset map δ \\\\delta is a hand-tuned static heuristic: a sample whose cold-start difficulty is mis-estimated keeps a biased prefix that the global loop cannot correct individually, and upgrading the offset to a genuine per-sample second loop would require denser per-sample k / G k/G estimates than G = 8 G{=}8 with sparse revisits provides. The cold-start calibration and difficulty probe, while one-time and amortized across runs, are not free; cheap or training-free per-sample difficulty estimation is an interesting direction for future work. The prefix source also matters (reference paths can be off-distribution), though our ablation suggests the controller is largely orthogonal to it. Small benchmarks (AIME, AMC) carry wide intervals, so we rely on several benchmarks and avg@8 sampling rather than any single number. Underlying per-step data and training code are not released.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Quantities quoted in the main text are measured as follows. Monotonicity violation rate ( ∼ \\\\sim 6%, Section 5.4 ): the fraction of adjacent ( ρ , κ ( ρ ) ) (\\\\rho,\\\\kappa(\\\\rho)) pairs with Δ κ < 0 \\\\Delta\\\\kappa<0 , pooled over the calibration sweep (512 problems × \\\\times 4 adjacent grid intervals = = 2,048 pairs) and all subsequent controller reads. Secant fallback rate ( ∼ \\\\sim 17%, Section 4.2 ): 4 of the 24 controller updates ( ∼ \\\\sim 240 steps, N = 10 N{=}10 ) triggered the bisection fallback, 3 of them in the first third of training. Prefill overhead ( < < 15%, Section 4 ): prefix tokens account for 14.6% of all training-time tokens (prefill + + generated), measured over the full AdaPrefix-GRPO run.\"\n }\n]","isConversational":false}}},{"rowIdx":96,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.07673"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"MedPMC: A Systematic Framework for Scaling High-Fidelity Medical Multimodal Data for Foundation Models"},"authors":{"kind":"list like","value":["Hyunjae Kim","Dain Kim","Pan Xiao","Serina S. Applebaum","Younjoon Chung","Xuguang Ai","Yu Yin","Roy Jiang","Yuexi Du","Yawen Wei","Yiming Kong","Tuo Guo","Zhiyuan Cao","Mengmeng Du","Yuelei Fu","Yan Hu","Rui Shi","Gui Yang","Kevin W. Jin","Yuntian Liu","Yuxuan Tian","Jonathan Marquez","Zhen Chen","Sheng Zhang","Hoifung Poon","Hua Xu","Jaewoo Kang","Qingyu Chen"],"string":"[\n \"Hyunjae Kim\",\n \"Dain Kim\",\n \"Pan Xiao\",\n \"Serina S. Applebaum\",\n \"Younjoon Chung\",\n \"Xuguang Ai\",\n \"Yu Yin\",\n \"Roy Jiang\",\n \"Yuexi Du\",\n \"Yawen Wei\",\n \"Yiming Kong\",\n \"Tuo Guo\",\n \"Zhiyuan Cao\",\n \"Mengmeng Du\",\n \"Yuelei Fu\",\n \"Yan Hu\",\n \"Rui Shi\",\n \"Gui Yang\",\n \"Kevin W. Jin\",\n \"Yuntian Liu\",\n \"Yuxuan Tian\",\n \"Jonathan Marquez\",\n \"Zhen Chen\",\n \"Sheng Zhang\",\n \"Hoifung Poon\",\n \"Hua Xu\",\n \"Jaewoo Kang\",\n \"Qingyu Chen\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.07673v1"},"published":{"kind":"timestamp","value":"2026-07-08T00:00:00","string":"2026-07-08T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"para","text":"\\ul"},{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Medicine is multimodal in nature [ 2 , 68 ] . Triage, diagnosis, prognosis, and treatment planning routinely require clinicians to integrate images, text, laboratory values, and other complementary data streams [ 20 , 35 ] . In principle, medical foundation models could unlock this potential by providing reusable backbones that generalize across medical modalities and can be adapted to local clinical settings with limited labeled data [ 31 , 80 ] . In practice, however, current medical AI systems are still largely developed and evaluated within single-modality, single-task settings rather than on the heterogeneous evidence used in clinical care [ 83 , 107 ] . This limitation stems not from model architecture alone, but from the absence of large-scale, high-quality multimodal data that are publicly accessible for training and evaluation, shareable across institutions, and suitable for reproducible research [ 107 , 83 , 1 , 34 ] . Unlike curated benchmark datasets, real-world medical data are fragmented across devices and health systems; constrained by privacy, licensing, and governance requirements; and costly to annotate at scale [ 47 , 55 , 63 ] . Although pioneering efforts have been made to make clinical datasets publicly accessible [ 40 , 33 , 5 ] , these resources remain much smaller and narrower than the data used to train general-domain foundation models. These barriers have led to two persistent challenges: a benchmark-to-bedside gap, in which models that perform well on public benchmarks may not deliver comparable utility in real clinical settings [ 87 , 42 , 8 ] ; and fragmented, institution-specific curation efforts that limit reproducibility, scalability, and reuse [ 37 , 36 ] ."},{"type":"para","text":"One practical strategy is to leverage publicly available biomedical resources—including web-scale data, biomedical literature, and medical textbooks—to develop public foundation model backbones that are shareable, reproducible, and ultimately adaptable to clinical settings with less reliance on locally curated data [ 12 , 34 , 109 , 18 ] . Among these sources, PubMed Central (PMC) has emerged as one of the most promising and widely used resources [ 34 , 12 ] . PMC hosts millions of expert-authored biomedical articles that pair diverse medical visuals with explanatory text across a broad range of specialties, and it continues to expand rapidly as new biomedical knowledge is published [ 71 ] . Prior studies have leveraged PMC figures and associated text to develop medical vision-language models [ 52 , 25 , 93 ] and, more recently, generative multimodal large language models (MLLMs) [ 50 , 78 ] ."},{"type":"para","text":"However, biomedical literature is created to communicate scientific findings, not to provide ready-to-use training data for multimodal foundation models. Repurposing it into high-fidelity multimodal training data poses substantial challenges. First, PMC contains a large fraction of visual content that is not directly useful for learning clinically relevant visual representations, including graphs, charts, schematics, flow diagrams, and molecular illustrations. Second, figures in the biomedical literature are often compound multi-panel images; without accurate decomposition of subfigures and alignment with their corresponding subcaptions, the image-text correspondence required for effective multimodal learning is substantially degraded. Finally, the continuous expansion of biomedical literature requires curation frameworks capable of dynamically incorporating newly published content. To date, existing PMC-derived datasets have not comprehensively addressed these challenges. For example, in a recent 24-million image-text pair dataset [ 62 ] , our analysis revealed that 80.3% of the images were non-medical. Moreover, compound figures accounted for 61% of the data but were not decomposed into individual panels. Furthermore, because most existing datasets are released as static snapshots, they become progressively outdated and fail to continuously integrate newly published knowledge. These limitations are further detailed in Extended Data Table 1 and the Discussion section."},{"type":"para","text":"To address this, we developed MedPMC, an automated and systematic framework for curating high-fidelity medical multimodal data from permissively licensed biomedical literature. MedPMC extends beyond a static dataset and instead provides reusable, continuously evolving data infrastructure for developing medical multimodal foundation models, including contrastive vision-language encoders and MLLMs. Our contributions operate at three levels. At the data resource level, MedPMC yields 11 million medical image-text pairs from 6.1 million PMC articles published through June 2024. The framework consists of five sequential stages—initial screening, multi-panel figure detection, multi-panel figure separation, caption separation and alignment, and medical figure classification—each handled by a purpose-built model to improve the fidelity of the resulting training data ( Fig. 1 a ). Manual evaluation by five annotators, including three medical experts, showed that 95.3% of curated images were medically relevant. The resulting corpus spans a broad range of specialties, including pathology, radiology, ophthalmology, dermatology, and endoscopy, and includes clinically important modalities that are less commonly represented at scale in public AI datasets, such as slit-lamp photography, skin photography, gross pathology photography, endoscopy, angiography, PET/SPECT, and fMRI. Importantly, MedPMC is designed for sustainable expansion rather than one-time dataset construction. Because each processing stage is modular and supported by a dedicated benchmark, individual components can be systematically evaluated, replaced, and refined as improved methods become available. This design allows MedPMC to continuously incorporate newly published biomedical literature, and the framework has identified more than one million new medically relevant image-text pairs per year since 2020."},{"type":"para","text":"At the foundation model level, we trained MedPMC-CLIP, a CLIP-style vision-language model [ 79 ] , as one representative instantiation to test whether high-fidelity MedPMC curation translates into stronger medical visual representations ( Fig. 1 b ). MedPMC-CLIP achieves state-of-the-art performance across 26 public medical benchmarks spanning 11 specialties, covering clinically relevant tasks such as pneumonia detection, diabetic retinopathy grading, lymph node metastasis detection, and colon cancer detection. The model improves average AUC by 7.1 percentage points over BMC-CLIP [ 62 ] , a strong biomedical vision-language baseline, despite being trained on less than half the training data. These gains are consistent across specialties, demonstrating that high-fidelity curation produces visual representations that generalize broadly across medical domains. To further assess whether these representations can support generative multimodal systems, we conducted a controlled experiment following the training strategy introduced by LLaVA-Med [ 50 ] . With all other components held fixed, replacing only the vision encoder with MedPMC-CLIP improves performance by 1.9 and 16.9 percentage points on two multimodal medical question-answering (QA) benchmarks, respectively."},{"type":"para","text":"At the clinical validation level, beyond standard benchmark evaluation, we further assessed transfer to downstream clinical applications. Using 10,524 dermatology patient images derived from a multi-site, hospital-based dermatology consultation cohort within the Yale New Haven Health System (YNHHS), we evaluated a morphology-to-image retrieval task in which textual descriptions of clinical findings were used to retrieve visually similar patient images, reflecting a clinically meaningful image-retrieval use case relevant to diagnostic decision-making [ 45 , 110 ] ( Fig. 1 c ). MedPMC-CLIP improves Recall@5 by 11.7 percentage points over BMC-CLIP, demonstrating that MedPMC-derived supervision can transfer effectively to clinically realistic settings. This translational benefit is further supported by embedding-space analysis, which shows that MedPMC better captures the visual distribution of real patient images from YNHHS, with broader coverage and closer alignment than existing dermatology datasets."},{"type":"para","text":"Collectively, these results position MedPMC as scalable data infrastructure for medical multimodal foundation models, linking high-fidelity literature-derived data curation to stronger visual representations, improved multimodal reasoning, and broader generalization across medical domains and downstream tasks. To support reuse and reproducibility, we publicly release the MedPMC framework, curated corpus, benchmark resources, and pretrained model checkpoints (see Data Availability and Code Availability)."},{"type":"heading","level":1,"text":"2 Results"},{"type":"heading","level":2,"text":"2.1 Pipeline Design and Data Curation"},{"type":"para","text":"Fig. 1 a illustrates the overall curation framework. The framework was designed to transform permissively licensed biomedical literature into high-fidelity medical image-text data through a sequence of automated, modular processing stages. Here, we summarize the purpose of each stage and the resulting data outputs; detailed methodological descriptions are provided in the Methods section."},{"type":"heading","level":4,"text":"Definition of medical images"},{"type":"para","text":"PMC contains a broad spectrum of biomedical visual content, ranging from clinical imaging studies to basic science experiments, molecular biology figures, statistical plots, and conceptual diagrams. In this study, we use “medical images” as an operational umbrella term to refer to visual content relevant to clinical medicine, human health, or translational biomedical research. Our goal was to curate visual evidence most relevant for clinical and translational medical AI applications, including image interpretation, diagnosis, disease assessment, treatment planning, healthcare workflows, and medical research involving human-derived specimens [ 2 , 81 ] . We therefore prioritized image types routinely encountered in clinical practice, as well as human-relevant biomedical imaging modalities [ 64 , 75 ] , including radiology, ophthalmology, pathology, microscopy, endoscopy, dermatology, physiological signals, surgical photographs, and other forms of human clinical photography. In contrast, we excluded visual content primarily depicting non-human organisms, basic cellular or molecular biology experiments without direct human medical or translational relevance, scientific plots, graphs, tables, conceptual diagrams, educational illustrations, and other materials outside this scope."},{"type":"heading","level":4,"text":"Training data construction"},{"type":"para","text":"The five-stage curation pipeline relies on purpose-built models for each component. We first leveraged existing labeled benchmarks, including ImageCLEF 2016 [ 27 ] , MedICaT [ 88 ] , and DocFigure [ 43 ] , to train these models. However, these resources did not fully cover all stages of the curation process and were insufficient for several tasks. For instance, no public benchmark was available for the initial screening stage, while only 1,600 labeled examples were available for caption separation and alignment despite the complexity of the task. We therefore supplemented existing benchmarks with PMC-derived synthetic annotations. From 39,360 sampled figures, we generated 99,321 labeled examples for the initial screening, multi-panel figure detection, and caption separation and alignment stages, complementing the 50,995 manually labeled samples available from existing benchmarks ( Extended Data Table 2 ). This expanded training set substantially improved performance across the affected pipeline stages. For example, augmenting with PMC-derived synthetic data improved the validation F1 score for the caption separation and alignment task by 3.82 percentage points over using the labeled data alone (see Supplementary Information, Section 1 for details). Importantly, the final reported performance of each curation component was assessed on held-out real instances with manual labels."},{"type":"heading","level":4,"text":"Initial screening"},{"type":"para","text":"The initial screening stage was designed to identify candidate medical figures while avoiding the need to download all PMC images, which is time-consuming and storage-intensive at scale. We first collected all open-access articles in PMC available as of June 2024, yielding 6,106,189 articles. Among these, 5,099,175 articles carried permissive licenses that allowed downstream use and redistribution, including CC BY (3,940,226), CC BY-NC (742,418), CC BY-NC-SA (270,631), CC0 (144,680), and CC BY-SA (1,220). We excluded articles licensed under CC BY-NC-ND (592,009) and CC BY-ND (8,188), as these licenses prohibit redistribution of derived resources, as well as records labeled as NO-CC CODE (406,817), which typically indicate missing, custom, or non-machine-readable license terms. We then trained a text classifier that used figure captions and associated inline reference text to distinguish candidate medical figures from non-medical figures. Applied to the 5.1 million permissively licensed articles, this stage retained 3.7 million candidate image-text pairs for downstream processing."},{"type":"heading","level":4,"text":"Multi-panel figure detection"},{"type":"para","text":"After initial screening, we downloaded the retained candidate images and determined whether each image was single-panel or multi-panel. Single-panel images could be directly retained for downstream curation, whereas multi-panel figures required further decomposition and caption alignment. We therefore trained an image classifier to distinguish single-panel from multi-panel figures. This stage identified 3.1 million multi-panel figures."},{"type":"heading","level":4,"text":"Multi-panel figure separation"},{"type":"para","text":"We next decomposed the identified multi-panel figures into individual subfigures. A figure separation model was trained to predict bounding boxes for panel-level image crops. In total, 3.1 million multi-panel figures were decomposed into an average of 9.2 sub-panels, yielding 29 million subfigures. This high panel density is consistent with the layouts commonly observed in biomedical articles, including microscopy grids, multi-condition comparisons, and multi-modality clinical figures."},{"type":"heading","level":4,"text":"Caption separation and alignment"},{"type":"para","text":"After figure separation, we processed the corresponding full captions to construct panel-level image-text pairs. This stage involved two linked tasks: caption separation, which decomposes a full figure caption into subcaptions, and subfigure-subcaption alignment, which associates each subcaption with its corresponding subfigure. Rather than performing these tasks sequentially [ 88 , 52 ] , we addressed them jointly using an MLLM. The model takes as input the original multi-panel figure, the extracted subfigures, and the full caption, and directly generates a list of subcaptions corresponding to the input subfigures. Generated subcaptions were aligned to subfigures according to their order. To prioritize pair fidelity, we removed samples in which the number of generated subcaptions did not match the number of input subfigures. These cases typically involved figures with many subpanels or ambiguous semantic boundaries, where reliable panel-level correspondence is inherently difficult to establish. Although this filtering step reduced the total number of retained pairs, it improved the fidelity of the resulting image-text pairs. After this process, we obtained 12.5 million high-quality subfigure-subcaption pairs."},{"type":"heading","level":4,"text":"Medical figure classification"},{"type":"para","text":"Because multi-panel figures often contain mixtures of medical and non-medical visual elements, such as charts, diagrams, or experimental schematics, we further filtered the decomposed subfigures. We trained an image classifier to determine whether each subfigure was medically relevant. After this final filtering step, we retained 7.3 million medical subfigure-subcaption pairs."},{"type":"heading","level":2,"text":"2.2 Stage-level Benchmark and Pipeline Performance"},{"type":"heading","level":4,"text":"Benchmark curation"},{"type":"para","text":"A central challenge in large-scale medical multimodal data curation is that errors introduced at intermediate stages can propagate through the pipeline and degrade the final image-text pairs [ 3 , 97 ] . To systematically evaluate and optimize each component of MedPMC, we constructed a benchmark suite for component-level assessment across the full curation pipeline. This benchmark enables direct comparison between our purpose-built models and prior approaches used in PMC-derived dataset construction, while also providing a standardized basis for future refinement of individual pipeline components. The benchmark integrates three complementary data sources: (i) repurposed existing labeled benchmarks, including ImageCLEF 2016, MedICaT, and DocFigure; (ii) newly constructed PMC-derived synthetic annotations; and (iii) newly annotated samples curated by human annotators (see Extended Data Table 2 for dataset statistics and composition). Similar to our strategy for supplementing pipeline training data, we added PMC-derived synthetic annotations and newly annotated samples because existing benchmarks did not fully cover all pipeline stages or provide sufficient examples for robust evaluation, particularly for the caption separation and alignment task, which requires structured caption parsing and subfigure-level alignment rather than simple image classification and is therefore substantially more difficult and costly to annotate. The validation set was designed to assess in-distribution performance relative to the pipeline training data and included 11,357 labeled examples and 19,562 synthetic examples across all stages. The held-out test set comprised 24,068 labeled samples from existing benchmarks and 2,010 newly annotated examples. Additional details on benchmark construction and annotation procedures are described in the Methods section."},{"type":"heading","level":4,"text":"Results"},{"type":"para","text":"Using this benchmark, we evaluated each component of the MedPMC pipeline against baseline methods, including previous PMC-based data curation approaches (e.g., MedICaT and PMC-OA [ 52 ] ), task-specific state-of-the-art models, and component-specific ablations. Detailed descriptions of all baseline methods and complete component-level results are provided in Supplementary Information, Section 1 . For the initial screening stage, we compared multiple biomedical language models as well as different textual inputs, including captions alone and captions combined with inline text. The PubMedBERT classifier [ 30 ] using both captions and inline text achieved the best performance (F1 = 93.2), substantially outperforming the keyword-matching baseline used in MedICaT (F1 = 61.7). For multi-panel figure detection, we compared text-based approaches, multiple vision model architectures, and the effect of synthetic training data. The Vision Transformer [ 24 ] trained using both labeled and synthetic training data achieved the best performance (F1 = 96.5). For figure separation, the previous state-of-the-art model, YOLO-OCR-D [ 67 ] , reported strong performance (mAP = 90.2), but it is not publicly available. We therefore trained a YOLOv10-based detector [ 91 ] using the ImageCLEF dataset, achieving comparable performance (mAP = 89.8) while enabling a fully reproducible pipeline. For caption separation and alignment, we compared the rule-based caption parsing and CLIP-based alignment strategy used by PMC-OA, GPT-4 Turbo (hereafter referred to as GPT-4T) in a zero-shot setting [ 38 ] , and supervised InternVL-2.5-4B models [ 17 ] trained using manually annotated MedICaT data with or without synthetic augmentation. The supervised InternVL-2.5-4B model achieved the best performance (F1 = 81.4, ROUGE-L = 85.3), substantially outperforming both the PMC-OA pipeline (F1 = 48.5, ROUGE-L = 44.4) and GPT-4T zero-shot (F1 = 75.2, ROUGE-L = 76.9). Finally, for medical figure classification, we compared four vision models, with the Vision Transformer again achieving the best performance (F1 = 96.5)."},{"type":"heading","level":2,"text":"2.3 Dataset Composition, Coverage, and Scalability"},{"type":"para","text":"The final MedPMC dataset comprises 3.1 million multi-panel figure-caption pairs, 0.6 million single-panel image-caption pairs, and 7.3 million subfigure-subcaption pairs extracted from multi-panel figures, totaling approximately 11 million medical image-text pairs. Although MedPMC is smaller than broader PMC-derived collections such as PMC-15M and BIOMEDICA [ 62 ] , those resources are not restricted to medical images and include a large proportion of non-medical content. To quantify this difference in dataset composition and assess the clinical relevance of the curated images, we conducted a manual evaluation with five annotators, including medical experts. We analyzed 2,906 samples from MedPMC and 432 samples from BIOMEDICA for comparison. Multi-panel images were excluded from this analysis because assigning a single image category to a compound figure is inherently ambiguous."},{"type":"para","text":"As shown in Fig. 2 a , BIOMEDICA contained a substantial proportion of non-medical images (80.3%), with statistical figures, graphs, and charts accounting for 42% of sampled images. In contrast, MedPMC was predominantly composed of medical images, with only 4.7% categorized as non-medical. A substantial amount of images covered clinically relevant modalities routinely encountered in clinical practice, including radiology, ophthalmology, pathology, endoscopy, dermatology, physiological signals, surgical imaging, and other forms of clinical photography. Additionally, non-pathology microscopy provides complementary coverage of medical imaging beyond routine clinical practice. Fig. 2 b further breaks down the radiology subset into finer-grained modalities. MRI (31.7%), CT (23.4%), and X-ray (15.1%) constituted the largest radiology categories, while ultrasound, PET/SPECT, fMRI, angiography, and mammography were also represented. Detailed distributions across additional domains are provided in Extended Data Table 3 . Fig. 2 c illustrates the scalability of the MedPMC framework by quantifying the annual number of publications and corresponding figure-caption pairs processed by the pipeline. The volume of biomedical publications increased steadily over time, accompanied by a proportional increase in extracted image-text pairs. Since 2020, the framework has identified more than one million image-text pairs per year, demonstrating its capacity to continuously incorporate a vast amount of newly published biomedical literature."},{"type":"heading","level":2,"text":"2.4 Foundation Model Evaluation and Clinical Transfer Validation"},{"type":"para","text":"To evaluate whether high-fidelity MedPMC curation translates into stronger medical visual representations and downstream multimodal reasoning, we evaluated MedPMC through two representative model instantiations. First, we trained MedPMC-CLIP, a CLIP-style vision-language model, following the exact training protocol of BMC-CLIP [ 62 ] , including identical architecture, initialization, configuration, hyperparameters, and training schedule. Specifically, we initialized MedPMC-CLIP from the same OpenCLIP ViT-L/14 weights used by BMC-CLIP and replaced only the BIOMEDICA 24M training corpus with MedPMC. This design enabled a head-to-head comparison that isolated the effect of dataset quality. Second, we examined whether the visual representation learned by MedPMC-CLIP could serve as a stronger visual backbone for generative multimodal systems. We adopted the LLaVA-Med training pipeline [ 50 ] , which builds on LLaVA v1.5 [ 56 ] and performs the visual alignment and instruction tuning stages. While keeping the training procedure unchanged, we replaced only the original vision encoder with MedPMC-CLIP. This controlled design tested whether improved visual pretraining transfers to downstream medical visual question answering. Detailed results with 95% confidence intervals are provided in Supplementary Information, Section 2 ."},{"type":"heading","level":4,"text":"MedPMC-CLIP improves performance across medical specialties"},{"type":"para","text":"We evaluated MedPMC-CLIP on image classification across 26 public medical benchmarks spanning 11 specialties. Extended Data Table 4 lists the benchmarks, specialties, and task definitions included in the evaluation. Together, these benchmarks cover disease detection and classification, severity grading, tissue classification, anatomical structure recognition, cellular phenotyping, and morphological or ultrastructural identification. Clinically relevant tasks include pneumonia detection, breast malignancy classification, diabetic retinopathy grading, colon adenocarcinoma detection, lymph node metastasis detection, heart failure classification, and skin lesion classification. Fig. 3 a summarizes performance across the 26 benchmarks. For each benchmark, performance was first averaged within specialty, and the final score was computed by averaging across 11 specialties. MedPMC-CLIP consistently outperformed baseline models under this evaluation. Compared with BMC-CLIP, MedPMC-CLIP improved the specialty-macro average accuracy by 7.1 percentage points (95% CI, 6.0–8.2), F1 score by 10.5 percentage points (95% CI, 9.2–11.6), and AUC by 7.1 percentage points (95% CI, 6.3–8.0; paired bootstrap with 10,000 replicates). Fig. 3 b presents performance by specialty. MedPMC-CLIP outperformed BMC-CLIP in 10 of 11 specialties, indicating that the gains were not confined to a single domain but were broadly distributed across diverse medical fields."},{"type":"heading","level":4,"text":"MedPMC-CLIP strengthens downstream multimodal medical QA"},{"type":"para","text":"Fig. 3 c shows that replacing the vision encoder with MedPMC-CLIP yielded an observed improvement of 1.9 percentage points on MMMU (95% CI, − 4.1 -4.1 to 8.0 percentage points) and a substantially larger improvement of 16.9 percentage points on OmniMedVQA (95% CI, 14.9–18.7 percentage points; paired bootstrap with 10,000 replicates). This finding highlights the role of the vision encoder in medical multimodal reasoning and is consistent with prior studies showing that vision encoder choice [ 66 ] , as well as domain-specific adaptation of the visual backbone [ 105 ] , can substantially affect downstream multimodal LLM performance. The larger gain on OmniMedVQA was primarily driven by tasks in which visual recognition plays a central role. We analyzed performance according to the query categories provided by OmniMedVQA. Performance improved from 46.9% to 75.2% for modality recognition, corresponding to an increase of 28.3 percentage points (95% CI, 24.2–32.4); from 34.7% to 57.0% for anatomy identification, an increase of 22.3 percentage points (95% CI, 17.6–26.9); and from 26.0% to 40.8% for the “other biological attributes” category, an increase of 14.8 percentage points (95% CI, 7.6–22.0). These findings suggest that a stronger medical vision encoder can directly improve the model’s ability to interpret medical images. Importantly, the improvement was not limited to visually straightforward recognition tasks. Performance on the combined “disease diagnosis and lesion grading” category increased from 34.2% to 45.4%, corresponding to an improvement of 11.1 percentage points (95% CI, 8.5–13.8). In contrast, MMMU questions are often framed around patient presentations and require complex clinical reasoning; thus, although the point estimate favored the stronger vision encoder, its confidence interval included zero, suggesting that further gains may require more effective end-to-end instruction tuning."},{"type":"heading","level":4,"text":"MedPMC-CLIP improves morphology-guided retrieval of clinical dermatology images"},{"type":"para","text":"Beyond standard benchmarking, we conducted a clinical evaluation using 10,524 dermatology patient images derived from a multi-site, hospital-based dermatology consultation cohort within the Yale New Haven Health System (YNHHS). Specifically, we evaluated MedPMC-CLIP in a clinically motivated morphology-to-skin-image retrieval task, in which textual morphology descriptions are used to retrieve relevant clinical images. This setting serves as a probe of representation learning and reflects a clinically meaningful retrieval use case, as retrieving similar medical images has long been studied as a form of decision support and education in dermatology and broader medical imaging contexts [ 82 , 28 , 70 , 59 ] . As shown in Fig. 3 d , MedPMC-CLIP outperformed BMC-CLIP, improving Recall@1 by 3.4 percentage points (95% CI, 2.7–4.1), Recall@5 by 11.7 percentage points (95% CI, 10.7–12.6), and Recall@10 by 10.7 percentage points (95% CI, 9.8–11.6; paired bootstrap with 10,000 replicates). These results indicate that improvements observed on public benchmarks transfer to a clinically realistic image retrieval setting using patient data."},{"type":"heading","level":4,"text":"MedPMC dermatology images better match the visual distribution of clinical images"},{"type":"para","text":"Although MedPMC images are derived from the biomedical literature rather than routine clinical practice, MedPMC-CLIP showed improved performance on downstream clinical dermatology retrieval. To better understand the factors driving this transferability, we examined whether dermatology images curated from MedPMC occupy a visual space similar to that of real-world clinical dermatology photographs. Specifically, we compared MedPMC with three widely used public dermatology datasets containing skin photographs (Fitzpatrick17k [ 29 ] , SCIN [ 94 ] , and DermNet [ 23 ] ) and assessed their respective visual similarity to the YNHHS clinical cohort. For each public dataset, we sampled 10,000 images; for MedPMC, we first identified skin photographs using the MedPMC-CLIP model and then randomly sampled 10,000 images from this curated subset."},{"type":"para","text":"In the DINOv2 embedding space, MedPMC demonstrated the greatest visual similarity to YNHHS clinical images among all evaluated external sources. Quantitatively, YNHHS clinical images yielded the lowest median nearest-neighbor cosine distance to MedPMC (0.239), compared with 0.262 for Fitzpatrick17k, 0.270 for SCIN, and 0.299 for DermNet ( Fig. 4 a ). Consistently, when each YNHHS clinical image was assigned to the external source of its nearest neighbor, MedPMC accounted for the majority of assignments (51.5%), followed by Fitzpatrick17k (21.2%), SCIN (20.7%), and DermNet (6.5%) ( Fig. 4 b ). UMAP projections further revealed that while YNHHS images formed a broad visual manifold that reflects the inherent variability of clinical photography, MedPMC showed extensive overlap across this space. In contrast, existing public datasets exhibited more uneven distributions, leaving substantial portions of the clinical manifold underrepresented ( Fig. 4 c–f ). These findings indicate that MedPMC captures a diverse visual distribution that closely aligns with real-world clinical data, providing a potential explanation for its downstream transfer to clinical dermatology tasks."},{"type":"heading","level":1,"text":"3 Discussion"},{"type":"para","text":"We developed MedPMC, an automated and systematic framework for curating high-fidelity medical multimodal data from permissively licensed biomedical literature, together with MedPMC-CLIP, a vision-language foundation model trained on the resulting MedPMC corpus to demonstrate its utility. The main finding is that biomedical literature, when transformed into medically relevant and panel-level aligned data, can support effective and reproducible development of medical multimodal models. These data-level improvements translated into consistent model-level gains. Using the exact training protocol of BMC-CLIP and changing only the training corpus, MedPMC-CLIP improved average zero-shot AUC by 7.1 percentage points over BMC-CLIP across 26 public medical benchmarks spanning 11 specialties. Gains in AUC were observed in 10 of the 11 specialties, despite MedPMC-CLIP being trained on fewer than half as many image-text pairs. These benefits also extended to MLLMs: within the LLaVA-Med framework, replacing only the vision encoder with MedPMC-CLIP—while holding the remaining model and training pipeline fixed—improved performance by 1.9 percentage points on MMMU and 16.9 percentage points on OmniMedVQA. Importantly, these improvements were not confined to public benchmarks. Using 10,524 dermatology patient records from a multi-site, hospital-based consultation cohort, MedPMC-CLIP improved performance in zero-shot retrieval. Embedding-space analysis further showed that MedPMC dermatology images were more closely aligned with clinical dermatology images than several widely used public dermatology datasets. These findings suggest that MedPMC can help address a central bottleneck in medical multimodal AI—the scarcity of large-scale, clinically meaningful multimodal data [ 107 , 83 , 1 , 34 ] . Beyond serving as a corpus, MedPMC provides a modular curation framework with component-level benchmarks that can be updated as new biomedical articles become available. Because it is derived from permissively licensed literature, the resulting resources can be shared more openly, supporting reproducibility, adaptation, and continued model development across both benchmark and clinically realistic settings."},{"type":"heading","level":4,"text":"Data curation as reusable infrastructure for medical AI"},{"type":"para","text":"A key implication of this study is that data curation should be treated as a core engineering problem in medical multimodal AI. PMC offers a scalable public source of biomedical image-text data, and the resource continues to grow as new biomedical knowledge is published [ 71 ] . However, directly repurposing PMC figures introduces several failure modes, including non-medical visual content, compound multi-panel figures, weak subfigure-subcaption correspondence, and dataset staleness. MedPMC addresses these failure modes through a modular five-stage pipeline. The initial screening step identifies medically relevant figures before image download, reducing unnecessary data transfer and storage. Multi-panel figure detection and separation convert compound figures into panel-level visual units. Caption separation and subfigure-subcaption alignment improve image-text correspondence at the level at which supervision is most meaningful. Medical figure classification further filters decomposed panels to remove remaining non-medical visual elements. Because each stage is supported by component-level benchmarks, the pipeline can be evaluated, audited, and improved rather than treated as an opaque preprocessing procedure. This infrastructure framing distinguishes MedPMC from a one-time dataset release. Biomedical literature continues to expand, and static datasets can become progressively incomplete as new diseases, technologies, imaging modalities, and clinical terminology emerge."},{"type":"para","text":"To keep MedPMC current, we plan to release updated corpus versions every six months, aligned with the semiannual PMC baseline snapshots. After the initial June 2024 release, each update will process newly available permissively licensed articles not included in prior versions and will be documented by article cutoff date, license filters, pipeline component checkpoints, and processing configuration. Based on recent publication trends, these updates are expected to identify about one million additional medical image-text pairs per year. The modular design of MedPMC also allows individual pipeline components to be replaced or improved over time."},{"type":"heading","level":4,"text":"Relationship to existing PMC-derived resources"},{"type":"para","text":"MedPMC builds on a growing literature that has established biomedical articles as a valuable source of medical multimodal supervision. Early resources such as ROCO [ 77 ] and MedICaT [ 88 ] showed that figures and captions from biomedical literature could support medical vision-language learning, although these datasets were relatively small or focused on narrower domains. More recent PMC-derived resources, including PMC-OA [ 52 ] , PMC-15M [ 108 ] , BIOMEDICA [ 62 ] , and Open-PMC-18M [ 6 ] , substantially expanded the scale of literature-derived image-text data, enabling medical vision-language models such as PubMedCLIP [ 25 ] , BiomedCLIP [ 108 ] , and BMC-CLIP [ 62 ] , as well as multimodal medical LLMs such as Med-Flamingo [ 69 ] , LLaVA-Med [ 50 ] , BiomedGPT [ 106 ] , HealthGPT [ 51 ] , and MedGemma [ 84 ] . However, as detailed in Extended Data Table 1 , these large-scale data resources remain limited in medical relevance, compound-figure processing, and reproducible curation pipelines."},{"type":"para","text":"MedPMC extends this line of work by emphasizing fidelity, reproducibility, and updatability rather than scale alone. First, many existing large-scale PMC-derived datasets include substantial non-medical visual content, such as graphs, charts, molecular illustrations, workflow diagrams, and system schematics. In the manual analysis, 80% of sampled BIOMEDICA figures were non-medical, compared with 4.7% for MedPMC. Filtering strategies based on manually curated keywords can miss medically relevant figures outside predefined terminology, whereas image-only filtering requires large-scale image download before screening and may not capture clinical context. MedPMC instead formulates medically relevant figure selection as a text classification problem using captions and inline reference text, which provides contextual information before image download and improves scalability. This text-based classifier reached a recall of 93.1 and an F1 score of 93.2, substantially outperforming keyword-based filtering. Second, existing resources differ substantially in how they handle compound figures. Some exclude multi-panel figures, which discards informative medical images; others retain compound figures as whole figure-caption pairs, which can weaken image-text correspondence when panels depict different modalities, disease states, time points, or experimental conditions. Panel-level approaches based on sequential caption splitting and heuristic or CLIP-based alignment can also introduce alignment errors, while approaches that run an MLLM separately for every subfigure [ 6 ] incur inference costs that scale with panel count. MedPMC instead separates compound figures and jointly performs caption separation and subfigure-subcaption alignment with a supervised 4B-parameter MLLM that considers all panels together. This approach reached 81.4 F1 and 85.3 ROUGE-L, outperforming rule- and CLIP-based matching used in prior pipelines as well as GPT-4, while remaining efficient at scale by avoiding per-subfigure inference. Third, most prior PMC-derived resources are released as static snapshots. Because their full curation pipelines, trained preprocessing models, and stage-specific evaluation resources are often not fully available, reproducing or extending the dataset construction process can be difficult. MedPMC was designed to make the curation process itself reusable and to provide a reproducible framework for generating, evaluating, and updating medical multimodal supervision at scale."},{"type":"heading","level":4,"text":"Complementary strengths of MedPMC and clinical data"},{"type":"para","text":"Importantly, MedPMC is not intended to replace clinical datasets. Literature-derived and clinical resources serve complementary roles in medical multimodal model development. Clinical datasets remain closest to deployment because they capture real patient populations, local imaging protocols, clinical workflows, and case mix. However, they are often concentrated within a limited number of institutions, specialties, and modalities, and their accompanying text is frequently limited to diagnostic labels or short reports rather than detailed descriptions of image findings. Despite several pioneering efforts [ 40 , 33 , 5 , 92 ] , such datasets remain difficult to assemble and share at scale because of privacy, governance, licensing, and annotation constraints."},{"type":"para","text":"PMC-derived data may offer a distinct but complementary form of supervision. Compared with many clinical datasets, literature-derived resources can provide greater scale, richer image-associated context, including experimental settings and diagnostic descriptions, and broader coverage across diseases, specialties, modalities, and emerging concepts, including rare conditions and technologies that may be sparsely represented in existing clinical resources. Because they are derived from publicly available and permissively licensed literature, they can also be shared and reproduced more readily than many institution-specific clinical datasets. One practical development pathway is therefore to use MedPMC-like resources for broad pretraining and representation learning, followed by clinical adaptation and validation in the intended target setting. In this workflow, literature-derived data provide scale, coverage, and descriptive supervision, while clinical data provide local grounding and direct alignment with real-world practice. This complementary role is further supported by preliminary few-shot adaptation results in Supplementary Information, Section 3 , which show a modest performance advantage over OpenCLIP and BMC-CLIP."},{"type":"para","text":"Our clinical dermatology evaluation demonstrates how literature-derived pretraining can support downstream clinical applications. Although MedPMC is derived from biomedical publications rather than routine clinical workflows, MedPMC-CLIP transferred to morphology-guided retrieval on internal patient data. The embedding-space analysis further suggested that MedPMC dermatology images overlap more broadly with internal clinical dermatology images than several widely used public dermatology datasets. This finding does not imply that literature-derived images are equivalent to clinical data or sufficient for deployment without target-domain validation. Rather, it indicates that carefully curated biomedical literature can provide pretraining supervision that is relevant to downstream clinical tasks and can complement, rather than replace, institution-specific clinical datasets."},{"type":"heading","level":4,"text":"Potential applications"},{"type":"para","text":"First, the corpus can be used to train or initialize medical vision-language encoders, visual backbones for MLLMs, and specialty-specific foundation models [ 21 , 78 ] . Second, because each image is linked to article text, figure captions, subcaptions, and medical figure categories, MedPMC can be filtered into specialty-, modality-, or concept-specific subsets. This may be useful when dedicated public datasets are limited, such as in rare diseases, surgical imaging, pathology subdomains, clinical skin photography, or specific imaging modalities [ 72 , 98 ] . Third, the structure of compound figures provides a source of supervision that is difficult to obtain from standard single-image datasets. Many biomedical figures show related images, such as disease progression, pre- and post-treatment comparisons, multi-view imaging, paired radiology-pathology findings, or contrasts between disease subtypes and controls. These structures could support models that reason over image sets rather than isolated images [ 18 ] . Fourth, the aligned image-text pairs can be indexed as a searchable corpus of biomedical visual evidence. Such an index could support similar-case search, visual example retrieval, and multimodal retrieval-augmented generation, in which MedPMC functions both as a training resource and as an external knowledge source for downstream multimodal systems [ 103 ] ."},{"type":"heading","level":4,"text":"Limitations and Future Work"},{"type":"para","text":"First, this study focuses on image-text data, whereas clinical practice routinely involves broader and more heterogeneous information streams, including structured tables, laboratory values, temporal clinical context, and longitudinal patient records. Some of these modalities, such as tables and structured content embedded in biomedical articles, could be incorporated into a similar literature-based curation framework. Extending MedPMC to capture these additional modalities is therefore an important direction for future work. Other data types, particularly longitudinal patient records and real-world clinical workflows, are rarely available in a consistent or shareable form in the public literature. Addressing these gaps will require complementary frameworks for curating and integrating clinical data under appropriate privacy, governance, and institutional constraints."},{"type":"para","text":"Second, because MedPMC is derived from biomedical articles, the corpus reflects both the strengths and limitations of the source literature. Biomedical figure captions vary in quality and may contain incomplete descriptions, inconsistent terminology, non-visual information, or diagnostic context that is not directly inferable from the image alone. In addition, biomedical publications introduce selection bias in what is visually reported. Published figures may preferentially show representative examples, diagnostically salient findings, unusual cases, positive or publishable results, or visually clear images, and may differ from routine clinical images in terms of case mix, acquisition setting, image quality, presentation style, and patient population [ 74 ] . Prior work has shown that improving caption quality or restructuring text for multimodal learning can improve model performance [ 100 , 15 ] ; these approaches are complementary to MedPMC because they depend on accurately aligned image-caption pairs as input. Future work could therefore combine MedPMC-style high-fidelity image-text alignment with caption refinement, structured report generation, and region- or concept-level supervision."},{"type":"para","text":"Third, although we systematically evaluated MedPMC across different model types, including contrastive vision-language encoders and MLLMs, and across multiple settings, including public benchmark evaluations, downstream tasks, and clinically realistic applications, our experiments do not exhaustively cover all possible model architectures or applications. This reflects the role of MedPMC as reusable data infrastructure rather than as a single task-specific model. Given the rapid development of medical foundation models, it is not feasible for one study to evaluate every emerging architecture or clinical use case [ 9 , 8 , 16 ] . In addition, potential overlap between large-scale pretraining corpora and public benchmarks is an increasingly recognized challenge for foundation-model and LLM evaluation [ 85 , 49 , 14 ] . We carefully selected public benchmarks that were largely derived from institutional clinical collections, established medical imaging repositories, or challenge datasets rather than directly from PMC articles, and we further included an internal YNHHS clinical cohort that was independent of the PMC-derived pretraining corpus. Nevertheless, exact overlap cannot be fully excluded. We therefore interpret the public benchmark results together with architecture-matched comparisons, downstream QA evaluation, and independent clinical transfer experiments."},{"type":"para","text":"Fourth, although our experiments show that models trained with MedPMC transfer to clinically motivated tasks, including morphology-guided dermatology image retrieval using internal patient data, these evaluations do not establish deployment readiness. MedPMC should be viewed as a source of broad pretraining supervision that complements, rather than replaces, target-domain clinical data. Additional clinical evaluations of model adaptation, such as further fine-tuning, and clinician-in-the-loop workflows will also be important for quantifying how public medical resources and models derived from them adapt to real-world clinical settings and support clinical decision-making [ 58 ] . By releasing the corpus, curation pipeline, component-level benchmarks, pretrained models, and source-license metadata, we aim to support continued external evaluation, model comparison, pipeline refinement, and clinically grounded adaptation by the research community."},{"type":"heading","level":1,"text":"4 Methods"},{"type":"heading","level":2,"text":"4.1 MedPMC Pipeline Implementation"},{"type":"para","text":"We trained specialized models for each stage of the MedPMC curation pipeline. For each task, we leveraged existing datasets when they were well aligned with the target prediction problem. However, available datasets were often limited in scale, restricted to specific domains, or not fully representative of PMC figure layouts and captioning styles. To improve robustness in the target curation setting, we additionally constructed new datasets. Dataset statistics for each pipeline component are provided in Extended Data Table 2 . The detailed prompts used for synthetic data generation are provided in Supplementary Information, Section 4 . Importantly, all datasets, trained models, and code for running and evaluating each pipeline component are publicly released, supporting reproducibility, reuse, benchmarking, and transparent inspection of implementation details."},{"type":"heading","level":4,"text":"Initial screening"},{"type":"para","text":"The initial screening model was designed to identify candidate medical figures before image download. For each figure, we concatenated the figure caption with the inline reference text from the article body and formulated the task as binary classification of medical relevance. To construct the training data, we randomly sampled 39,360 figures from articles indexed by Open-i ( https://openi.nlm.nih.gov ), an open-access biomedical image search engine, using the “Exclude Graphics” option to reduce the retrieval of plots, charts, and other non-image graphics. We then generated synthetic medical-relevance labels for the sampled figures using GPT-4T ( gpt-4-turbo-2024-04-09 ). Following our definition of medical figures, GPT-4T assigned labels after jointly considering both the image and its associated text. We initialized the classifier with PubMedBERT [ 30 ] , which was pretrained on PubMed abstracts and full-text articles ( https://huggingface.co/microsoft/BiomedNLP-BiomedBERT-base-uncased-abstract-fulltext ). The model was fine-tuned with a batch size of 64 and a learning rate of 3e-5."},{"type":"heading","level":4,"text":"Multi-panel figure detection"},{"type":"para","text":"The multi-panel figure detection model was designed to distinguish single-panel from multi-panel figures before subfigure separation. We formulated this step as binary image classification and trained the model using 16,775 samples from the compound figure detection subtask of the ImageCLEFmed challenge, together with 39,360 additional synthetically labeled examples. The synthetically labeled examples were drawn from the same Open-i sample used for initial screening, but were re-annotated for panel structure rather than medical relevance. GPT-4 was provided with each figure and its caption and assigned a binary label indicating whether the figure contained a single panel or multiple panels. We adopted a Vision Transformer ( https://huggingface.co/timm/vit_base_patch16_rope_reg1_gap_256.sbb_in1k ) as the backbone model [ 24 ] and fine-tuned the model using the PyTorch Image Models repository (timm) [ 95 ] for 100 epochs with a batch size of 64. All other hyperparameters followed the default settings of the training framework, with the learning rate automatically scaled according to the effective global batch size."},{"type":"heading","level":4,"text":"Multi-panel figure separation"},{"type":"para","text":"The multi-panel figure separation model took figures classified as multi-panel as input and predicted bounding boxes corresponding to individual subfigures. For training, we used the dataset provided for the figure separation task in the ImageCLEFmed challenge [ 27 ] . We did not incorporate additional synthetic annotations at this stage because they did not further improve validation performance. We used YOLOv10 as the backbone model [ 91 ] . The model was initialized from pretrained weights and fine-tuned using the default training configuration in Ultralytics v8.2.73 ( https://github.com/ultralytics/ultralytics ). Training was performed for 100 epochs with a batch size of 16 and an input resolution of 640 × 640; automatic optimizer selection chose AdamW with an initial learning rate of 2e-3 [ 60 ] ."},{"type":"heading","level":4,"text":"Caption separation and alignment"},{"type":"para","text":"The caption separation and alignment model took as input the original multi-panel figure, its full caption, and the separated subfigures, and generated the corresponding subcaptions. For training, we used 1,361 manually annotated samples from the MedICaT dataset [ 88 ] . To improve scale and domain diversity beyond the predominantly radiology-focused MedICaT dataset, we generated an additional 20,601 synthetic annotation examples. Specifically, we applied the trained multi-panel detection model to the Open-i figures used in the previous stage, selected those predicted as multi-panel, and separated them into subfigures using the trained YOLOv10 model. GPT-4 was then provided with the original figure, the separated subfigures, and the full figure caption to generate aligned subcaptions for each subfigure. We adopted InternVL-2.5-4B-MPO as the backbone model [ 17 ] . The model was then trained using the official implementation of InternVL ( https://github.com/OpenGVLab/InternVL ) with the default training configuration, using three training epochs, a total batch size of 128, a learning rate of 4e-5, and BF16 mixed-precision training."},{"type":"heading","level":4,"text":"Medical figure classification"},{"type":"para","text":"The medical figure classification model performed binary image classification to determine whether an input image or subfigure was medically relevant. We constructed a training set of 26,453 samples by combining data from ImageCLEF, MedICaT, and DocFigure [ 43 ] . A Vision Transformer was used as the backbone model. We trained the model using the timm library with the same setup as the multi-panel figure detection model, except that training was performed for 300 epochs."},{"type":"heading","level":4,"text":"Validation and model selection"},{"type":"para","text":"For each pipeline stage, we evaluated multiple backbone models, input configurations, and other task-specific design choices, and selected the final component based on validation-set performance. The validation sets combined existing benchmark data with PMC-derived synthetic annotations generated using the same procedures as the corresponding training data. To prevent overlap between training and validation samples originating from the same source article, synthetic annotations were partitioned at the article level. Validation-set composition and sample counts are provided in Extended Data Table 2 . Initial screening, multi-panel figure detection, and medical figure classification were evaluated using precision, recall, and F1 score. Following prior work [ 67 ] , multi-panel figure separation was evaluated using accuracy and mean average precision (mAP): accuracy measured agreement between the predicted and reference panel structure, whereas mAP quantified the localization of predicted subfigure bounding boxes relative to the ground-truth annotations. Following MedICaT, the caption separation and alignment task was evaluated using F1 score and ROUGE-L, both computed between the predicted and reference subcaptions. The selected models substantially outperformed the comparison methods across the pipeline stages. Detailed descriptions of the baseline models and comparative analyses are provided in Supplementary Information, Section 1 ."},{"type":"heading","level":2,"text":"4.2 Benchmark Annotation"},{"type":"para","text":"To enable systematic evaluation of the five components of the MedPMC pipeline, we constructed dedicated test sets using existing benchmark data and newly annotated examples from PMC. Whereas the validation sets were used for model selection, hyperparameter tuning, and other development decisions, test sets were kept fully held out and used only for final performance evaluation. Newly annotated PMC samples were drawn from articles indexed by Open-i, following the same initial sampling strategy used for training-data construction. However, all articles represented in the test sets were excluded from both the training and validation sets to prevent article-level overlap."},{"type":"para","text":"The manual annotation covered the first four stages: (1) initial screening, (2) multi-panel figure detection, (3) multi-panel figure separation, and (4) caption separation and alignment. For the medical figure classification task, manual annotation was bypassed because a substantial dataset of approximately 19,000 samples had already been secured from public resources. New annotations were performed by 13 annotators (S.S.A., Y.W., Y.K., T.G., Z. Cao., M.D., Y.F., Y.H., R.S., G.Y., K.W.J., Y.L., and Y.T.). Before annotation, all annotators completed a training session covering the annotation guidelines and representative examples. Each sample was independently annotated by two annotators, and disagreements were adjudicated by a senior annotator (H.K.). After each annotation batch, the adjudicated decisions and their rationales were shared with the annotators to promote consistency across subsequent batches."},{"type":"para","text":"For the initial screening, four annotators (S.S.A., Z. Cao., Y.F., and Y.H.) reviewed image-caption pairs to annotate each sample as either medical or non-medical. Under our operational definition of medical figures, non-human-centered subjects, such as animal models, were strictly classified as non-medical. For this task, a total of 600 samples—comprising a mixture of single-panel and multi-panel figures to faithfully reflect real-world data distributions—were partitioned into batches of 100. Each annotator evaluated three separate batches, culminating in 300 annotations per annotator. For multi-panel figure detection, two annotators (R.S. and G.Y.) labeled a total of 500 figures as either single-panel or multi-panel. This dataset was partitioned into five batches of 100 figures each. The multi-panel figure separation and caption separation and alignment stages were performed concurrently by the same cohort of annotators within a unified interface. For this joint task, seven annotators (Y.W., Y.K., T.G., M.D., K.W.J., Y.L., and Y.T.) processed 455 compound figures across six batches. For each figure, annotators delineated the bounding boxes of individual subfigures and subsequently extracted and aligned the corresponding subcaptions. If a figure was determined not to be multi-panel, this workflow was skipped."},{"type":"heading","level":2,"text":"4.3 Dataset Composition Analysis"},{"type":"para","text":"MedPMC consists of multi-panel figures, single-panel figures, and subfigures extracted from multi-panel figures. For dataset composition analysis, we excluded multi-panel figures because assigning a single category to a compound figure is inherently ambiguous. We randomly sampled 1,232 single-panel figures and 1,674 subfigures from MedPMC, totaling 2,906 samples. Each group was evaluated separately, and the final distribution was obtained by weighting the results according to the proportions of single-panel figures and subfigures in the full dataset. For BIOMEDICA, we similarly excluded multi-panel figures and restricted the analysis to single-panel images. To do so, we applied our compound figure detection model to identify and filter out multi-panel images, resulting in 9.3 million images (38.7%) retained for analysis. From this subset, we randomly sampled 432 images for manual evaluation."},{"type":"para","text":"A total of five annotators (P.X., S.S.A., R.J., Y.D., and J.M.) participated in the study, including two hospital residents and one final-year medical student. Each annotator was presented with both the image and its corresponding caption through a dedicated annotation interface. Annotators were asked to assign each sample to one of the predefined categories. If a sample did not fit any predefined category, annotators could select an “Other” option and provide a brief specification. The three medical experts additionally reviewed and resolved ambiguous cases that were initially marked as uncertain. The category schema was designed in collaboration with the medical experts, and the definition of non-medical images was informed by prior work [ 27 ] ."},{"type":"heading","level":2,"text":"4.4 Model Training and Downstream Evaluations"},{"type":"heading","level":4,"text":"Baseline models"},{"type":"para","text":"We compared MedPMC-CLIP against a range of baseline vision-language models for image classification and retrieval. These include general-domain foundation models such as (1) CoCa [ 102 ] , a contrastive-captioning model trained on web-scale datasets, and (2) OpenCLIP [ 19 ] , a standard CLIP architecture trained on LAION-2B. Within the medical domain, we evaluated (3) PMC-CLIP [ 52 ] , specifically the version initialized from the OpenAI CLIP ViT-L/14 checkpoint and subsequently fine-tuned on PMC-OA and ROCO data, (4) BiomedCLIP [ 108 ] pretrained on the PMC-15M corpus, and (5) MedSigLIP [ 84 ] , a sample-efficient medical encoder optimized via sigmoid pairwise loss. Finally, we compared our model against (6) BMC-CLIP [ 62 ] , which was trained on the BIOMEDICA-24M dataset; this model serves as the most direct baseline for evaluating MedPMC-CLIP, as we maintained the same model architecture and training pipeline configuration while isolating the pretraining dataset as the sole variable. For medical visual QA, we used LLaVA-Med [ 50 ] , a biomedical adaptation of LLaVA (v1.5) [ 56 ] that connects a CLIP vision encoder to a language model through a linear projection layer and is trained through a two-stage curriculum of visual alignment and instruction tuning."},{"type":"heading","level":4,"text":"Training details"},{"type":"para","text":"For MedPMC-CLIP, we followed the experimental setup of BIOMEDICA using the official code repository ( https://github.com/minwoosun/biomedica-etl ). Specifically, we initialized the model from OpenCLIP (ViT-L/14; checkpoint: commonpool_xl_clip_s13b_b90k and adopted the same configuration and hyperparameters (a batch size of 8,192 and a learning rate of 1e-6), while replacing the training data with MedPMC (11 million image-text pairs) For MLLM evaluation, we adopted the same training data and hyperparameters as LLaVA-Med [ 50 ] , following the official implementation ( https://github.com/microsoft/LLaVA-Med/tree/main ). Stage 1 (biomedical concept feature alignment) was performed on 467K samples using a batch size of 128 and a learning rate of 2e-3. Stage 2 (end-to-end instruction tuning) was then conducted on 56K samples for 3 epochs using a batch size of 128 and a learning rate of 2e-5. We used the LLaVA training codebase ( https://github.com/haotian-liu/llava )."},{"type":"heading","level":4,"text":"26 image classification benchmarks"},{"type":"para","text":"We used a comprehensive suite of medical benchmarks covering a diverse range of clinical modalities and specialties, including radiology, ophthalmology, pathology, dermatology, surgery, and microscopy. Detailed specifications regarding benchmark specialties, core tasks, and sample sizes are provided in Extended Data Table 4 . For each task, prompt templates were adopted directly from those supported by Micro-Bench [ 61 ] ; for domains not covered by existing frameworks, we custom-designed the evaluation prompts, all of which are publicly verifiable via our released evaluation code ( https://github.com/Yale-BIDS-Chen-Lab/MedPMC )."},{"type":"para","text":"Performance was evaluated using accuracy, F1 score, and area under the receiver operating characteristic curve (AUC). For binary and multiclass tasks, accuracy was calculated using the top-scoring class, F1 score was macro-averaged across classes, and AUC was calculated by pooling the one-hot-encoded labels and corresponding scores across all sample-class pairs. For multi-label tasks, predictions were obtained using a threshold of 0.5, and accuracy was calculated across all sample-label pairs. Positive-class F1 scores and AUCs were calculated independently for each label and then macro-averaged across labels; labels without both positive and negative samples were excluded from AUC calculation."},{"type":"heading","level":4,"text":"Downstream medical QA"},{"type":"para","text":"We evaluated MLLMs on MMMU [ 104 ] and OmniMedVQA [ 32 ] in a zero-shot setting. These benchmarks were selected because they are highly popular and widely recognized standards within the community, providing an exceptionally broad coverage of the medical domain rather than focusing on a single, narrow specialty. MMMU spans a diverse set of 30 domains; in our experiments, we focused on the clinical medicine subset and included only single-image questions, consisting of 314 samples. OmniMedVQA originally comprises more than 120,000 samples from 73 medical datasets, subdivided into 42 open-access and 31 restricted-access repositories. In this study, we used only the open-access datasets and excluded four COVID-19-related datasets curated from the literature to minimize the risk of data contamination (see “Contamination-aware evaluation design”), resulting in 38 open-access datasets. To ensure balanced and computationally efficient evaluation across these various source datasets, we randomly sampled a maximum of 100 examples per dataset. We then removed DRIMDB samples that assessed about image quality (good, bad, or outlier) but were incorrectly categorized as lesion grading, resulting in a final evaluation set of 3,593 samples. For all inference generations, we utilized greedy decoding with the temperature fixed at 0.0 to ensure deterministic results. Models were evaluated using only the image and the original question, with no prompting engineering."},{"type":"heading","level":4,"text":"Downstream clinical application: morphology-to-skin-image retrieval"},{"type":"para","text":"Retrieving similar medical images has long been studied as a form of decision support and education in dermatology and broader medical imaging contexts [ 82 , 28 , 70 , 59 ] . In this study, textual morphology descriptions were used as queries to retrieve relevant clinical dermatology images. The study was approved by the Yale Institutional Review Board (IRB) under protocol number 2000041054. We constructed the evaluation dataset from 10,524 real-world dermatology patient images from a multi-site, hospital-based dermatology consultation cohort within the Yale New Haven Health System (YNHHS). The dataset included multiple images from some patients, although each image was treated as an individual retrieval sample. All models were evaluated using the same set of images. For each sample, a clinical image was originally paired with a raw morphology description extracted from the corresponding clinical note. Because multiple clinical images can share the same morphological characteristics, using only the original, unnormalized image-description pairs as ground truth would underestimate clinically relevant matches. To address this and standardize the textual descriptions, the raw morphology descriptions were automatically mapped to standardized morphology concepts based on the SkinCon taxonomy [ 22 ] (e.g., papule, plaque, macule, and erythema) using an open-sourced large language model, gpt-oss-120b [ 4 ] . To ensure data quality and evaluate the reliability of this automated normalization, a hospital resident in dermatology (R.J.) manually reviewed and verified a sampled subset of the generated mappings. The detailed prompt templates for this LLM-based parsing are provided in Supplementary Information, Section 4 . This standardization mapping enabled images sharing the same set of morphology concepts to be grouped as clinically related. For retrieval evaluation, we adopted an established set-based matching criterion [ 54 , 48 ] . A retrieved image was considered correct if its associated morphology concept set contained the ground-truth concept set of the query image. This formulation allowed multiple retrieved images to be treated as correct, reflecting cases in which different images shared the same core morphological features even when additional attributes differed."},{"type":"heading","level":4,"text":"Embedding-space comparison with clinical dermatology images"},{"type":"para","text":"We further examined whether dermatology images curated from MedPMC occupy a visual space similar to that of real-world clinical dermatology images. To identify dermatology images within MedPMC, we used the dermatology-related categories and text prompts described in Extended Data Table 3 . All MedPMC images were scored using MedPMC-CLIP based on image-text similarity, and images classified as skin photographs were used to construct the MedPMC dermatology subset. We compared this subset against three widely used public dermatology datasets: Fitzpatrick17k [ 29 ] , a collection of community-contributed dermatology photographs annotated with skin type and disease labels; SCIN [ 94 ] , a dataset of smartphone-acquired skin images collected from diverse participants in real-world settings; and DermNet [ 23 ] , a large collection of educational dermatology images curated from the DermNet resource. To enable balanced comparisons across sources, 10,000 images were randomly sampled from each external dataset. YNHHS clinical dermatology images served as the reference distribution."},{"type":"para","text":"All images were embedded using DINOv2 (ViT-B/14) [ 76 ] . To quantify similarity between each external source and the YNHHS clinical image distribution, we computed the cosine distance from each YNHHS clinical image to its nearest neighbor within each external dataset. We then summarized the resulting nearest-neighbor distance distributions and compared median distances across datasets. In addition, each YNHHS clinical image was assigned to the external source containing its overall nearest neighbor across all external datasets, and the fraction of images assigned to each source was calculated. For visualization, DINOv2 embeddings were projected into two dimensions using UMAP [ 65 ] . The UMAP projection was fit on the combined image set and used to visualize the distribution of YNHHS clinical dermatology images together with images from each external source."},{"type":"heading","level":4,"text":"Statistical analysis"},{"type":"para","text":"We estimated 95% CIs for AUC using nonparametric bootstrap resampling with 10,000 replicates. In each replicate, samples were drawn with replacement from the corresponding test set, and AUC was recalculated using the same evaluation procedure as for the point estimate. Binary and multiclass datasets were resampled within each class to preserve the original class distribution, whereas multilabel datasets were resampled at the sample level while retaining all labels associated with each selected sample. To compare MedPMC-CLIP with BMC-CLIP, identical bootstrap samples were applied to both models within each benchmark. For each bootstrap replicate, benchmark-level differences in accuracy, macro F1 score, and AUC were calculated, averaged within each specialty, and then macro-averaged across the 11 specialties. For the medical QA experiments, individual model accuracies were reported with Wilson 95% CIs. Between-model accuracy differences were evaluated using paired bootstrap resampling with 10,000 replicates, with identical question samples applied to baseline LLaVA-Med and LLaVA-Med with the MedPMC-CLIP encoder. Paired differences were calculated for the overall MMMU and OmniMedVQA results and for each reported OmniMedVQA query category. For the morphology-guided retrieval experiment, differences in Recall@1, Recall@5, and Recall@10 were evaluated using query-level paired bootstrap resampling with 10,000 replicates. All 95% CIs were defined by the 2.5th and 97.5th percentiles of the corresponding bootstrap distributions. The full results are provided in Supplementary Information, Section 2 . We also release the evaluation code in our official GitHub repository to support reproducibility."},{"type":"heading","level":4,"text":"Contamination-aware evaluation design"},{"type":"para","text":"Potential data contamination between large-scale pretraining corpora and public evaluation benchmarks is an increasingly recognized challenge, especially in the evaluation of foundation models and large language models [ 85 , 49 , 14 ] . We therefore considered dataset provenance when designing the evaluation strategy. Most evaluation datasets used across the 26 benchmark tasks and downstream medical QA experiments were derived from institutional clinical collections, established medical imaging repositories, or challenge tasks rather than directly from PMC articles, which reduces the likelihood of systematic overlap with the MedPMC pretraining corpus. For OmniMedVQA, we further excluded four open-access subsets (“Covid CT,” “CoronaHack,” “Covid19 heywhale,” and “COVIDx CXR-4”) that were curated from the literature, as such datasets are more likely to contain images from preprints or publications that may overlap with PMC-indexed articles. However, provenance alone cannot guarantee complete image-level independence. Public and web-curated benchmarks may aggregate images from heterogeneous sources, and individual images can be reproduced across preprints, publications, repositories, and benchmark datasets."},{"type":"para","text":"For this reason, we evaluated MedPMC-derived models across multiple complementary settings rather than relying solely on public benchmark performance. These included controlled architecture-matched comparisons on public benchmarks, downstream medical QA, and a clinical transfer evaluation using 10,524 dermatology patient images from the Yale New Haven Health System. The YNHHS cohort was not part of the PMC-derived pretraining corpus and therefore provides an independent assessment of transfer to clinically realistic patient data. The consistent gains observed across these evaluation settings suggest that MedPMC improves medical visual representations beyond potential benchmark-specific overlap."},{"type":"heading","level":1,"text":"Code Availability"},{"type":"para","text":"The code used for data curation and pipeline evaluation is publicly available at: https://github.com/Yale-BIDS-Chen-Lab/MedPMC ."}],"string":"[\n {\n \"type\": \"para\",\n \"text\": \"\\\\ul\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Medicine is multimodal in nature [ 2 , 68 ] . Triage, diagnosis, prognosis, and treatment planning routinely require clinicians to integrate images, text, laboratory values, and other complementary data streams [ 20 , 35 ] . In principle, medical foundation models could unlock this potential by providing reusable backbones that generalize across medical modalities and can be adapted to local clinical settings with limited labeled data [ 31 , 80 ] . In practice, however, current medical AI systems are still largely developed and evaluated within single-modality, single-task settings rather than on the heterogeneous evidence used in clinical care [ 83 , 107 ] . This limitation stems not from model architecture alone, but from the absence of large-scale, high-quality multimodal data that are publicly accessible for training and evaluation, shareable across institutions, and suitable for reproducible research [ 107 , 83 , 1 , 34 ] . Unlike curated benchmark datasets, real-world medical data are fragmented across devices and health systems; constrained by privacy, licensing, and governance requirements; and costly to annotate at scale [ 47 , 55 , 63 ] . Although pioneering efforts have been made to make clinical datasets publicly accessible [ 40 , 33 , 5 ] , these resources remain much smaller and narrower than the data used to train general-domain foundation models. These barriers have led to two persistent challenges: a benchmark-to-bedside gap, in which models that perform well on public benchmarks may not deliver comparable utility in real clinical settings [ 87 , 42 , 8 ] ; and fragmented, institution-specific curation efforts that limit reproducibility, scalability, and reuse [ 37 , 36 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"One practical strategy is to leverage publicly available biomedical resources—including web-scale data, biomedical literature, and medical textbooks—to develop public foundation model backbones that are shareable, reproducible, and ultimately adaptable to clinical settings with less reliance on locally curated data [ 12 , 34 , 109 , 18 ] . Among these sources, PubMed Central (PMC) has emerged as one of the most promising and widely used resources [ 34 , 12 ] . PMC hosts millions of expert-authored biomedical articles that pair diverse medical visuals with explanatory text across a broad range of specialties, and it continues to expand rapidly as new biomedical knowledge is published [ 71 ] . Prior studies have leveraged PMC figures and associated text to develop medical vision-language models [ 52 , 25 , 93 ] and, more recently, generative multimodal large language models (MLLMs) [ 50 , 78 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"However, biomedical literature is created to communicate scientific findings, not to provide ready-to-use training data for multimodal foundation models. Repurposing it into high-fidelity multimodal training data poses substantial challenges. First, PMC contains a large fraction of visual content that is not directly useful for learning clinically relevant visual representations, including graphs, charts, schematics, flow diagrams, and molecular illustrations. Second, figures in the biomedical literature are often compound multi-panel images; without accurate decomposition of subfigures and alignment with their corresponding subcaptions, the image-text correspondence required for effective multimodal learning is substantially degraded. Finally, the continuous expansion of biomedical literature requires curation frameworks capable of dynamically incorporating newly published content. To date, existing PMC-derived datasets have not comprehensively addressed these challenges. For example, in a recent 24-million image-text pair dataset [ 62 ] , our analysis revealed that 80.3% of the images were non-medical. Moreover, compound figures accounted for 61% of the data but were not decomposed into individual panels. Furthermore, because most existing datasets are released as static snapshots, they become progressively outdated and fail to continuously integrate newly published knowledge. These limitations are further detailed in Extended Data Table 1 and the Discussion section.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To address this, we developed MedPMC, an automated and systematic framework for curating high-fidelity medical multimodal data from permissively licensed biomedical literature. MedPMC extends beyond a static dataset and instead provides reusable, continuously evolving data infrastructure for developing medical multimodal foundation models, including contrastive vision-language encoders and MLLMs. Our contributions operate at three levels. At the data resource level, MedPMC yields 11 million medical image-text pairs from 6.1 million PMC articles published through June 2024. The framework consists of five sequential stages—initial screening, multi-panel figure detection, multi-panel figure separation, caption separation and alignment, and medical figure classification—each handled by a purpose-built model to improve the fidelity of the resulting training data ( Fig. 1 a ). Manual evaluation by five annotators, including three medical experts, showed that 95.3% of curated images were medically relevant. The resulting corpus spans a broad range of specialties, including pathology, radiology, ophthalmology, dermatology, and endoscopy, and includes clinically important modalities that are less commonly represented at scale in public AI datasets, such as slit-lamp photography, skin photography, gross pathology photography, endoscopy, angiography, PET/SPECT, and fMRI. Importantly, MedPMC is designed for sustainable expansion rather than one-time dataset construction. Because each processing stage is modular and supported by a dedicated benchmark, individual components can be systematically evaluated, replaced, and refined as improved methods become available. This design allows MedPMC to continuously incorporate newly published biomedical literature, and the framework has identified more than one million new medically relevant image-text pairs per year since 2020.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"At the foundation model level, we trained MedPMC-CLIP, a CLIP-style vision-language model [ 79 ] , as one representative instantiation to test whether high-fidelity MedPMC curation translates into stronger medical visual representations ( Fig. 1 b ). MedPMC-CLIP achieves state-of-the-art performance across 26 public medical benchmarks spanning 11 specialties, covering clinically relevant tasks such as pneumonia detection, diabetic retinopathy grading, lymph node metastasis detection, and colon cancer detection. The model improves average AUC by 7.1 percentage points over BMC-CLIP [ 62 ] , a strong biomedical vision-language baseline, despite being trained on less than half the training data. These gains are consistent across specialties, demonstrating that high-fidelity curation produces visual representations that generalize broadly across medical domains. To further assess whether these representations can support generative multimodal systems, we conducted a controlled experiment following the training strategy introduced by LLaVA-Med [ 50 ] . With all other components held fixed, replacing only the vision encoder with MedPMC-CLIP improves performance by 1.9 and 16.9 percentage points on two multimodal medical question-answering (QA) benchmarks, respectively.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"At the clinical validation level, beyond standard benchmark evaluation, we further assessed transfer to downstream clinical applications. Using 10,524 dermatology patient images derived from a multi-site, hospital-based dermatology consultation cohort within the Yale New Haven Health System (YNHHS), we evaluated a morphology-to-image retrieval task in which textual descriptions of clinical findings were used to retrieve visually similar patient images, reflecting a clinically meaningful image-retrieval use case relevant to diagnostic decision-making [ 45 , 110 ] ( Fig. 1 c ). MedPMC-CLIP improves Recall@5 by 11.7 percentage points over BMC-CLIP, demonstrating that MedPMC-derived supervision can transfer effectively to clinically realistic settings. This translational benefit is further supported by embedding-space analysis, which shows that MedPMC better captures the visual distribution of real patient images from YNHHS, with broader coverage and closer alignment than existing dermatology datasets.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Collectively, these results position MedPMC as scalable data infrastructure for medical multimodal foundation models, linking high-fidelity literature-derived data curation to stronger visual representations, improved multimodal reasoning, and broader generalization across medical domains and downstream tasks. To support reuse and reproducibility, we publicly release the MedPMC framework, curated corpus, benchmark resources, and pretrained model checkpoints (see Data Availability and Code Availability).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Results\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.1 Pipeline Design and Data Curation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fig. 1 a illustrates the overall curation framework. The framework was designed to transform permissively licensed biomedical literature into high-fidelity medical image-text data through a sequence of automated, modular processing stages. Here, we summarize the purpose of each stage and the resulting data outputs; detailed methodological descriptions are provided in the Methods section.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Definition of medical images\"\n },\n {\n \"type\": \"para\",\n \"text\": \"PMC contains a broad spectrum of biomedical visual content, ranging from clinical imaging studies to basic science experiments, molecular biology figures, statistical plots, and conceptual diagrams. In this study, we use “medical images” as an operational umbrella term to refer to visual content relevant to clinical medicine, human health, or translational biomedical research. Our goal was to curate visual evidence most relevant for clinical and translational medical AI applications, including image interpretation, diagnosis, disease assessment, treatment planning, healthcare workflows, and medical research involving human-derived specimens [ 2 , 81 ] . We therefore prioritized image types routinely encountered in clinical practice, as well as human-relevant biomedical imaging modalities [ 64 , 75 ] , including radiology, ophthalmology, pathology, microscopy, endoscopy, dermatology, physiological signals, surgical photographs, and other forms of human clinical photography. In contrast, we excluded visual content primarily depicting non-human organisms, basic cellular or molecular biology experiments without direct human medical or translational relevance, scientific plots, graphs, tables, conceptual diagrams, educational illustrations, and other materials outside this scope.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Training data construction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The five-stage curation pipeline relies on purpose-built models for each component. We first leveraged existing labeled benchmarks, including ImageCLEF 2016 [ 27 ] , MedICaT [ 88 ] , and DocFigure [ 43 ] , to train these models. However, these resources did not fully cover all stages of the curation process and were insufficient for several tasks. For instance, no public benchmark was available for the initial screening stage, while only 1,600 labeled examples were available for caption separation and alignment despite the complexity of the task. We therefore supplemented existing benchmarks with PMC-derived synthetic annotations. From 39,360 sampled figures, we generated 99,321 labeled examples for the initial screening, multi-panel figure detection, and caption separation and alignment stages, complementing the 50,995 manually labeled samples available from existing benchmarks ( Extended Data Table 2 ). This expanded training set substantially improved performance across the affected pipeline stages. For example, augmenting with PMC-derived synthetic data improved the validation F1 score for the caption separation and alignment task by 3.82 percentage points over using the labeled data alone (see Supplementary Information, Section 1 for details). Importantly, the final reported performance of each curation component was assessed on held-out real instances with manual labels.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Initial screening\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The initial screening stage was designed to identify candidate medical figures while avoiding the need to download all PMC images, which is time-consuming and storage-intensive at scale. We first collected all open-access articles in PMC available as of June 2024, yielding 6,106,189 articles. Among these, 5,099,175 articles carried permissive licenses that allowed downstream use and redistribution, including CC BY (3,940,226), CC BY-NC (742,418), CC BY-NC-SA (270,631), CC0 (144,680), and CC BY-SA (1,220). We excluded articles licensed under CC BY-NC-ND (592,009) and CC BY-ND (8,188), as these licenses prohibit redistribution of derived resources, as well as records labeled as NO-CC CODE (406,817), which typically indicate missing, custom, or non-machine-readable license terms. We then trained a text classifier that used figure captions and associated inline reference text to distinguish candidate medical figures from non-medical figures. Applied to the 5.1 million permissively licensed articles, this stage retained 3.7 million candidate image-text pairs for downstream processing.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Multi-panel figure detection\"\n },\n {\n \"type\": \"para\",\n \"text\": \"After initial screening, we downloaded the retained candidate images and determined whether each image was single-panel or multi-panel. Single-panel images could be directly retained for downstream curation, whereas multi-panel figures required further decomposition and caption alignment. We therefore trained an image classifier to distinguish single-panel from multi-panel figures. This stage identified 3.1 million multi-panel figures.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Multi-panel figure separation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We next decomposed the identified multi-panel figures into individual subfigures. A figure separation model was trained to predict bounding boxes for panel-level image crops. In total, 3.1 million multi-panel figures were decomposed into an average of 9.2 sub-panels, yielding 29 million subfigures. This high panel density is consistent with the layouts commonly observed in biomedical articles, including microscopy grids, multi-condition comparisons, and multi-modality clinical figures.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Caption separation and alignment\"\n },\n {\n \"type\": \"para\",\n \"text\": \"After figure separation, we processed the corresponding full captions to construct panel-level image-text pairs. This stage involved two linked tasks: caption separation, which decomposes a full figure caption into subcaptions, and subfigure-subcaption alignment, which associates each subcaption with its corresponding subfigure. Rather than performing these tasks sequentially [ 88 , 52 ] , we addressed them jointly using an MLLM. The model takes as input the original multi-panel figure, the extracted subfigures, and the full caption, and directly generates a list of subcaptions corresponding to the input subfigures. Generated subcaptions were aligned to subfigures according to their order. To prioritize pair fidelity, we removed samples in which the number of generated subcaptions did not match the number of input subfigures. These cases typically involved figures with many subpanels or ambiguous semantic boundaries, where reliable panel-level correspondence is inherently difficult to establish. Although this filtering step reduced the total number of retained pairs, it improved the fidelity of the resulting image-text pairs. After this process, we obtained 12.5 million high-quality subfigure-subcaption pairs.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Medical figure classification\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Because multi-panel figures often contain mixtures of medical and non-medical visual elements, such as charts, diagrams, or experimental schematics, we further filtered the decomposed subfigures. We trained an image classifier to determine whether each subfigure was medically relevant. After this final filtering step, we retained 7.3 million medical subfigure-subcaption pairs.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.2 Stage-level Benchmark and Pipeline Performance\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Benchmark curation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A central challenge in large-scale medical multimodal data curation is that errors introduced at intermediate stages can propagate through the pipeline and degrade the final image-text pairs [ 3 , 97 ] . To systematically evaluate and optimize each component of MedPMC, we constructed a benchmark suite for component-level assessment across the full curation pipeline. This benchmark enables direct comparison between our purpose-built models and prior approaches used in PMC-derived dataset construction, while also providing a standardized basis for future refinement of individual pipeline components. The benchmark integrates three complementary data sources: (i) repurposed existing labeled benchmarks, including ImageCLEF 2016, MedICaT, and DocFigure; (ii) newly constructed PMC-derived synthetic annotations; and (iii) newly annotated samples curated by human annotators (see Extended Data Table 2 for dataset statistics and composition). Similar to our strategy for supplementing pipeline training data, we added PMC-derived synthetic annotations and newly annotated samples because existing benchmarks did not fully cover all pipeline stages or provide sufficient examples for robust evaluation, particularly for the caption separation and alignment task, which requires structured caption parsing and subfigure-level alignment rather than simple image classification and is therefore substantially more difficult and costly to annotate. The validation set was designed to assess in-distribution performance relative to the pipeline training data and included 11,357 labeled examples and 19,562 synthetic examples across all stages. The held-out test set comprised 24,068 labeled samples from existing benchmarks and 2,010 newly annotated examples. Additional details on benchmark construction and annotation procedures are described in the Methods section.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Using this benchmark, we evaluated each component of the MedPMC pipeline against baseline methods, including previous PMC-based data curation approaches (e.g., MedICaT and PMC-OA [ 52 ] ), task-specific state-of-the-art models, and component-specific ablations. Detailed descriptions of all baseline methods and complete component-level results are provided in Supplementary Information, Section 1 . For the initial screening stage, we compared multiple biomedical language models as well as different textual inputs, including captions alone and captions combined with inline text. The PubMedBERT classifier [ 30 ] using both captions and inline text achieved the best performance (F1 = 93.2), substantially outperforming the keyword-matching baseline used in MedICaT (F1 = 61.7). For multi-panel figure detection, we compared text-based approaches, multiple vision model architectures, and the effect of synthetic training data. The Vision Transformer [ 24 ] trained using both labeled and synthetic training data achieved the best performance (F1 = 96.5). For figure separation, the previous state-of-the-art model, YOLO-OCR-D [ 67 ] , reported strong performance (mAP = 90.2), but it is not publicly available. We therefore trained a YOLOv10-based detector [ 91 ] using the ImageCLEF dataset, achieving comparable performance (mAP = 89.8) while enabling a fully reproducible pipeline. For caption separation and alignment, we compared the rule-based caption parsing and CLIP-based alignment strategy used by PMC-OA, GPT-4 Turbo (hereafter referred to as GPT-4T) in a zero-shot setting [ 38 ] , and supervised InternVL-2.5-4B models [ 17 ] trained using manually annotated MedICaT data with or without synthetic augmentation. The supervised InternVL-2.5-4B model achieved the best performance (F1 = 81.4, ROUGE-L = 85.3), substantially outperforming both the PMC-OA pipeline (F1 = 48.5, ROUGE-L = 44.4) and GPT-4T zero-shot (F1 = 75.2, ROUGE-L = 76.9). Finally, for medical figure classification, we compared four vision models, with the Vision Transformer again achieving the best performance (F1 = 96.5).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.3 Dataset Composition, Coverage, and Scalability\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The final MedPMC dataset comprises 3.1 million multi-panel figure-caption pairs, 0.6 million single-panel image-caption pairs, and 7.3 million subfigure-subcaption pairs extracted from multi-panel figures, totaling approximately 11 million medical image-text pairs. Although MedPMC is smaller than broader PMC-derived collections such as PMC-15M and BIOMEDICA [ 62 ] , those resources are not restricted to medical images and include a large proportion of non-medical content. To quantify this difference in dataset composition and assess the clinical relevance of the curated images, we conducted a manual evaluation with five annotators, including medical experts. We analyzed 2,906 samples from MedPMC and 432 samples from BIOMEDICA for comparison. Multi-panel images were excluded from this analysis because assigning a single image category to a compound figure is inherently ambiguous.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As shown in Fig. 2 a , BIOMEDICA contained a substantial proportion of non-medical images (80.3%), with statistical figures, graphs, and charts accounting for 42% of sampled images. In contrast, MedPMC was predominantly composed of medical images, with only 4.7% categorized as non-medical. A substantial amount of images covered clinically relevant modalities routinely encountered in clinical practice, including radiology, ophthalmology, pathology, endoscopy, dermatology, physiological signals, surgical imaging, and other forms of clinical photography. Additionally, non-pathology microscopy provides complementary coverage of medical imaging beyond routine clinical practice. Fig. 2 b further breaks down the radiology subset into finer-grained modalities. MRI (31.7%), CT (23.4%), and X-ray (15.1%) constituted the largest radiology categories, while ultrasound, PET/SPECT, fMRI, angiography, and mammography were also represented. Detailed distributions across additional domains are provided in Extended Data Table 3 . Fig. 2 c illustrates the scalability of the MedPMC framework by quantifying the annual number of publications and corresponding figure-caption pairs processed by the pipeline. The volume of biomedical publications increased steadily over time, accompanied by a proportional increase in extracted image-text pairs. Since 2020, the framework has identified more than one million image-text pairs per year, demonstrating its capacity to continuously incorporate a vast amount of newly published biomedical literature.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.4 Foundation Model Evaluation and Clinical Transfer Validation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To evaluate whether high-fidelity MedPMC curation translates into stronger medical visual representations and downstream multimodal reasoning, we evaluated MedPMC through two representative model instantiations. First, we trained MedPMC-CLIP, a CLIP-style vision-language model, following the exact training protocol of BMC-CLIP [ 62 ] , including identical architecture, initialization, configuration, hyperparameters, and training schedule. Specifically, we initialized MedPMC-CLIP from the same OpenCLIP ViT-L/14 weights used by BMC-CLIP and replaced only the BIOMEDICA 24M training corpus with MedPMC. This design enabled a head-to-head comparison that isolated the effect of dataset quality. Second, we examined whether the visual representation learned by MedPMC-CLIP could serve as a stronger visual backbone for generative multimodal systems. We adopted the LLaVA-Med training pipeline [ 50 ] , which builds on LLaVA v1.5 [ 56 ] and performs the visual alignment and instruction tuning stages. While keeping the training procedure unchanged, we replaced only the original vision encoder with MedPMC-CLIP. This controlled design tested whether improved visual pretraining transfers to downstream medical visual question answering. Detailed results with 95% confidence intervals are provided in Supplementary Information, Section 2 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"MedPMC-CLIP improves performance across medical specialties\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluated MedPMC-CLIP on image classification across 26 public medical benchmarks spanning 11 specialties. Extended Data Table 4 lists the benchmarks, specialties, and task definitions included in the evaluation. Together, these benchmarks cover disease detection and classification, severity grading, tissue classification, anatomical structure recognition, cellular phenotyping, and morphological or ultrastructural identification. Clinically relevant tasks include pneumonia detection, breast malignancy classification, diabetic retinopathy grading, colon adenocarcinoma detection, lymph node metastasis detection, heart failure classification, and skin lesion classification. Fig. 3 a summarizes performance across the 26 benchmarks. For each benchmark, performance was first averaged within specialty, and the final score was computed by averaging across 11 specialties. MedPMC-CLIP consistently outperformed baseline models under this evaluation. Compared with BMC-CLIP, MedPMC-CLIP improved the specialty-macro average accuracy by 7.1 percentage points (95% CI, 6.0–8.2), F1 score by 10.5 percentage points (95% CI, 9.2–11.6), and AUC by 7.1 percentage points (95% CI, 6.3–8.0; paired bootstrap with 10,000 replicates). Fig. 3 b presents performance by specialty. MedPMC-CLIP outperformed BMC-CLIP in 10 of 11 specialties, indicating that the gains were not confined to a single domain but were broadly distributed across diverse medical fields.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"MedPMC-CLIP strengthens downstream multimodal medical QA\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fig. 3 c shows that replacing the vision encoder with MedPMC-CLIP yielded an observed improvement of 1.9 percentage points on MMMU (95% CI, − 4.1 -4.1 to 8.0 percentage points) and a substantially larger improvement of 16.9 percentage points on OmniMedVQA (95% CI, 14.9–18.7 percentage points; paired bootstrap with 10,000 replicates). This finding highlights the role of the vision encoder in medical multimodal reasoning and is consistent with prior studies showing that vision encoder choice [ 66 ] , as well as domain-specific adaptation of the visual backbone [ 105 ] , can substantially affect downstream multimodal LLM performance. The larger gain on OmniMedVQA was primarily driven by tasks in which visual recognition plays a central role. We analyzed performance according to the query categories provided by OmniMedVQA. Performance improved from 46.9% to 75.2% for modality recognition, corresponding to an increase of 28.3 percentage points (95% CI, 24.2–32.4); from 34.7% to 57.0% for anatomy identification, an increase of 22.3 percentage points (95% CI, 17.6–26.9); and from 26.0% to 40.8% for the “other biological attributes” category, an increase of 14.8 percentage points (95% CI, 7.6–22.0). These findings suggest that a stronger medical vision encoder can directly improve the model’s ability to interpret medical images. Importantly, the improvement was not limited to visually straightforward recognition tasks. Performance on the combined “disease diagnosis and lesion grading” category increased from 34.2% to 45.4%, corresponding to an improvement of 11.1 percentage points (95% CI, 8.5–13.8). In contrast, MMMU questions are often framed around patient presentations and require complex clinical reasoning; thus, although the point estimate favored the stronger vision encoder, its confidence interval included zero, suggesting that further gains may require more effective end-to-end instruction tuning.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"MedPMC-CLIP improves morphology-guided retrieval of clinical dermatology images\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Beyond standard benchmarking, we conducted a clinical evaluation using 10,524 dermatology patient images derived from a multi-site, hospital-based dermatology consultation cohort within the Yale New Haven Health System (YNHHS). Specifically, we evaluated MedPMC-CLIP in a clinically motivated morphology-to-skin-image retrieval task, in which textual morphology descriptions are used to retrieve relevant clinical images. This setting serves as a probe of representation learning and reflects a clinically meaningful retrieval use case, as retrieving similar medical images has long been studied as a form of decision support and education in dermatology and broader medical imaging contexts [ 82 , 28 , 70 , 59 ] . As shown in Fig. 3 d , MedPMC-CLIP outperformed BMC-CLIP, improving Recall@1 by 3.4 percentage points (95% CI, 2.7–4.1), Recall@5 by 11.7 percentage points (95% CI, 10.7–12.6), and Recall@10 by 10.7 percentage points (95% CI, 9.8–11.6; paired bootstrap with 10,000 replicates). These results indicate that improvements observed on public benchmarks transfer to a clinically realistic image retrieval setting using patient data.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"MedPMC dermatology images better match the visual distribution of clinical images\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Although MedPMC images are derived from the biomedical literature rather than routine clinical practice, MedPMC-CLIP showed improved performance on downstream clinical dermatology retrieval. To better understand the factors driving this transferability, we examined whether dermatology images curated from MedPMC occupy a visual space similar to that of real-world clinical dermatology photographs. Specifically, we compared MedPMC with three widely used public dermatology datasets containing skin photographs (Fitzpatrick17k [ 29 ] , SCIN [ 94 ] , and DermNet [ 23 ] ) and assessed their respective visual similarity to the YNHHS clinical cohort. For each public dataset, we sampled 10,000 images; for MedPMC, we first identified skin photographs using the MedPMC-CLIP model and then randomly sampled 10,000 images from this curated subset.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In the DINOv2 embedding space, MedPMC demonstrated the greatest visual similarity to YNHHS clinical images among all evaluated external sources. Quantitatively, YNHHS clinical images yielded the lowest median nearest-neighbor cosine distance to MedPMC (0.239), compared with 0.262 for Fitzpatrick17k, 0.270 for SCIN, and 0.299 for DermNet ( Fig. 4 a ). Consistently, when each YNHHS clinical image was assigned to the external source of its nearest neighbor, MedPMC accounted for the majority of assignments (51.5%), followed by Fitzpatrick17k (21.2%), SCIN (20.7%), and DermNet (6.5%) ( Fig. 4 b ). UMAP projections further revealed that while YNHHS images formed a broad visual manifold that reflects the inherent variability of clinical photography, MedPMC showed extensive overlap across this space. In contrast, existing public datasets exhibited more uneven distributions, leaving substantial portions of the clinical manifold underrepresented ( Fig. 4 c–f ). These findings indicate that MedPMC captures a diverse visual distribution that closely aligns with real-world clinical data, providing a potential explanation for its downstream transfer to clinical dermatology tasks.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Discussion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We developed MedPMC, an automated and systematic framework for curating high-fidelity medical multimodal data from permissively licensed biomedical literature, together with MedPMC-CLIP, a vision-language foundation model trained on the resulting MedPMC corpus to demonstrate its utility. The main finding is that biomedical literature, when transformed into medically relevant and panel-level aligned data, can support effective and reproducible development of medical multimodal models. These data-level improvements translated into consistent model-level gains. Using the exact training protocol of BMC-CLIP and changing only the training corpus, MedPMC-CLIP improved average zero-shot AUC by 7.1 percentage points over BMC-CLIP across 26 public medical benchmarks spanning 11 specialties. Gains in AUC were observed in 10 of the 11 specialties, despite MedPMC-CLIP being trained on fewer than half as many image-text pairs. These benefits also extended to MLLMs: within the LLaVA-Med framework, replacing only the vision encoder with MedPMC-CLIP—while holding the remaining model and training pipeline fixed—improved performance by 1.9 percentage points on MMMU and 16.9 percentage points on OmniMedVQA. Importantly, these improvements were not confined to public benchmarks. Using 10,524 dermatology patient records from a multi-site, hospital-based consultation cohort, MedPMC-CLIP improved performance in zero-shot retrieval. Embedding-space analysis further showed that MedPMC dermatology images were more closely aligned with clinical dermatology images than several widely used public dermatology datasets. These findings suggest that MedPMC can help address a central bottleneck in medical multimodal AI—the scarcity of large-scale, clinically meaningful multimodal data [ 107 , 83 , 1 , 34 ] . Beyond serving as a corpus, MedPMC provides a modular curation framework with component-level benchmarks that can be updated as new biomedical articles become available. Because it is derived from permissively licensed literature, the resulting resources can be shared more openly, supporting reproducibility, adaptation, and continued model development across both benchmark and clinically realistic settings.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Data curation as reusable infrastructure for medical AI\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A key implication of this study is that data curation should be treated as a core engineering problem in medical multimodal AI. PMC offers a scalable public source of biomedical image-text data, and the resource continues to grow as new biomedical knowledge is published [ 71 ] . However, directly repurposing PMC figures introduces several failure modes, including non-medical visual content, compound multi-panel figures, weak subfigure-subcaption correspondence, and dataset staleness. MedPMC addresses these failure modes through a modular five-stage pipeline. The initial screening step identifies medically relevant figures before image download, reducing unnecessary data transfer and storage. Multi-panel figure detection and separation convert compound figures into panel-level visual units. Caption separation and subfigure-subcaption alignment improve image-text correspondence at the level at which supervision is most meaningful. Medical figure classification further filters decomposed panels to remove remaining non-medical visual elements. Because each stage is supported by component-level benchmarks, the pipeline can be evaluated, audited, and improved rather than treated as an opaque preprocessing procedure. This infrastructure framing distinguishes MedPMC from a one-time dataset release. Biomedical literature continues to expand, and static datasets can become progressively incomplete as new diseases, technologies, imaging modalities, and clinical terminology emerge.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To keep MedPMC current, we plan to release updated corpus versions every six months, aligned with the semiannual PMC baseline snapshots. After the initial June 2024 release, each update will process newly available permissively licensed articles not included in prior versions and will be documented by article cutoff date, license filters, pipeline component checkpoints, and processing configuration. Based on recent publication trends, these updates are expected to identify about one million additional medical image-text pairs per year. The modular design of MedPMC also allows individual pipeline components to be replaced or improved over time.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Relationship to existing PMC-derived resources\"\n },\n {\n \"type\": \"para\",\n \"text\": \"MedPMC builds on a growing literature that has established biomedical articles as a valuable source of medical multimodal supervision. Early resources such as ROCO [ 77 ] and MedICaT [ 88 ] showed that figures and captions from biomedical literature could support medical vision-language learning, although these datasets were relatively small or focused on narrower domains. More recent PMC-derived resources, including PMC-OA [ 52 ] , PMC-15M [ 108 ] , BIOMEDICA [ 62 ] , and Open-PMC-18M [ 6 ] , substantially expanded the scale of literature-derived image-text data, enabling medical vision-language models such as PubMedCLIP [ 25 ] , BiomedCLIP [ 108 ] , and BMC-CLIP [ 62 ] , as well as multimodal medical LLMs such as Med-Flamingo [ 69 ] , LLaVA-Med [ 50 ] , BiomedGPT [ 106 ] , HealthGPT [ 51 ] , and MedGemma [ 84 ] . However, as detailed in Extended Data Table 1 , these large-scale data resources remain limited in medical relevance, compound-figure processing, and reproducible curation pipelines.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"MedPMC extends this line of work by emphasizing fidelity, reproducibility, and updatability rather than scale alone. First, many existing large-scale PMC-derived datasets include substantial non-medical visual content, such as graphs, charts, molecular illustrations, workflow diagrams, and system schematics. In the manual analysis, 80% of sampled BIOMEDICA figures were non-medical, compared with 4.7% for MedPMC. Filtering strategies based on manually curated keywords can miss medically relevant figures outside predefined terminology, whereas image-only filtering requires large-scale image download before screening and may not capture clinical context. MedPMC instead formulates medically relevant figure selection as a text classification problem using captions and inline reference text, which provides contextual information before image download and improves scalability. This text-based classifier reached a recall of 93.1 and an F1 score of 93.2, substantially outperforming keyword-based filtering. Second, existing resources differ substantially in how they handle compound figures. Some exclude multi-panel figures, which discards informative medical images; others retain compound figures as whole figure-caption pairs, which can weaken image-text correspondence when panels depict different modalities, disease states, time points, or experimental conditions. Panel-level approaches based on sequential caption splitting and heuristic or CLIP-based alignment can also introduce alignment errors, while approaches that run an MLLM separately for every subfigure [ 6 ] incur inference costs that scale with panel count. MedPMC instead separates compound figures and jointly performs caption separation and subfigure-subcaption alignment with a supervised 4B-parameter MLLM that considers all panels together. This approach reached 81.4 F1 and 85.3 ROUGE-L, outperforming rule- and CLIP-based matching used in prior pipelines as well as GPT-4, while remaining efficient at scale by avoiding per-subfigure inference. Third, most prior PMC-derived resources are released as static snapshots. Because their full curation pipelines, trained preprocessing models, and stage-specific evaluation resources are often not fully available, reproducing or extending the dataset construction process can be difficult. MedPMC was designed to make the curation process itself reusable and to provide a reproducible framework for generating, evaluating, and updating medical multimodal supervision at scale.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Complementary strengths of MedPMC and clinical data\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Importantly, MedPMC is not intended to replace clinical datasets. Literature-derived and clinical resources serve complementary roles in medical multimodal model development. Clinical datasets remain closest to deployment because they capture real patient populations, local imaging protocols, clinical workflows, and case mix. However, they are often concentrated within a limited number of institutions, specialties, and modalities, and their accompanying text is frequently limited to diagnostic labels or short reports rather than detailed descriptions of image findings. Despite several pioneering efforts [ 40 , 33 , 5 , 92 ] , such datasets remain difficult to assemble and share at scale because of privacy, governance, licensing, and annotation constraints.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"PMC-derived data may offer a distinct but complementary form of supervision. Compared with many clinical datasets, literature-derived resources can provide greater scale, richer image-associated context, including experimental settings and diagnostic descriptions, and broader coverage across diseases, specialties, modalities, and emerging concepts, including rare conditions and technologies that may be sparsely represented in existing clinical resources. Because they are derived from publicly available and permissively licensed literature, they can also be shared and reproduced more readily than many institution-specific clinical datasets. One practical development pathway is therefore to use MedPMC-like resources for broad pretraining and representation learning, followed by clinical adaptation and validation in the intended target setting. In this workflow, literature-derived data provide scale, coverage, and descriptive supervision, while clinical data provide local grounding and direct alignment with real-world practice. This complementary role is further supported by preliminary few-shot adaptation results in Supplementary Information, Section 3 , which show a modest performance advantage over OpenCLIP and BMC-CLIP.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our clinical dermatology evaluation demonstrates how literature-derived pretraining can support downstream clinical applications. Although MedPMC is derived from biomedical publications rather than routine clinical workflows, MedPMC-CLIP transferred to morphology-guided retrieval on internal patient data. The embedding-space analysis further suggested that MedPMC dermatology images overlap more broadly with internal clinical dermatology images than several widely used public dermatology datasets. This finding does not imply that literature-derived images are equivalent to clinical data or sufficient for deployment without target-domain validation. Rather, it indicates that carefully curated biomedical literature can provide pretraining supervision that is relevant to downstream clinical tasks and can complement, rather than replace, institution-specific clinical datasets.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Potential applications\"\n },\n {\n \"type\": \"para\",\n \"text\": \"First, the corpus can be used to train or initialize medical vision-language encoders, visual backbones for MLLMs, and specialty-specific foundation models [ 21 , 78 ] . Second, because each image is linked to article text, figure captions, subcaptions, and medical figure categories, MedPMC can be filtered into specialty-, modality-, or concept-specific subsets. This may be useful when dedicated public datasets are limited, such as in rare diseases, surgical imaging, pathology subdomains, clinical skin photography, or specific imaging modalities [ 72 , 98 ] . Third, the structure of compound figures provides a source of supervision that is difficult to obtain from standard single-image datasets. Many biomedical figures show related images, such as disease progression, pre- and post-treatment comparisons, multi-view imaging, paired radiology-pathology findings, or contrasts between disease subtypes and controls. These structures could support models that reason over image sets rather than isolated images [ 18 ] . Fourth, the aligned image-text pairs can be indexed as a searchable corpus of biomedical visual evidence. Such an index could support similar-case search, visual example retrieval, and multimodal retrieval-augmented generation, in which MedPMC functions both as a training resource and as an external knowledge source for downstream multimodal systems [ 103 ] .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Limitations and Future Work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"First, this study focuses on image-text data, whereas clinical practice routinely involves broader and more heterogeneous information streams, including structured tables, laboratory values, temporal clinical context, and longitudinal patient records. Some of these modalities, such as tables and structured content embedded in biomedical articles, could be incorporated into a similar literature-based curation framework. Extending MedPMC to capture these additional modalities is therefore an important direction for future work. Other data types, particularly longitudinal patient records and real-world clinical workflows, are rarely available in a consistent or shareable form in the public literature. Addressing these gaps will require complementary frameworks for curating and integrating clinical data under appropriate privacy, governance, and institutional constraints.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Second, because MedPMC is derived from biomedical articles, the corpus reflects both the strengths and limitations of the source literature. Biomedical figure captions vary in quality and may contain incomplete descriptions, inconsistent terminology, non-visual information, or diagnostic context that is not directly inferable from the image alone. In addition, biomedical publications introduce selection bias in what is visually reported. Published figures may preferentially show representative examples, diagnostically salient findings, unusual cases, positive or publishable results, or visually clear images, and may differ from routine clinical images in terms of case mix, acquisition setting, image quality, presentation style, and patient population [ 74 ] . Prior work has shown that improving caption quality or restructuring text for multimodal learning can improve model performance [ 100 , 15 ] ; these approaches are complementary to MedPMC because they depend on accurately aligned image-caption pairs as input. Future work could therefore combine MedPMC-style high-fidelity image-text alignment with caption refinement, structured report generation, and region- or concept-level supervision.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Third, although we systematically evaluated MedPMC across different model types, including contrastive vision-language encoders and MLLMs, and across multiple settings, including public benchmark evaluations, downstream tasks, and clinically realistic applications, our experiments do not exhaustively cover all possible model architectures or applications. This reflects the role of MedPMC as reusable data infrastructure rather than as a single task-specific model. Given the rapid development of medical foundation models, it is not feasible for one study to evaluate every emerging architecture or clinical use case [ 9 , 8 , 16 ] . In addition, potential overlap between large-scale pretraining corpora and public benchmarks is an increasingly recognized challenge for foundation-model and LLM evaluation [ 85 , 49 , 14 ] . We carefully selected public benchmarks that were largely derived from institutional clinical collections, established medical imaging repositories, or challenge datasets rather than directly from PMC articles, and we further included an internal YNHHS clinical cohort that was independent of the PMC-derived pretraining corpus. Nevertheless, exact overlap cannot be fully excluded. We therefore interpret the public benchmark results together with architecture-matched comparisons, downstream QA evaluation, and independent clinical transfer experiments.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Fourth, although our experiments show that models trained with MedPMC transfer to clinically motivated tasks, including morphology-guided dermatology image retrieval using internal patient data, these evaluations do not establish deployment readiness. MedPMC should be viewed as a source of broad pretraining supervision that complements, rather than replaces, target-domain clinical data. Additional clinical evaluations of model adaptation, such as further fine-tuning, and clinician-in-the-loop workflows will also be important for quantifying how public medical resources and models derived from them adapt to real-world clinical settings and support clinical decision-making [ 58 ] . By releasing the corpus, curation pipeline, component-level benchmarks, pretrained models, and source-license metadata, we aim to support continued external evaluation, model comparison, pipeline refinement, and clinically grounded adaptation by the research community.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Methods\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 MedPMC Pipeline Implementation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We trained specialized models for each stage of the MedPMC curation pipeline. For each task, we leveraged existing datasets when they were well aligned with the target prediction problem. However, available datasets were often limited in scale, restricted to specific domains, or not fully representative of PMC figure layouts and captioning styles. To improve robustness in the target curation setting, we additionally constructed new datasets. Dataset statistics for each pipeline component are provided in Extended Data Table 2 . The detailed prompts used for synthetic data generation are provided in Supplementary Information, Section 4 . Importantly, all datasets, trained models, and code for running and evaluating each pipeline component are publicly released, supporting reproducibility, reuse, benchmarking, and transparent inspection of implementation details.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Initial screening\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The initial screening model was designed to identify candidate medical figures before image download. For each figure, we concatenated the figure caption with the inline reference text from the article body and formulated the task as binary classification of medical relevance. To construct the training data, we randomly sampled 39,360 figures from articles indexed by Open-i ( https://openi.nlm.nih.gov ), an open-access biomedical image search engine, using the “Exclude Graphics” option to reduce the retrieval of plots, charts, and other non-image graphics. We then generated synthetic medical-relevance labels for the sampled figures using GPT-4T ( gpt-4-turbo-2024-04-09 ). Following our definition of medical figures, GPT-4T assigned labels after jointly considering both the image and its associated text. We initialized the classifier with PubMedBERT [ 30 ] , which was pretrained on PubMed abstracts and full-text articles ( https://huggingface.co/microsoft/BiomedNLP-BiomedBERT-base-uncased-abstract-fulltext ). The model was fine-tuned with a batch size of 64 and a learning rate of 3e-5.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Multi-panel figure detection\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The multi-panel figure detection model was designed to distinguish single-panel from multi-panel figures before subfigure separation. We formulated this step as binary image classification and trained the model using 16,775 samples from the compound figure detection subtask of the ImageCLEFmed challenge, together with 39,360 additional synthetically labeled examples. The synthetically labeled examples were drawn from the same Open-i sample used for initial screening, but were re-annotated for panel structure rather than medical relevance. GPT-4 was provided with each figure and its caption and assigned a binary label indicating whether the figure contained a single panel or multiple panels. We adopted a Vision Transformer ( https://huggingface.co/timm/vit_base_patch16_rope_reg1_gap_256.sbb_in1k ) as the backbone model [ 24 ] and fine-tuned the model using the PyTorch Image Models repository (timm) [ 95 ] for 100 epochs with a batch size of 64. All other hyperparameters followed the default settings of the training framework, with the learning rate automatically scaled according to the effective global batch size.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Multi-panel figure separation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The multi-panel figure separation model took figures classified as multi-panel as input and predicted bounding boxes corresponding to individual subfigures. For training, we used the dataset provided for the figure separation task in the ImageCLEFmed challenge [ 27 ] . We did not incorporate additional synthetic annotations at this stage because they did not further improve validation performance. We used YOLOv10 as the backbone model [ 91 ] . The model was initialized from pretrained weights and fine-tuned using the default training configuration in Ultralytics v8.2.73 ( https://github.com/ultralytics/ultralytics ). Training was performed for 100 epochs with a batch size of 16 and an input resolution of 640 × 640; automatic optimizer selection chose AdamW with an initial learning rate of 2e-3 [ 60 ] .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Caption separation and alignment\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The caption separation and alignment model took as input the original multi-panel figure, its full caption, and the separated subfigures, and generated the corresponding subcaptions. For training, we used 1,361 manually annotated samples from the MedICaT dataset [ 88 ] . To improve scale and domain diversity beyond the predominantly radiology-focused MedICaT dataset, we generated an additional 20,601 synthetic annotation examples. Specifically, we applied the trained multi-panel detection model to the Open-i figures used in the previous stage, selected those predicted as multi-panel, and separated them into subfigures using the trained YOLOv10 model. GPT-4 was then provided with the original figure, the separated subfigures, and the full figure caption to generate aligned subcaptions for each subfigure. We adopted InternVL-2.5-4B-MPO as the backbone model [ 17 ] . The model was then trained using the official implementation of InternVL ( https://github.com/OpenGVLab/InternVL ) with the default training configuration, using three training epochs, a total batch size of 128, a learning rate of 4e-5, and BF16 mixed-precision training.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Medical figure classification\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The medical figure classification model performed binary image classification to determine whether an input image or subfigure was medically relevant. We constructed a training set of 26,453 samples by combining data from ImageCLEF, MedICaT, and DocFigure [ 43 ] . A Vision Transformer was used as the backbone model. We trained the model using the timm library with the same setup as the multi-panel figure detection model, except that training was performed for 300 epochs.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Validation and model selection\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For each pipeline stage, we evaluated multiple backbone models, input configurations, and other task-specific design choices, and selected the final component based on validation-set performance. The validation sets combined existing benchmark data with PMC-derived synthetic annotations generated using the same procedures as the corresponding training data. To prevent overlap between training and validation samples originating from the same source article, synthetic annotations were partitioned at the article level. Validation-set composition and sample counts are provided in Extended Data Table 2 . Initial screening, multi-panel figure detection, and medical figure classification were evaluated using precision, recall, and F1 score. Following prior work [ 67 ] , multi-panel figure separation was evaluated using accuracy and mean average precision (mAP): accuracy measured agreement between the predicted and reference panel structure, whereas mAP quantified the localization of predicted subfigure bounding boxes relative to the ground-truth annotations. Following MedICaT, the caption separation and alignment task was evaluated using F1 score and ROUGE-L, both computed between the predicted and reference subcaptions. The selected models substantially outperformed the comparison methods across the pipeline stages. Detailed descriptions of the baseline models and comparative analyses are provided in Supplementary Information, Section 1 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 Benchmark Annotation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To enable systematic evaluation of the five components of the MedPMC pipeline, we constructed dedicated test sets using existing benchmark data and newly annotated examples from PMC. Whereas the validation sets were used for model selection, hyperparameter tuning, and other development decisions, test sets were kept fully held out and used only for final performance evaluation. Newly annotated PMC samples were drawn from articles indexed by Open-i, following the same initial sampling strategy used for training-data construction. However, all articles represented in the test sets were excluded from both the training and validation sets to prevent article-level overlap.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The manual annotation covered the first four stages: (1) initial screening, (2) multi-panel figure detection, (3) multi-panel figure separation, and (4) caption separation and alignment. For the medical figure classification task, manual annotation was bypassed because a substantial dataset of approximately 19,000 samples had already been secured from public resources. New annotations were performed by 13 annotators (S.S.A., Y.W., Y.K., T.G., Z. Cao., M.D., Y.F., Y.H., R.S., G.Y., K.W.J., Y.L., and Y.T.). Before annotation, all annotators completed a training session covering the annotation guidelines and representative examples. Each sample was independently annotated by two annotators, and disagreements were adjudicated by a senior annotator (H.K.). After each annotation batch, the adjudicated decisions and their rationales were shared with the annotators to promote consistency across subsequent batches.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For the initial screening, four annotators (S.S.A., Z. Cao., Y.F., and Y.H.) reviewed image-caption pairs to annotate each sample as either medical or non-medical. Under our operational definition of medical figures, non-human-centered subjects, such as animal models, were strictly classified as non-medical. For this task, a total of 600 samples—comprising a mixture of single-panel and multi-panel figures to faithfully reflect real-world data distributions—were partitioned into batches of 100. Each annotator evaluated three separate batches, culminating in 300 annotations per annotator. For multi-panel figure detection, two annotators (R.S. and G.Y.) labeled a total of 500 figures as either single-panel or multi-panel. This dataset was partitioned into five batches of 100 figures each. The multi-panel figure separation and caption separation and alignment stages were performed concurrently by the same cohort of annotators within a unified interface. For this joint task, seven annotators (Y.W., Y.K., T.G., M.D., K.W.J., Y.L., and Y.T.) processed 455 compound figures across six batches. For each figure, annotators delineated the bounding boxes of individual subfigures and subsequently extracted and aligned the corresponding subcaptions. If a figure was determined not to be multi-panel, this workflow was skipped.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3 Dataset Composition Analysis\"\n },\n {\n \"type\": \"para\",\n \"text\": \"MedPMC consists of multi-panel figures, single-panel figures, and subfigures extracted from multi-panel figures. For dataset composition analysis, we excluded multi-panel figures because assigning a single category to a compound figure is inherently ambiguous. We randomly sampled 1,232 single-panel figures and 1,674 subfigures from MedPMC, totaling 2,906 samples. Each group was evaluated separately, and the final distribution was obtained by weighting the results according to the proportions of single-panel figures and subfigures in the full dataset. For BIOMEDICA, we similarly excluded multi-panel figures and restricted the analysis to single-panel images. To do so, we applied our compound figure detection model to identify and filter out multi-panel images, resulting in 9.3 million images (38.7%) retained for analysis. From this subset, we randomly sampled 432 images for manual evaluation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A total of five annotators (P.X., S.S.A., R.J., Y.D., and J.M.) participated in the study, including two hospital residents and one final-year medical student. Each annotator was presented with both the image and its corresponding caption through a dedicated annotation interface. Annotators were asked to assign each sample to one of the predefined categories. If a sample did not fit any predefined category, annotators could select an “Other” option and provide a brief specification. The three medical experts additionally reviewed and resolved ambiguous cases that were initially marked as uncertain. The category schema was designed in collaboration with the medical experts, and the definition of non-medical images was informed by prior work [ 27 ] .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.4 Model Training and Downstream Evaluations\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Baseline models\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We compared MedPMC-CLIP against a range of baseline vision-language models for image classification and retrieval. These include general-domain foundation models such as (1) CoCa [ 102 ] , a contrastive-captioning model trained on web-scale datasets, and (2) OpenCLIP [ 19 ] , a standard CLIP architecture trained on LAION-2B. Within the medical domain, we evaluated (3) PMC-CLIP [ 52 ] , specifically the version initialized from the OpenAI CLIP ViT-L/14 checkpoint and subsequently fine-tuned on PMC-OA and ROCO data, (4) BiomedCLIP [ 108 ] pretrained on the PMC-15M corpus, and (5) MedSigLIP [ 84 ] , a sample-efficient medical encoder optimized via sigmoid pairwise loss. Finally, we compared our model against (6) BMC-CLIP [ 62 ] , which was trained on the BIOMEDICA-24M dataset; this model serves as the most direct baseline for evaluating MedPMC-CLIP, as we maintained the same model architecture and training pipeline configuration while isolating the pretraining dataset as the sole variable. For medical visual QA, we used LLaVA-Med [ 50 ] , a biomedical adaptation of LLaVA (v1.5) [ 56 ] that connects a CLIP vision encoder to a language model through a linear projection layer and is trained through a two-stage curriculum of visual alignment and instruction tuning.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Training details\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For MedPMC-CLIP, we followed the experimental setup of BIOMEDICA using the official code repository ( https://github.com/minwoosun/biomedica-etl ). Specifically, we initialized the model from OpenCLIP (ViT-L/14; checkpoint: commonpool_xl_clip_s13b_b90k and adopted the same configuration and hyperparameters (a batch size of 8,192 and a learning rate of 1e-6), while replacing the training data with MedPMC (11 million image-text pairs) For MLLM evaluation, we adopted the same training data and hyperparameters as LLaVA-Med [ 50 ] , following the official implementation ( https://github.com/microsoft/LLaVA-Med/tree/main ). Stage 1 (biomedical concept feature alignment) was performed on 467K samples using a batch size of 128 and a learning rate of 2e-3. Stage 2 (end-to-end instruction tuning) was then conducted on 56K samples for 3 epochs using a batch size of 128 and a learning rate of 2e-5. We used the LLaVA training codebase ( https://github.com/haotian-liu/llava ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"26 image classification benchmarks\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We used a comprehensive suite of medical benchmarks covering a diverse range of clinical modalities and specialties, including radiology, ophthalmology, pathology, dermatology, surgery, and microscopy. Detailed specifications regarding benchmark specialties, core tasks, and sample sizes are provided in Extended Data Table 4 . For each task, prompt templates were adopted directly from those supported by Micro-Bench [ 61 ] ; for domains not covered by existing frameworks, we custom-designed the evaluation prompts, all of which are publicly verifiable via our released evaluation code ( https://github.com/Yale-BIDS-Chen-Lab/MedPMC ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Performance was evaluated using accuracy, F1 score, and area under the receiver operating characteristic curve (AUC). For binary and multiclass tasks, accuracy was calculated using the top-scoring class, F1 score was macro-averaged across classes, and AUC was calculated by pooling the one-hot-encoded labels and corresponding scores across all sample-class pairs. For multi-label tasks, predictions were obtained using a threshold of 0.5, and accuracy was calculated across all sample-label pairs. Positive-class F1 scores and AUCs were calculated independently for each label and then macro-averaged across labels; labels without both positive and negative samples were excluded from AUC calculation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Downstream medical QA\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluated MLLMs on MMMU [ 104 ] and OmniMedVQA [ 32 ] in a zero-shot setting. These benchmarks were selected because they are highly popular and widely recognized standards within the community, providing an exceptionally broad coverage of the medical domain rather than focusing on a single, narrow specialty. MMMU spans a diverse set of 30 domains; in our experiments, we focused on the clinical medicine subset and included only single-image questions, consisting of 314 samples. OmniMedVQA originally comprises more than 120,000 samples from 73 medical datasets, subdivided into 42 open-access and 31 restricted-access repositories. In this study, we used only the open-access datasets and excluded four COVID-19-related datasets curated from the literature to minimize the risk of data contamination (see “Contamination-aware evaluation design”), resulting in 38 open-access datasets. To ensure balanced and computationally efficient evaluation across these various source datasets, we randomly sampled a maximum of 100 examples per dataset. We then removed DRIMDB samples that assessed about image quality (good, bad, or outlier) but were incorrectly categorized as lesion grading, resulting in a final evaluation set of 3,593 samples. For all inference generations, we utilized greedy decoding with the temperature fixed at 0.0 to ensure deterministic results. Models were evaluated using only the image and the original question, with no prompting engineering.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Downstream clinical application: morphology-to-skin-image retrieval\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Retrieving similar medical images has long been studied as a form of decision support and education in dermatology and broader medical imaging contexts [ 82 , 28 , 70 , 59 ] . In this study, textual morphology descriptions were used as queries to retrieve relevant clinical dermatology images. The study was approved by the Yale Institutional Review Board (IRB) under protocol number 2000041054. We constructed the evaluation dataset from 10,524 real-world dermatology patient images from a multi-site, hospital-based dermatology consultation cohort within the Yale New Haven Health System (YNHHS). The dataset included multiple images from some patients, although each image was treated as an individual retrieval sample. All models were evaluated using the same set of images. For each sample, a clinical image was originally paired with a raw morphology description extracted from the corresponding clinical note. Because multiple clinical images can share the same morphological characteristics, using only the original, unnormalized image-description pairs as ground truth would underestimate clinically relevant matches. To address this and standardize the textual descriptions, the raw morphology descriptions were automatically mapped to standardized morphology concepts based on the SkinCon taxonomy [ 22 ] (e.g., papule, plaque, macule, and erythema) using an open-sourced large language model, gpt-oss-120b [ 4 ] . To ensure data quality and evaluate the reliability of this automated normalization, a hospital resident in dermatology (R.J.) manually reviewed and verified a sampled subset of the generated mappings. The detailed prompt templates for this LLM-based parsing are provided in Supplementary Information, Section 4 . This standardization mapping enabled images sharing the same set of morphology concepts to be grouped as clinically related. For retrieval evaluation, we adopted an established set-based matching criterion [ 54 , 48 ] . A retrieved image was considered correct if its associated morphology concept set contained the ground-truth concept set of the query image. This formulation allowed multiple retrieved images to be treated as correct, reflecting cases in which different images shared the same core morphological features even when additional attributes differed.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Embedding-space comparison with clinical dermatology images\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We further examined whether dermatology images curated from MedPMC occupy a visual space similar to that of real-world clinical dermatology images. To identify dermatology images within MedPMC, we used the dermatology-related categories and text prompts described in Extended Data Table 3 . All MedPMC images were scored using MedPMC-CLIP based on image-text similarity, and images classified as skin photographs were used to construct the MedPMC dermatology subset. We compared this subset against three widely used public dermatology datasets: Fitzpatrick17k [ 29 ] , a collection of community-contributed dermatology photographs annotated with skin type and disease labels; SCIN [ 94 ] , a dataset of smartphone-acquired skin images collected from diverse participants in real-world settings; and DermNet [ 23 ] , a large collection of educational dermatology images curated from the DermNet resource. To enable balanced comparisons across sources, 10,000 images were randomly sampled from each external dataset. YNHHS clinical dermatology images served as the reference distribution.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All images were embedded using DINOv2 (ViT-B/14) [ 76 ] . To quantify similarity between each external source and the YNHHS clinical image distribution, we computed the cosine distance from each YNHHS clinical image to its nearest neighbor within each external dataset. We then summarized the resulting nearest-neighbor distance distributions and compared median distances across datasets. In addition, each YNHHS clinical image was assigned to the external source containing its overall nearest neighbor across all external datasets, and the fraction of images assigned to each source was calculated. For visualization, DINOv2 embeddings were projected into two dimensions using UMAP [ 65 ] . The UMAP projection was fit on the combined image set and used to visualize the distribution of YNHHS clinical dermatology images together with images from each external source.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Statistical analysis\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We estimated 95% CIs for AUC using nonparametric bootstrap resampling with 10,000 replicates. In each replicate, samples were drawn with replacement from the corresponding test set, and AUC was recalculated using the same evaluation procedure as for the point estimate. Binary and multiclass datasets were resampled within each class to preserve the original class distribution, whereas multilabel datasets were resampled at the sample level while retaining all labels associated with each selected sample. To compare MedPMC-CLIP with BMC-CLIP, identical bootstrap samples were applied to both models within each benchmark. For each bootstrap replicate, benchmark-level differences in accuracy, macro F1 score, and AUC were calculated, averaged within each specialty, and then macro-averaged across the 11 specialties. For the medical QA experiments, individual model accuracies were reported with Wilson 95% CIs. Between-model accuracy differences were evaluated using paired bootstrap resampling with 10,000 replicates, with identical question samples applied to baseline LLaVA-Med and LLaVA-Med with the MedPMC-CLIP encoder. Paired differences were calculated for the overall MMMU and OmniMedVQA results and for each reported OmniMedVQA query category. For the morphology-guided retrieval experiment, differences in Recall@1, Recall@5, and Recall@10 were evaluated using query-level paired bootstrap resampling with 10,000 replicates. All 95% CIs were defined by the 2.5th and 97.5th percentiles of the corresponding bootstrap distributions. The full results are provided in Supplementary Information, Section 2 . We also release the evaluation code in our official GitHub repository to support reproducibility.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Contamination-aware evaluation design\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Potential data contamination between large-scale pretraining corpora and public evaluation benchmarks is an increasingly recognized challenge, especially in the evaluation of foundation models and large language models [ 85 , 49 , 14 ] . We therefore considered dataset provenance when designing the evaluation strategy. Most evaluation datasets used across the 26 benchmark tasks and downstream medical QA experiments were derived from institutional clinical collections, established medical imaging repositories, or challenge tasks rather than directly from PMC articles, which reduces the likelihood of systematic overlap with the MedPMC pretraining corpus. For OmniMedVQA, we further excluded four open-access subsets (“Covid CT,” “CoronaHack,” “Covid19 heywhale,” and “COVIDx CXR-4”) that were curated from the literature, as such datasets are more likely to contain images from preprints or publications that may overlap with PMC-indexed articles. However, provenance alone cannot guarantee complete image-level independence. Public and web-curated benchmarks may aggregate images from heterogeneous sources, and individual images can be reproduced across preprints, publications, repositories, and benchmark datasets.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For this reason, we evaluated MedPMC-derived models across multiple complementary settings rather than relying solely on public benchmark performance. These included controlled architecture-matched comparisons on public benchmarks, downstream medical QA, and a clinical transfer evaluation using 10,524 dermatology patient images from the Yale New Haven Health System. The YNHHS cohort was not part of the PMC-derived pretraining corpus and therefore provides an independent assessment of transfer to clinically realistic patient data. The consistent gains observed across these evaluation settings suggest that MedPMC improves medical visual representations beyond potential benchmark-specific overlap.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Code Availability\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The code used for data curation and pipeline evaluation is publicly available at: https://github.com/Yale-BIDS-Chen-Lab/MedPMC .\"\n }\n]","isConversational":false}}},{"rowIdx":97,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.07670"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Does Bielik Know What It Doesn't Know? Activation Dispersion Separates Entity Familiarity from Factual Reliability Across Model Scale"},"authors":{"kind":"list like","value":["Grzegorz Brzezinka"],"string":"[\n \"Grzegorz Brzezinka\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.07670v1"},"published":{"kind":"timestamp","value":"2026-07-08T00:00:00","string":"2026-07-08T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"When an LLM is asked about something it does not know, does its internal state look different from when it is on familiar ground? A physically motivated intuition says yes: knowledge retrieval should behave like a localized excitation: a compact, selective activation pattern (memory retrieval), whereas confabulation should look delocalized , with activation smeared broadly across the network. As a motivating analogy, no more than that, the dichotomy recalls localized versus extended eigenstates in disordered media ( Anderson , 1958 ) , and it suggests cheap, unsupervised concentration statistics: the inverse participation ratio (IPR) and the entropy of the activation distribution, computed from a single forward pass with no labels, no sampling, and no auxiliary model."},{"type":"para","text":"The literature warns that the naive version of this intuition is almost certainly false. First, large language models are always strongly contextually sparse : for any given token, most attention heads and most Multi-Layer Perceptron (MLP) parameters can be silenced with little effect on the output, regardless of whether the model knows the answer ( Liu et al. , 2023 ; Li et al. , 2023 ) . “The whole network lights up uniformly” is not an operating regime that actually occurs; if there is a signal, it must be a quantitative and geometric difference in how concentrated the activity is, not a qualitative one. Second, mechanistic work locates knowledge awareness in low-dimensional directions rather than in global activation volume: sparse-autoencoder features that fire on known versus unknown entities causally steer refusal and hallucination behavior ( Ferrando et al. , 2025 ) , and circuit-tracing analyses find “known answer” features whose misfiring produces hallucinations ( Lindsey et al. , 2025 ) . A global dispersion measure could easily average such a directional signal into noise."},{"type":"para","text":"This leaves a concrete, if narrower than it once was, empirical gap. Unsupervised single-pass dispersion statistics for hallucination detection have recently appeared: D 2 HScore computes intra-layer dispersion of token representations plus inter-layer drift ( Ding et al. , 2025 ) , EigenTrack tracks covariance-spectrum statistics of hidden states, including spectral entropy, during generation ( Ettori et al. , 2025 ) , and MIND builds an unsupervised internal-state detector from pseudo-labeled generations ( Su et al. , 2024 ) . All of these, however, score generated answers using token-span or covariance-level quantities. What remains untested, to our knowledge, is closed-form, per-neuron dispersion of a single activation vector (IPR and per-neuron spectral entropy - the physics-flavored localization statistics) read at the prompt point, before any answer token , as a detector of knowledge presence rather than answer-level hallucination - benchmarked simultaneously against the supervised ceiling (a linear probe), a selection-aware permutation floor, a token-count lexical baseline, and the standard multi-sample baseline (semantic entropy), across a scale sweep. Nor has any internal-state hallucination study been carried out on a Polish model family: the multilingual shared task Mu-SHROOM spans 14 languages without Polish ( Vázquez et al. , 2025 ) , and existing multilingual internal-state probing covers French, Bangla, and Amharic ( Alvi et al. , 2026 ) - while tokenization of Polish named entities and knowledge of Polish domains are exactly the conditions under which a Polish-first model like Bielik ( Ociepa et al. , 2025a , b ) is deployed."},{"type":"para","text":"We run this test across four Bielik v3.0 instruction-tuned models (1.5B, 4.5B, an 11B, and a 7B Minitron variant obtained by pruning and distillation), on a three-condition dataset spanning four entity domains: Polish athletes, cities, writers, and musicians. Each condition is holding famous ( known ), real but obscure ( unknown-real ), and fabricated but morphologically plausible ( fabricated ) entities, with fabricated names approximately length-matched to famous ones (the residual mismatch is measured and reported as an explicit per-domain lexical baseline). The four domains let us test whether the familiarity signal is a property of a single entity type or generalizes across them (cf. Ferrando et al. , 2025 ) . Beyond detection, we measure (on the athletes domain) the model’s behavioral reliability on the same entities: how often it actually answers correctly about entities it plausibly knows, which lets us put the representational signal and the behavioral phenotype on the same scale axis."},{"type":"heading","level":4,"text":"Contributions."},{"type":"para","text":"A systematic test of closed-form, per-neuron activation dispersion at the prompt point as an unsupervised knowledge-presence detector, with full uncertainty quantification. One-pass IPR and spectral entropy over post-SwiGLU MLP activations separate known from fabricated athletes with Area Under the Receiver Operating Characteristic (AUROC) 0.94–0.98 on every model in the sweep; a supervised linear probe reaches 0.99–1.00. All values clear selection-aware permutation floors of ≈ \\approx 0.70–0.74 (empirical p ≤ 10 − 3 p\\leq 10^{-3} ) and survive held-out layer selection at 0.93–0.99 (Section 4.1 ). On the fabricated-string-free control contrast ( known vs. unknown-real ), dispersion reaches 0.97–1.00 (Section 4.2 )."},{"type":"para","text":"Generalization across four entity types. The same one-pass detector separates known from fabricated across athletes, cities, writers, and musicians (best dispersion 0.95–1.00, probe 0.99–1.00), and probes transfer across domains at mean off-diagonal AUROC 0.987/0.959/0.919/0.931 (1.5B/4.5B/7B/11B). The only large transfer drops involve cities, the one domain whose prompt template differs; a matched-template counterfactual shows the target-side drop is template-caused. Under a shared neutral stem, transfer into cities recovers to 0.999–1.000 at every scale, with a residual cities-as-source asymmetry (0.82–0.98) consistent with a tighter places manifold. A per-head analysis shows the attention-entropy signal is diffuse, with only 1–2 heads shared across all four domains’ top-20 (Section 4.8 )."},{"type":"para","text":"Two dissociated scaling curves, measured on one dataset. The representational familiarity signal (best of the metric pair) is at ceiling on this contrast already at 1.5B, while behavioral factual reliability rises sharply with scale: 0 → \\rightarrow 2 → \\rightarrow 10 → \\rightarrow 19 of 42 known athletes answered fully correctly under a strict judge, 6 → \\rightarrow 16 → \\rightarrow 24 → \\rightarrow 33 under a soft key-facts rubric (Section 4.3 ). This ordering is what frequency-based accounts of factual recall predict ( Mallen et al. , 2023 ; Kandpal et al. , 2023 ) ; the contribution is the clean same-dataset, same-entities measurement of both axes across scale."},{"type":"para","text":"An honest mostly-negative result on the behavioral axis. Within known athletes, correct-vs-hallucination is far harder than known-vs-fabricated, and the full model × \\times threshold grid is inconsistent: dispersion sometimes beats the probe, the probe once falls below its own permutation floor, and the strongest cell (11B probe 0.929) is suggestive rather than established. No metric we tested reads factual reliability the way dispersion reads familiarity (Section 4.6 )."},{"type":"para","text":"A complementarity analysis against semantic entropy. Discrete semantic entropy over five samples reaches AUROC 0.71–0.83 on known-vs-fabricated - below the one-pass dispersion signal with non-overlapping CIs - but wins the correctness contrast (up to 0.87) where dispersion fails; the two consume different information (prompt-side familiarity vs. generation-side consistency) (Section 4.7 )."},{"type":"para","text":"A near-zero-refusal regime, quantified by an answer-level LLM audit. Across 2,520 sampled athlete answers (42 entities × \\times 3 conditions × \\times 5 samples × \\times 4 models), the models decline twice and hedge once (99.88% direct assertions). Only the largest model ever abstains - despite carrying a near-perfectly decodable internal familiarity signal at every scale. This is a concrete, measurable alignment gap for Polish LLM deployments (Section 5 )."},{"type":"heading","level":1,"text":"2 Related work"},{"type":"heading","level":4,"text":"White-box detectors on internal states."},{"type":"para","text":"Azaria and Mitchell ( 2023 ) first showed that a classifier on hidden activations predicts statement truthfulness, establishing the supervised-probe paradigm; Orgad et al. ( 2025 ) sharpened it, finding truthfulness information concentrated in exact-answer tokens, and notably probes that fail to generalize across datasets. Unsupervised and dispersion-flavored detectors are an active line. INSIDE ( Chen et al. , 2024 ) performs eigen-analysis of the covariance of hidden states across multiple sampled responses (EigenScore); LLM-Check ( Sriramanan et al. , 2024 ) makes the covariance line single-pass - its Hidden Score is the log-determinant of the Gram matrix of hidden states over response tokens, alongside logit-entropy and attention-kernel scores; D 2 HScore ( Ding et al. , 2025 ) scores hallucination from training-free intra-layer dispersion of token representations plus inter-layer drift; EigenTrack ( Ettori et al. , 2025 ) tracks covariance-spectrum statistics, including spectral entropy, during generation; MIND ( Su et al. , 2024 ) trains an internal-state detector without human labels from pseudo-labeled generations; HaloScope ( Du et al. , 2024 ) finds a truth subspace in activation space from unlabeled generations. Semantic Entropy Probes ( Kossen et al. , 2024 ) distill the multi-sample semantic-entropy signal into a cheap linear probe on hidden states; Lookback Lens ( Chuang et al. , 2024 ) detects contextual hallucination from attention-allocation ratios alone; Binkowski et al. ( 2025 ) use spectral features (Laplacian eigenvalues) of attention maps; TSV ( Park et al. , 2025 ) steers latents with learned truthful directions. All of the unsupervised detectors above score generated answers from token-span or covariance-level statistics. Our contribution is deliberately minimalist and differently aimed: two closed-form per-neuron dispersion statistics of a single activation vector, read at the prompt point before generation, as a detector of knowledge presence - no training, one forward pass - evaluated against the probe that upper-bounds what the activations contain."},{"type":"heading","level":4,"text":"Black-box baselines."},{"type":"para","text":"Semantic entropy ( Kuhn et al. , 2023 ; Farquhar et al. , 2024 ) clusters multiple sampled answers by bidirectional entailment and measures entropy over meaning classes; SelfCheckGPT ( Manakul et al. , 2023 ) checks sampled answers for mutual consistency. These methods are model-agnostic but require several forward passes plus auxiliary inference; they are our cost reference point (Section 4.7 )."},{"type":"heading","level":4,"text":"Self-knowledge, long-tail knowledge, and mechanistic accounts."},{"type":"para","text":"Kadavath et al. ( 2022 ) showed that models can largely predict whether they know the answer to a question ( P ( IK ) P(\\mathrm{IK}) ), and Yin et al. ( 2023 ) benchmarked whether models know what they do not know at the behavioral level - both foreshadowing the familiarity/reliability split we measure internally and externally on one dataset. Popularity and pretraining-frequency effects bear directly on that split: factual recall tracks entity popularity ( Mallen et al. , 2023 ) and models struggle to learn long-tail knowledge without many supporting documents ( Kandpal et al. , 2023 ) , which predicts exactly the pattern we observe: entity recognition is arriving long before reliable facts about the entity. Mechanistically, Ferrando et al. ( 2025 ) identified causal known-entity/unknown-entity SAE directions on the entity token, which modulate attribute-extraction attention heads; Obeso et al. ( 2025 ) carried that entity-familiarity signal into real-time detection of hallucinated entities in long-form generation; Lindsey et al. ( 2025 ) traced known-answer circuits whose failure modes produce hallucination; and Wu et al. ( 2025 ) showed that long-context factuality is carried by a sparse ( < < 5%), causally necessary set of retrieval heads. Closest to our probe, Moreno Cencerrado et al. ( 2025 ) predict answer accuracy from question-only linear probes - supervised pre-generation prediction, evaluated against answer-level correctness across 7B–70B models; our supervised probe replicates that pre-generation readability on a new family and language, while our headline detector needs no labels at all. Our dispersion results are a coarse-grained, label-free shadow of this machinery."},{"type":"heading","level":4,"text":"Non-English and multilingual hallucination detection."},{"type":"para","text":"The multilingual state of the art bounds our language claim. Mu-SHROOM, the SemEval-2025 shared task on multilingual hallucination detection, spans 14 languages and 43 teams - without Polish - and operates at the output-span level ( Vázquez et al. , 2025 ) . MultiHaluDet probes hidden states multilingually, covering French, Bangla, and Amharic ( Alvi et al. , 2026 ) ; HalluVerse25 offers a fine-grained multilingual benchmark (English, Arabic, Turkish) with no internal-state analysis ( Abdaljalil et al. , 2025 ) . We find no internal-state hallucination study on a Polish-native model family; to our knowledge this work is the first."},{"type":"heading","level":4,"text":"Pitfalls for global measures."},{"type":"para","text":"Massive activations - a handful of neurons with values orders of magnitude larger than the rest, largely input-independent ( Sun et al. , 2024 ) - can dominate any participation-ratio statistic; we winsorize activations before computing dispersion (Section 3.5 ). The effective rank of a representation matrix ( Roy and Vetterli , 2007 ) extends dispersion from single vectors to token spans and is a one-pass relative of EigenScore; we include it in the extended metric set. Contextual-sparsity results ( Liu et al. , 2023 ; Li et al. , 2023 ) calibrate expectations: sparsity is ubiquitous and input-dependent, so only relative dispersion differences between conditions are meaningful."},{"type":"heading","level":1,"text":"3 Method"},{"type":"heading","level":2,"text":"3.1 Dataset: three conditions across four entity domains"},{"type":"para","text":"We construct the dataset in four entity domains: athletes, cities, writers, and musicians, each holding three conditions of 42 entities, 1 1 1 The count of 42 per condition is a deliberate nod to Adams ( 1979 ) . for 126 entities per domain and 504 per model:"},{"type":"para","text":"known : famous entities the models can be presumed to have seen abundantly in pretraining (e.g., Robert Lewandowski and Adam Małysz for athletes, Warszawa for cities, Adam Mickiewicz for writers, Fryderyk Chopin for musicians);"},{"type":"para","text":"unknown-real : real but obscure entities (lower-league athletes, small villages, minor authors and performers) that the models almost certainly cannot describe;"},{"type":"para","text":"fabricated : invented names, morphologically well-formed in Polish (in the spirit of “Roman Lewandowicz”), constructed to be approximately length-matched to the known set of the same domain."},{"type":"para","text":"The athletes domain is our primary domain: it alone carries behavioral labels, semantic-entropy scores, and the refusal audit (Sections 3.2 – 3.7 ); the other three domains carry condition-based detection labels only. The athletes set extends the original 41-per-condition list of the round-1 study by one entity per condition."},{"type":"para","text":"Two Polish prompt templates are used, differing only in the question stem so that a person is asked who and a place what :"},{"type":"para","text":"Kim jest { entity } ? Odpowiedz jednym zdaniem. (“Who is {entity}? Answer in one sentence.”) - athletes, writers, musicians; Czym jest { entity } ? Odpowiedz jednym zdaniem. (“What is {entity}? Answer in one sentence.”) - cities."},{"type":"para","text":"Within a domain the template is uniform, so all variation between prompts is confined to the entity span; the stem difference between the cities template and the three people templates is a controlled confound we return to in Section 4.8 . For the template-causality control reported there, the cities and writers domains were additionally re-extracted under a third, neutral stem grammatical for both people and places - Co wiesz o { entity } ? Odpowiedz jednym zdaniem. (“What do you know about {entity}? Answer in one sentence.”), on all four models."},{"type":"para","text":"The length matching is only approximate, and the model family uses two tokenizers, not one: 1.5B and 4.5B share a tokenizer under which athlete names span 2–8 tokens ( known mean 3.19, unknown-real 3.98, fabricated 3.90), while Minitron-7B and 11B share a different one under which the same names span 4–11 tokens (means 6.17, 6.02, 6.76). Entity token count alone therefore separates known from fabricated athletes at AUROC 0.681 (small-model tokenizer) and 0.612 (large-model tokenizer), a weak but nonzero lexical baseline that we report explicitly and control for in Section 4.2 . It is however far below the dispersion signal. The three new domains were matched at least as tightly as the athletes baseline: token-count-only K-vs-F AUROC is 0.637/0.494 (cities), 0.500/0.500 (writers), and 0.509/0.505 (musicians) under the small/large tokenizers - writers and musicians at chance, cities saturating at 0.637 under the small tokenizer because a handful of single-token major cities cannot be matched by any invented multi-token toponym. Full per-domain screening, replacement logs, and web-verification coverage are documented with the release. The experiments were orchestrated with AI coding assistance. All results were produced and verified by the author."},{"type":"heading","level":2,"text":"3.2 Behavioral labeling (athletes)"},{"type":"para","text":"Behavioral labeling is carried out on the athletes domain only. For each entity and model we sample five answers (chat template, at most 64 new tokens each). Answers to known entities are graded by an LLM judge (Claude Opus 4.8) in two regimes:"},{"type":"para","text":"Strict : a deliberately minimal judge - a bare one-line prompt asking whether the answer is factually correct about the real world, with no rubric, no tolerance for answers truncated by the 64-token generation cap, and unparseable judge outputs defaulting to incorrect . An entity is fully correct if all 5 of 5 samples pass. Small models fail this bar mostly by decorating true identities with invented achievements (e.g., a world-championship title Lewandowski never won), but a part of the strict failure count reflects truncation and parse defaults rather than confabulation - one motivation for the soft rubric below."},{"type":"para","text":"Soft : a key-facts rubric - identity and discipline must be correct; fabricated significant achievements still count as errors; harmless omissions and truncations are tolerated. We report thresholds of 5/5, ≥ \\geq 4/5, and ≥ \\geq 3/5 correct samples."},{"type":"para","text":"The main detection contrast ( known vs. fabricated ) is condition-based and therefore independent of the judge; judges enter only through the behavioral axis (Section 4.6 ). In manual spot checks of 615 soft-judge verdicts (5 samples × \\times 41 known entities × \\times three dense models, on the round-1 athletes list) we found 2 anomalous verdicts ( ≈ \\approx 0.3%); see Section 6 ."},{"type":"para","text":"Refusal and hedging are audited at the answer level over all three athlete conditions: an LLM classifier (Claude Haiku 4.5) with randomized human control labels each of the 2,520 stored answers (42 entities × \\times 3 conditions × \\times 5 samples × \\times 4 models) as a direct assertion, a refusal, or a hedged non-answer (Section 5 ). An earlier six-substring marker heuristic proved brittle in both directions: its single hit was a false positive on a song title, and it missed both refusals the audit found, so we report only the LLM-audited counts."},{"type":"heading","level":2,"text":"3.3 Models"},{"type":"para","text":"We sweep four instruction-tuned Bielik v3.0 checkpoints ( Ociepa et al. , 2025a , b ) : 1.5B (32 transformer layers, d = 1536 d=1536 ), 4.5B (60 layers, d = 2048 d=2048 ), 11B (50 layers, d = 4096 d=4096 ), and Minitron-7B (40 layers, d = 4096 d=4096 ) ( Kinas et al. , 2026 ) , the last obtained by Minitron-style pruning and knowledge distillation ( Muralidharan et al. , 2024 ) from a larger Bielik model rather than trained densely from scratch. The Minitron point therefore extends the scale curve between 4.5B and 11B but is architecturally not a pure member of the dense family; we flag it explicitly wherever it behaves differently. All models share a Mistral-like decoder architecture with SwiGLU MLPs."},{"type":"heading","level":2,"text":"3.4 Signal extraction and measurement points"},{"type":"para","text":"For every prompt we run a single forward pass and record, at each layer: (i) the post-SwiGLU gate activations of the MLP (via forward hooks on act_fn ); (ii) residual-stream hidden states; (iii) attention maps; and (iv) logit-lens distributions (hidden state decoded through the output head). Signals are read at two token positions:"},{"type":"para","text":"the entity point : the last token of the entity name, where mechanistic work locates entity-recognition features ( Ferrando et al. , 2025 ) ;"},{"type":"para","text":"the prompt point : the last token of the prompt, several tokens downstream of the entity."},{"type":"para","text":"The entity point yields nominally higher separability, but it is also maximally exposed to a lexical confound: a rare fabricated string can produce atypical activation statistics for reasons of token frequency alone, independent of knowledge. The prompt point, where the model has integrated the question and lexical identity is no longer the input, is the more conservative measurement, and all headline numbers in this paper are prompt-point numbers . Entity-token indices are computed from the very encoding fed to the model, avoiding Beginning of Sequence (BOS) off-by-one misalignment (verified empirically)."},{"type":"heading","level":2,"text":"3.5 Dispersion metrics"},{"type":"para","text":"Let a ∈ ℝ d mlp a\\in\\mathbb{R}^{d_{\\mathrm{mlp}}} be the vector of post-SwiGLU MLP activations of layer ℓ \\ell at the measurement token. Define the normalized weights"},{"type":"para","text":"and from them"},{"type":"para","text":"1 / IPR 1/\\mathrm{IPR} is the effective number of active neurons; S S is the Shannon entropy of the activation distribution (labelled spectral entropy in the figures). Localized (concentrated) activity means high Inverse Participation Ration (IPR) and low S S ; delocalized activity the reverse. Because a few massive-activation outliers can dominate these statistics irrespective of content ( Sun et al. , 2024 ) , activations are winsorized (nearest-value clipping above the q = 0.99 q=0.99 quantile) before computing p p . Since SwiGLU activations have no exact zeros ( Li et al. , 2023 ) , both measures are continuous rather than combinatorial, which is irrelevant for IPR and entropy."},{"type":"para","text":"The winsorization quantile is a researcher degree of freedom sitting inside the headline metric, so we probed it directly on the 1.5B model by re-extracting raw activations and sweeping q ∈ { 0.95 , 0.99 , 0.999 , 1.0 } q\\in\\{0.95,0.99,0.999,1.0\\} (the last effectively skips winsorization). A strong signal exists at every setting - best-layer AUROC 0.899/0.933/0.836/0.889 for IPR and 0.910/0.944/0.890/0.905 for entropy - but the paper’s setting q = 0.99 q=0.99 is the most favorable of the four, the best layer moves with q q , and the per-layer value at the layer reported in Table 1 degrades substantially at other settings (entropy at layer 22: 0.509 at q = 0.95 q=0.95 , 0.782 at q = 0.999 q=0.999 , 0.731 without winsorization). Dispersion-as-familiarity is robust to the choice, while specific layer-level values are not. Only the 1.5B model was swept."},{"type":"para","text":"The extended metric set comprises of: the effective rank ( Roy and Vetterli , 2007 ) of the matrix of hidden states over the prompt tokens (per layer) - the exponential of the entropy of the normalized singular-value spectrum, a one-pass relative of EigenScore ( Chen et al. , 2024 ) ; attention entropy per head, averaged per layer, in raw and length-normalized variants; and logit-lens entropy ( nostalgebraist , 2020 ) per layer, measuring how early in depth the model commits to an output distribution. As a strictly cheaper output-side baseline we use the entropy of the first answer-token logprobs ."},{"type":"heading","level":2,"text":"3.6 Probe, controls, and evaluation"},{"type":"para","text":"For each residual layer we train an ℓ 2 \\ell_{2} -regularized logistic-regression probe on the raw hidden state (5-fold cross-validated), giving a per-layer supervised upper bound on the linearly decodable signal."},{"type":"para","text":"Because every headline number is a maximum over layers (and, for dispersion, over the {IPR, entropy} pair), we calibrate selection optimism explicitly with three complementary tools. Permutation null with selection: we permute condition labels and rerun the entire pipeline, including the layer (and metric) re-selection, per permutation - 10,000 permutations for dispersion, 1,000 for the probe - and take the 95th percentile of the null maxima as the floor. These selection-aware floors land at 0.701–0.715 (dispersion) and 0.721–0.742 (probe) across the four models, with null means around 0.65; a single label permutation, by contrast, yields 0.62–0.64 and understates the null’s upper range. Any metric must clear these floors, not 0.5. Bootstrap confidence intervals (CI): 95% CIs from 10,000 stratified resamples with the layer fixed at the observed best layer; these quantify sampling noise, not selection - and because they condition on the observed best layer, they are mildly optimistic post-selection (the held-out analysis below is the out-of-sample complement). Held-out layer selection: 200 stratified half-splits in which the layer (and metric) is selected on one half of the entities and the AUROC is evaluated on the other half."},{"type":"para","text":"All discrimination results are AUROC, reported as separability max ( AUC , 1 − AUC ) \\max(\\mathrm{AUC},\\,1-\\mathrm{AUC}) ; the primary contrast is known (0) vs. fabricated (1) with n = 84 n=84 per contrast (42 per class; likewise 84 per contrast in each of the three new domains)."},{"type":"heading","level":2,"text":"3.7 Semantic-entropy baseline"},{"type":"para","text":"From the five stored samples per athlete entity we compute discrete semantic entropy ( Kuhn et al. , 2023 ; Farquhar et al. , 2024 ) . The answers are clustered into meaning-equivalence classes by an LLM clustering judge (Claude Sonnet 5, bidirectional-entailment criterion), and the entropy of the resulting cluster distribution is the score. Clustering succeeded for every row (zero fallback rows in all 4 × 126 4\\times 126 athlete entities). This baseline costs five forward passes of the subject model plus clustering calls, versus one forward pass and no auxiliary model for dispersion."},{"type":"heading","level":1,"text":"4 Results"},{"type":"heading","level":2,"text":"4.1 Dispersion separates known from fabricated at every scale"},{"type":"para","text":"Table 1 and Figure 1 (left) give the headline result for athletes. On every model in the sweep, at least one unsupervised dispersion metric separates known from fabricated with AUROC between 0.946 [0.887, 0.988] (1.5B) and 0.984 [0.952, 1.000] (4.5B), read from a single forward pass at the prompt point with no labels. The supervised probe confirms that the information content of the activations is near-ceiling everywhere (0.990–1.000)."},{"type":"para","text":"Neither number is an artifact of layer-maximum selection (Table 2 ). In 10,000 label permutations that re-select the best layer and metric, the null maximum never reached the observed dispersion values (95th percentiles 0.701–0.715; empirical p ≤ 10 − 4 p\\leq 10^{-4} ); the probe behaves identically against its own null (95th percentiles 0.721–0.742; p ≤ 10 − 3 p\\leq 10^{-3} at 1,000 permutations). The signal also survives held-out selection: choosing the layer (and metric) on a random half of the entities and evaluating on the other half yields out-of-sample dispersion AUROCs of 0.928–0.970 and probe AUROCs of 0.977–0.993. The unsupervised metrics recover most of the gap between the selection floor and the probe ceiling, so the hypothesis in its volumetric form holds: the global dispersion measures work, not only low-dimensional directions. Section 4.8 shows the same detector holds across three further entity domains."},{"type":"heading","level":2,"text":"4.2 Control contrasts: the signal is familiarity, with a measurable lexical residue"},{"type":"para","text":"The three-condition design supports two control contrasts that bound how much of the headline could be lexical rather than epistemic (Table 3 )."},{"type":"para","text":"Known vs. unknown-real (real names only). Dispersion separates famous from obscure real athletes at 0.971–1.000 - as strong as or stronger than the headline contrast (probe: 0.998–1.000) - on names with natural Polish subword statistics, where the token-count baseline is 0.694 under the small-model tokenizer and near chance (0.518) under the large-model one. This is the strongest single piece of evidence that the signal tracks familiarity rather than fabricated-string weirdness."},{"type":"para","text":"Unknown-real vs. fabricated (both unfamiliar). If dispersion were a pure familiarity readout, this contrast should sit near 0.5. But it does not, the dispersion reaches 0.699–0.824 (probe 0.819–0.913), while token count alone is at chance for the small-model tokenizer (0.515) and 0.638 for the large-model one. Some residual property of the fabricated strings (lexical statistics beyond raw length, or morphological plausibility) is detectable in the activations. We treat 0.70–0.82 as an upper bound on the lexical/plausibility component of the headline contrast (it may also reflect genuine partial familiarity with real name-forms): large enough that the headline should not be read as purely epistemic, far too small to account for 0.94–0.98, and cleanly excluded from the known -vs- unknown-real control above."},{"type":"heading","level":2,"text":"4.3 Two scaling curves: signal at ceiling, rising reliability"},{"type":"para","text":"Figure 1 juxtaposes the two axes for athletes. The activation signal (panel a) is flat and at or near ceiling for the best-of-pair statistic max ( IPR , S ) \\max(\\mathrm{IPR},S) : it peaks at 4.5B (0.984 [0.952, 1.000]) and dips at 11B (0.972 [0.939, 0.994]); with bootstrap CIs spanning roughly ± \\pm 0.03–0.06, the dip is uninterpretable as a trend, so we read the statistic as at ceiling on this contrast at every scale - noting that famous-vs-fabricated is an easy contrast, and a ceiling here does not rule out further growth of familiarity itself on a graded-popularity axis. The individual metrics are less stable than the pair: the better metric flips from entropy (1.5B, 4.5B, 7B) to IPR (11B), and 11B entropy falls to 0.896 [0.827, 0.954], whose CI barely touches the 4.5B entropy CI [0.952, 1.000]; held-out selection shows the same picture (11B entropy 0.836 ± 0.056 0.836\\pm 0.056 out-of-sample vs. IPR 0.972 ± 0.016 0.972\\pm 0.016 ). The claim that survives scrutiny is that some member of the dispersion pair is near ceiling on this contrast at every scale - not that either metric alone is scale-stable. Behavioral reliability (panel b) tells the opposite story: fully correct known answers go 0 → \\rightarrow 2 → \\rightarrow 10 → \\rightarrow 19 of 42 (strict) and 6 → \\rightarrow 16 → \\rightarrow 24 → \\rightarrow 33 (soft 5/5) from 1.5B to 11B. Small models know which entities are familiar - their activations say so almost perfectly - yet routinely decorate answers about those entities with fabricated details."},{"type":"para","text":"The Minitron-7B point fits the behavioral curve cleanly between 4.5B and 11B on the strict and soft-5/5 criteria (10 and 24 of 42), which is itself noteworthy: pruning and distillation preserved both the familiarity geometry (dispersion 0.957) and a scale-appropriate level of factual reliability. The one exception is the most lenient threshold, where Minitron is non-monotonic (soft ≥ \\geq 3/5: 34 vs. 36 for the 4.5B model) - a two-entity difference we do not over-interpret here."},{"type":"heading","level":2,"text":"4.4 Depth structure of the signal"},{"type":"para","text":"Figure 2 shows the per-layer picture on a shared relative-depth axis (layer index divided by the deepest layer of its family, which sidesteps the one-index offset between MLP and residual layer numbering). The probe saturates from roughly 20% depth onward in every model: familiarity is linearly decodable through most of the network. Dispersion is choppier - individual layers alternate between strongly expressing and not expressing the signal - but its envelope forms a recognizable mid-depth band."},{"type":"para","text":"A band analysis (band = layers within 90% of the per-model maximum AUROC) quantifies which metrics occupy a consistent relative depth across all four models: spectral entropy shares a common band at relative depth [ 0.48 , 0.77 ] [0.48,0.77] , the probe at [ 0.34 , 1.00 ] [0.34,1.00] , logit-lens entropy at [ 0.58 , 0.68 ] [0.58,0.68] , and mean attention entropy overlaps only at the single depth 0.74 0.74 . IPR and effective rank have no common band across the four models. The mid-to-late-depth “knowledge band” is therefore a real, scale-stable feature for entropy-flavored measures, but not a universal property of every dispersion statistic - consistent with the picture that different layers of different models express the underlying directional signal ( Ferrando et al. , 2025 ) to different volumetric degrees."},{"type":"heading","level":2,"text":"4.5 Extended metrics: not all global measures are equal"},{"type":"para","text":"Table 4 and Figure 3 extend the comparison beyond MLP dispersion. The following three findings stand out. Attention entropy is robust across scale (0.839–0.991), consistent with the mechanistic account in which unknown entities suppress attribute-extraction attention, diffusing attention distributions ( Ferrando et al. , 2025 ; Chuang et al. , 2024 ; Binkowski et al. , 2025 ) . Logit-lens entropy grows with scale (0.722 at 1.5B, 0.937 at 4.5B, then ≈ \\approx 0.98 at 7B and 11B): in larger models, how early the network commits to an answer becomes an increasingly reliable familiarity readout. Effective rank is unstable (0.673–0.942 with no scale trend and no common depth band): extending dispersion from a single activation vector to the token-span covariance does not survive as a reliable one-pass statistic in this setting, despite its kinship to multi-sample EigenScore ( Chen et al. , 2024 ) . Global measures are not interchangeable; the physically motivated single-vector measures and attention entropy transfer across scale, effective rank does not."},{"type":"heading","level":2,"text":"4.6 The behavioral axis is much harder"},{"type":"para","text":"If dispersion measured factual reliability rather than mere familiarity, it should also separate correct answers from hallucinations within the known condition. Table 5 shows the full model × \\times threshold grid, and its eight adequately powered cells (both classes ≥ 8 \\geq 8 ) are inconsistent in every direction that matters. At the Minitron-7B soft-5/5 cell, dispersion (0.748) nominally beats the probe (0.706, which is itself below the single-permutation floor of 0.819); at the 4.5B soft-5/5 cell the ordering reverses (dispersion 0.748 vs. probe 0.762), and its dispersion maximum sits at layer 0. At the 1.5B ≥ \\geq 3/5 cell (the best-balanced cell in the grid, 23/19), the probe (0.584) falls below even its own single-permutation floor (0.677): at 1.5B no reliable behavioral signal is readable at all. At the 11B soft-5/5 cell, the probe reaches 0.929 against a single-permutation floor of 0.744, while dispersion (0.681) is indistinguishable from the first-token-entropy baseline (0.701) and its best layer is layer 1 - embedding-adjacent, a flag that the number is noise rather than signal, which itself supports the negative reading. Given that selection-aware nulls reach 0.72–0.74 at n = 84 n=84 on the detection contrast (Section 3.6 ) and n n is only 42 here, we read the 11B probe cell as suggestive of some weak, linearly decodable correctness signal, not as an established one; the remaining high values in the grid live in underpowered cells (minority class ≤ 7 \\leq 7 ) and should not be read as signal."},{"type":"para","text":"We frame this as a mostly-negative result and, we believe, the paper’s most useful distinction: activation dispersion detects entity familiarity, not factual reliability . Putting this into words, a one-pass dispersion gate can tell you the model has never heard of “Roman Lewandowicz”, while it cannot tell you whether the model is about to attach a fabricated world championship to Robert Lewandowski."},{"type":"heading","level":2,"text":"4.7 Semantic-entropy baseline: a complementary information source"},{"type":"para","text":"Table 6 and Figure 4 compare against the standard multi-sample baseline. On the familiarity contrast, semantic entropy reaches 0.71–0.83 - squarely in the range reported for SE-style methods ( Kuhn et al. , 2023 ; Farquhar et al. , 2024 ) - while one-pass dispersion reaches 0.94–0.98; the 95% CIs of the two do not overlap for any model (e.g., 11B: SE 0.832 [0.743, 0.911] vs. IPR 0.972 [0.939, 0.994]), and SE additionally costs five forward passes of the subject model plus clustering inference."},{"type":"para","text":"The comparison is, however, not apples-to-apples, and the two methods are better read as complementary information sources than as competitors. Dispersion consumes the deterministic prefill and measures prompt-side familiarity. SE on the other hand, consumes the generation distribution (five samples at temperature 0.7) and measures answer-side consistency. On SE’s home turf - the judge-labeled correctness contrast over all 126 athlete entities - SE reaches 0.78–0.87 (0.872 on 11B), precisely where dispersion fails (Section 4.6 ). SE is also strong on known vs. unknown-real (0.83–0.93). And five samples is the low end for SE (ten or more are typical), so the SE numbers here should be read as a floor. Dispersion wins the familiarity contrast; SE wins the correctness contrast; a deployment stack may well want both."},{"type":"para","text":"The per-condition means expose a clean scaling effect: mean SE on known falls from 0.91 (1.5B) through 0.59 and 0.67 to 0.26 (11B), while remaining high on fabricated and unknown-real . Larger models become increasingly self-consistent about entities they know - even when the consistent answer still contains a fabricated detail, which is precisely why SE and factual correctness are not the same axis either."},{"type":"heading","level":2,"text":"4.8 Generalization across entity types"},{"type":"para","text":"The results so far are all on athletes. To test whether the familiarity signal is a property of one entity type or of entity familiarity in general, we replicate the detection contrast on three further domains - cities, writers, and musicians - each with 42 known / unknown-real / fabricated entities (condition-based labels only). Figure 5 summarizes; Table 7 gives the per-domain, per-model separability."},{"type":"heading","level":4,"text":"The signal transfers across domains."},{"type":"para","text":"Training the probe on one domain and evaluating on another (Figure 5 , right) preserves most of the separability: the mean off-diagonal transfer AUROC is 0.987 (1.5B), 0.959 (4.5B), 0.919 (7B), and 0.931 (11B). Transfer among the three people domains (athletes, writers, musicians) is near-perfect at every scale (0.985–0.998). This parallels the cross-entity-type generalization reported by Ferrando et al. ( 2025 ) for their known-entity direction, while extending it to a global, unsupervised statistic on a different model family and language - and it is the opposite of the cross-dataset generalization failures reported for supervised truthfulness probes ( Orgad et al. , 2025 ) , consistent with familiarity being a more portable representation-level quantity than answer truthfulness."},{"type":"heading","level":4,"text":"The transfer drops are template-caused: a matched-template counterfactual."},{"type":"para","text":"Every large off-diagonal drop involves the cities domain, which is the only domain using the Czym jest (“what is”) stem rather than Kim jest (“who is”); the worst cells are cities → \\rightarrow athletes at 0.637 (Minitron) and cities → \\rightarrow musicians at 0.652 (11B), and the people-to-people versus cities-involving gap is + 0.078 +0.078 (4.5B), + 0.133 +0.133 (Minitron), and + 0.129 +0.129 (11B), but only − 0.005 -0.005 at 1.5B. Template and entity type are confounded for cities in the main matrix, so we ran a counterfactual control: cities and writers re-extracted under the shared neutral stem Co wiesz o { entity } ? (Section 3.1 ), which matches the template while varying entity type, on all four models. The neutral extractions are internally valid (within-domain K-vs-F probe 0.99–1.00, dispersion 0.96–1.00), and the decisive entity-type-varying cell recovers to ceiling: writers → \\rightarrow cities under the matched stem reaches probe transfer 0.999/1.000/1.000/1.000 (1.5B/4.5B/7B/11B), indistinguishable from the people → \\rightarrow people reference (0.98–1.00). The target-side drop into cities is therefore template-caused, not an entity-type effect . One asymmetry survives the matched template: cities as source transfers outward less well (cities → \\rightarrow writers probe 0.978/0.918/0.825/0.816), consistent with a tighter, lower-variance places manifold whose fitted probes generalize outward less well - not with places being harder to represent (within-cities separability remains the highest of any domain, Table 7 ) - and matching the original worst cells, which all have cities as the training domain. The template caveat is one Ferrando et al. ( 2025 ) did not encounter, since their cross-entity-type transfer held the prompt frame fixed."},{"type":"heading","level":4,"text":"The knowledge band sits at a common relative depth across domains."},{"type":"para","text":"A layer-band analysis (band = relative depths within 90% of the per-domain maximum AUROC) shows the detector occupies a shared mid-depth band across all four domains: for the probe, the cross-domain intersection is relative depth [ 0.16 , 1.00 ] [0.16,1.00] – [ 0.38 , 1.00 ] [0.38,1.00] with overlap/union rising from 0.74 (1.5B) to 0.91 (11B); for dispersion the common band is roughly [ 0.4 , 0.85 ] [0.4,0.85] , loosest on the 11B dispersion metric (overlap/union 0.28, best-depth spread 0.45). The band is thus a scale-stable, largely domain-independent feature for the probe and for entropy-flavored dispersion, and noisiest exactly where the individual dispersion metric was already least stable (11B, Section 4.3 )."},{"type":"heading","level":4,"text":"The signal is diffuse across attention heads."},{"type":"para","text":"Ranking every head by the known -vs- fabricated separability of its prompt-point attention entropy (Figure 6 ), only 1–2 heads sit in all four domains’ top-20 at any scale, and the mean pairwise cross-domain Spearman correlation of the per-head AUROC vectors rises only modestly with scale: 0.27 (1.5B, 384 heads), 0.50 (4.5B, 960), 0.52 (Minitron, 1280), 0.56 (11B, 1600). The best cross-domain heads sit at relative depth 0.63–0.80 (e.g., 11B L31H0, mean 0.986). The attention-entropy familiarity signal is therefore largely diffuse - spread across many heads that overlap only partially between domains - with the overlap strengthening but staying modest as models grow. This does not contradict the concentrated circuits reported in causal work - the attribute-extraction heads modulated by entity-recognition directions ( Ferrando et al. , 2025 ) , or the sparse, causally necessary retrieval heads of Wu et al. ( 2025 ) : those claims are mediational, ours is marginal correlational separability, and a concentrated causal circuit is fully compatible with diffuse marginal separability once its effects propagate downstream. What the diffuseness does establish is that our global statistics read a broadly distributed correlate of familiarity, not the circuit itself. As on athletes, effective rank remains the weakest metric across domains."},{"type":"heading","level":1,"text":"5 Discussion"},{"type":"heading","level":4,"text":"Two phenomena on different scaling curves."},{"type":"para","text":"The central empirical claim is a dissociation. Entity familiarity at the representation level is already at ceiling on our contrast at 1.5B: cheap, label-free dispersion statistics read it out at 0.94+, and a linear probe at ≈ \\approx 0.99 (whether familiarity as a graded quantity still improves with scale is not measurable on a famous-vs-fabricated contrast that every model aces). Behavioral factual reliability - answering about familiar entities without invented decorations - is nowhere near solved at 1.5B and improves steeply through 11B. These curves cannot be two views of one underlying quantity; a detector of the first is not a detector of the second, as Section 4.6 shows directly. This refines the localization intuition that motivated the study: the delocalized-activation signature marks missing entities , not missing facts . It also gives a coarse-grained, purely volumetric confirmation of the mechanistic picture in which entity-recognition circuitry ( Ferrando et al. , 2025 ; Lindsey et al. , 2025 ) is present and functional even in small models, while the knowledge those circuits gate remains scale-limited - consistent with frequency-based accounts of factual recall, in which long-tail facts arrive only with capacity and data repetition ( Mallen et al. , 2023 ; Kandpal et al. , 2023 ) while entity recognition arrives much earlier. The familiarity axis is moreover not an athletes-specific curiosity: it reproduces across cities, writers, and musicians and transfers between them (Section 4.8 ), so the detector reads a general property of entity familiarity rather than a domain-specific lexical quirk. Though, being diffuse across heads, it is a coarse volumetric shadow of the circuitry, not a localization of it. Nor is it a Bielik-specific quirk. A compact control on a non-Polish-centric family (Gemma-4 at two sizes, identical dataset and pipeline; Appendix A ) replicates both the familiarity signal and the dissociation qualitatively: weaker, and still rising with scale, weakest exactly in the domains whose famous tier is most Polish-local, which is what a signal that tracks actual model familiarity should do."},{"type":"heading","level":4,"text":"The near-zero-refusal regime."},{"type":"para","text":"An answer-level LLM audit (Claude Haiku 4.5) of all 2,520 sampled athlete answers - including 42 entities that do not exist and 42 the models demonstrably know nothing about - finds exactly 2 refusals and 1 hedged answer: 99.88% of answers are direct assertions. The models fluently biografize fabricated athletes while their own activations carry a near-perfectly decodable “I don’t know this entity” signal. Strikingly, both refusals come from the 11B model (one unknown-real , one fabricated entity; the single hedge is from Minitron-7B): only the largest model ever declines , and even it at a per-mille rate. This sharpens rather than weakens the abstention-gap point - the capacity to refuse emerges with scale, exactly as the steering results of Ferrando et al. ( 2025 ) (where activating the unknown-entity direction causes refusal) and the default-refusal circuits of Lindsey et al. ( 2025 ) would lead one to expect, yet it remains behaviorally negligible at 11B while the internal familiarity signal is already near-ceiling at 1.5B. One qualifier is owed: the prompt commands a one-sentence answer ( “Odpowiedz jednym zdaniem.” ), which plausibly pushes an instruction-following model away from abstaining, so the regime is partly a property of the prompt (Section 6 ). The Gemma-4 control bounds that qualifier: under the identical prompts, Gemma-4-12B abstains condition-selectively - 24/42 fabricated and 28/42 unknown-real athletes draw all-five-sample refusals, versus 7/42 known (Appendix A ) - so the prompt does not preclude abstention, and a model of this scale range can surface the familiarity representation behaviorally. That Bielik does not is therefore a tuning/alignment property , not an inevitability of the representational signal or the scale range. The representational precondition for calibrated abstention exists at every scale; the behavior barely exists at any Bielik scale. That is an actionable alignment gap for Bielik family rather than a capability gap ."},{"type":"heading","level":4,"text":"Practical implication: one-pass familiarity gating."},{"type":"para","text":"For Polish LLM deployments, the immediate application is a cheap gate. Compute IPR/spectral entropy at the prompt’s last token during the prefill pass (essentially free), and route high-dispersion queries to retrieval, abstention, or escalation. Unlike SelfCheckGPT or semantic entropy ( Manakul et al. , 2023 ; Farquhar et al. , 2024 ) , this requires no extra samples and no auxiliary model, and unlike probe-based detectors ( Azaria and Mitchell , 2023 ; Kossen et al. , 2024 ) it needs no labeled training data - though where labels are available, the probe’s near-ceiling performance argues for using both. Two caveats apply. The gate reads familiarity, not existence: it scores real-but-obscure and fabricated entities similarly ( unknown-real vs. fabricated 0.65–0.82 across domains), which is acceptable for routing - both belong in retrieval or abstention - but rules out using it to tell “obscure but real” from “nonexistent.” And, per Section 4.6 , the gate flags unfamiliar entities ; it must not be sold as a hallucination detector for content about familiar ones."},{"type":"heading","level":1,"text":"6 Limitations"},{"type":"para","text":"Lexical confound (C1), quantified. Fabricated names are rare strings, and rare tokens alone perturb activation statistics. The length matching between fabricated and known is only approximate: for athletes, token count alone reaches K-vs-F AUROC 0.681/0.612 (small/large tokenizer), and the unknown-real -vs- fabricated dispersion separation of 0.70–0.82 upper-bounds the lexical-plus-plausibility component (Section 4.2 ). The known -vs- unknown-real control (0.97–1.00), free of fabricated strings, shows familiarity dominates; residual lexical contributions to the headline cannot be excluded, but they cannot be the story. The three new domains are matched at least as tightly (token-count K-vs-F AUROC 0.49–0.64; Section 3.1 ), so the generalization result is not carried by a lexical artifact either. Sample size. Each detection contrast has n = 84 n=84 (42 per class; 42 per class on the behavioral axis); bootstrap CIs on headline cells span roughly ± \\pm 0.03–0.06 and selection-aware permutation nulls reach 0.70–0.74, so single-model AUROC differences below ≈ \\approx 0.05 should not be interpreted. Behavioral, SE, and refusal claims are athletes-only. The behavioral axis, semantic-entropy comparison, and refusal audit are measured on athletes; the cities, writers, and musicians domains carry condition-based detection labels only, so the dissociation and abstention-gap claims are demonstrated on one domain even though the detection signal itself is shown to generalize. Template effects in cross-domain transfer. The cities domain uses a different prompt stem ( Czym jest rather than Kim jest ), confounding template and entity type in the main transfer matrix. The matched-template control (Section 4.8 ) resolves the target-side attribution causally - transfer into cities recovers to ceiling under a shared neutral stem - but two caveats remain: the control re-extracted only cities and writers under a single alternative stem, and the residual cities-as-source asymmetry (probe 0.82–0.98 under matched templates, widening with scale) is characterized, not explained; we read it as a manifold-tightness effect, which a targeted geometry analysis could test directly. Language and model family. The main study is one Polish family. A compact control on a non-Polish-centric family (Gemma-4 at two sizes, same Polish dataset, pipeline, and judges; Appendix A ) qualitatively replicates the familiarity signal and the dissociation, which removes the sharpest version of this concern - but it is two checkpoints of one additional family on one language; the full multi-family, multi-language treatment (including an English translation of the dataset) is future work (already in progress). Per-head diffuseness. The attention-entropy signal is spread across many partially overlapping heads (only 1–2 shared across all four domains’ top-20; cross-domain Spearman 0.27–0.56), which qualifies any mechanistic reading: our global statistics are a coarse volumetric shadow of the entity-recognition circuitry ( Ferrando et al. , 2025 ) , not a localization of a small head set. Fabricated non-existence. The 168 invented names (42 per domain) were screened. Automated pool-stage filtering, manual morphology/confusability/notability checks, and, for the three new domains, per-name search-engine verification with a logged replacement trail - but not exhaustively verified against all real persons and places; rare collisions with obscure real individuals are possible, though the models would, with high likelihood, not know such individuals either, so labels are robust to this failure mode. Judge stack and circularity. One model family occupies every rung of the evaluation ladder: strict and soft judging (Claude Opus 4.8), semantic-entropy clustering (Claude Sonnet 5), and the refusal audit (Claude Haiku 4.5). A family-level blind spot about Polish sports biography would contaminate the behavioral counts, the SE correctness column, and the refusal counts simultaneously; no non-Claude judge was run (but is planned for future work). This exposure extends to the behavioral scaling claim itself: the 0 → \\rightarrow 2 → \\rightarrow 10 → \\rightarrow 19 strict and 6 → \\rightarrow 16 → \\rightarrow 24 → \\rightarrow 33 soft curves rest entirely on one judge family: the monotone trend is corroborated by two independent rubrics and is too large to plausibly be judge noise, but it has not been confirmed by a second judge family. The strict judge is additionally a bare one-line prompt whose parse failures default to incorrect (Section 3.2 ), so strict counts conflate confabulation with truncation and parsing artifacts - the soft rubric mitigates but does not eliminate this. Manual inspection of 615 soft-judge verdicts (the known answers of the three dense models) surfaced 2 anomalous cases ( ≈ \\approx 0.3%): one flag of a 4.5B answer crediting Ewa Swoboda with a 2018 European indoor title, where the flag was arguably correct (no European championships were held in 2018) and the strict judge’s pass was the error; and one false flag of an 11B answer about Agnieszka Radwańska that was truncated mid-sentence by the 64-token cap. We report this as a manual-inspection finding, not an on-disk artifact. Refusal operationalization and prompt pressure. Refusal counts are answer-level LLM classifications, not human annotations; and the one-sentence instruction in the prompt plausibly discourages abstention, so the near-zero refusal rate is partly a property of the prompt. The Gemma-4 control bounds this concern - under the identical one-sentence prompt, Gemma-4-12B refuses condition-selectively (Appendix A ), so the prompt does not preclude abstention - but a prompt-variant ablation on Bielik itself (without the instruction, or with an explicit “say so if you are unsure”) was still not run. Winsorization sensitivity. The dispersion quantile q = 0.99 q=0.99 is the most favorable of the four settings swept, the sweep covered only the 1.5B model on the round-1 athletes list (41 per condition), and per-layer values shift substantially with q q (Section 3.5 ); the familiarity signal itself persists at every setting. Underpowered behavioral cells. Four of the twelve grid cells have a minority class below 8 (1.5B 5/5; 4.5B ≥ \\geq 3/5; 11B ≥ \\geq 4/5 and ≥ \\geq 3/5); conclusions rest on the eight powered cells, which are themselves mutually inconsistent (Section 4.6 ) - the honest summary is that no reliable behavioral signal has been demonstrated, not that one was ruled out. Minitron. The 7B point is a pruned/distilled variant ( Muralidharan et al. , 2024 ) , not a dense family member; it tracks the dense scale curve on strict and soft-5/5 criteria but is non-monotonic at soft ≥ \\geq 3/5 (34 vs. 36 for 4.5B), and its layer count and depth profile are not directly comparable."},{"type":"heading","level":1,"text":"7 Reproducibility"},{"type":"para","text":"The full pipeline is included in the aforementioned public code repository: dataset construction, answer sampling and judging, activation extraction, and analysis, is a single command per model. Experiments ran on one Apple-silicon workstation (MPS backend, bf16); no GPU cluster is required at these scales. With Python tooling via uv :"},{"type":"para","text":"Hugging Face access to the gated Bielik v3.0 checkpoints and an Anthropic API key (judging, semantic-entropy clustering) are configured via environment variables . Per-model artifacts are written to results/<model>/ ( signals.parquet , hidden_states.npz , extended_signals.parquet , semantic_entropy.parquet , per-layer AUROC figures) and data/<model>/ ( labeled.parquet with all five sampled answers per entity, plus soft-judge labels); the three additional domains live under data/<model>/domains/<domain>/ and results/<model>/domains/<domain>/ with condition-based signals only. Judged with Claude Opus 4.8 (strict and soft rubrics); semantic-entropy clustering with Claude Sonnet 5; refusal audit with Claude Haiku 4.5. One reproducibility asymmetry should be stated plainly: answer sampling was unseeded (temperature 0.7), so the behavioral counts, SE values, and refusal audit are properties of the frozen answer artifacts shipped with the repository rather than of a fresh run; the activation/dispersion side is deterministic and exactly re-derivable. All numbers in this paper are recomputed from the raw artifacts by included consolidation scripts."},{"type":"heading","level":1,"text":"8 Conclusion"},{"type":"para","text":"Bielik knows what it doesn’t know. At every scale we tested, and legibly enough that two closed-form, unsupervised dispersion statistics read the distinction from a single forward pass at 0.94–0.98 AUROC. What Bielik does not know, below 11B, is the facts themselves: behavioral reliability about familiar athletes climbs from 0/42 to 19/42 across the same sweep, and no volumetric statistic we tested tracks it. Between these two curves sits the deployment problem: a model family that says “I don’t know” twice in 2,520 answers while its activations say exactly that, almost perfectly, at every scale and in every entity domain we tested. Closing that gap - by steering, fine-tuning, or simple prefill-time gating - looks less like a research puzzle and more like unfinished engineering."},{"type":"para","text":"To test whether the findings are Bielik-specific, we ran the unchanged Polish pipeline - identical entity lists, prompts, judges (strict: Claude Opus 4.8), measurement point, and statistics on two checkpoints of a non-Polish-centric family: google/gemma-4-E4B-it (MatFormer-heritage, 42 layers; paired with Bielik-4.5B) and google/gemma-4-12b-it (48 layers; paired with Bielik-11B), identified here by their Hugging Face model IDs. The only adaptation is architectural: the decoder lives under model.language_model and dispersion hooks attach to the per-layer mlp.act_fn , mirroring the Bielik hook path. One interpretive caveat is pre-registered in the analysis: the known tier was designed to be famous for Bielik , and Gemma’s strict known correctness is only 7–17% (Table 8 ), so some “known” entities are genuinely borderline for Gemma; the fabricated-string-free known -vs- unknown-real contrast is therefore the primary familiarity claim for this family, with known -vs- fabricated reported alongside. Refusal counts here are entity-level, marker-based all-five-sample refusals per condition (not the LLM audit used for Bielik)."},{"type":"para","text":"The familiarity signal replicates qualitatively - weaker, patchier, and, unlike Bielik, still rising with scale (mean K-vs-F dispersion 0.83 → \\rightarrow 0.91 from E4B to 12B; on this family the signal is not at ceiling at the smaller scale). The per-domain structure is the informative part: cities and writers are Bielik-grade (K-vs-F dispersion 0.92–1.00), while athletes and musicians (the domains whose famous tier is most Polish-local) drop to 0.66–0.87, exactly where Gemma’s actual familiarity with the known tier is demonstrably low (strict correctness 3/42 and 7/42). The signal dilutes where genuine familiarity is low, as a frequency-based account predicts: it tracks what the model knows, not what the dataset labels as famous. Cross-domain transfer is correspondingly weaker (mean off-diagonal probe 0.74/0.77 vs. Bielik’s 0.92–0.99), bounded by the weak source signal in the people domains rather than by the cities template. The familiarity-vs-reliability dissociation replicates (K-vs-UR probe 0.94 at both scales against strict correctness of 7–17%) - with the abstention twist reported in Section 5 : Gemma-4-12B, unlike every Bielik model and unlike Gemma-4-E4B, refuses condition-selectively in exactly the direction its internal familiarity signal points. The full multi-family treatment is deliberately out of scope here and reserved for follow-up work."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"When an LLM is asked about something it does not know, does its internal state look different from when it is on familiar ground? A physically motivated intuition says yes: knowledge retrieval should behave like a localized excitation: a compact, selective activation pattern (memory retrieval), whereas confabulation should look delocalized , with activation smeared broadly across the network. As a motivating analogy, no more than that, the dichotomy recalls localized versus extended eigenstates in disordered media ( Anderson , 1958 ) , and it suggests cheap, unsupervised concentration statistics: the inverse participation ratio (IPR) and the entropy of the activation distribution, computed from a single forward pass with no labels, no sampling, and no auxiliary model.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The literature warns that the naive version of this intuition is almost certainly false. First, large language models are always strongly contextually sparse : for any given token, most attention heads and most Multi-Layer Perceptron (MLP) parameters can be silenced with little effect on the output, regardless of whether the model knows the answer ( Liu et al. , 2023 ; Li et al. , 2023 ) . “The whole network lights up uniformly” is not an operating regime that actually occurs; if there is a signal, it must be a quantitative and geometric difference in how concentrated the activity is, not a qualitative one. Second, mechanistic work locates knowledge awareness in low-dimensional directions rather than in global activation volume: sparse-autoencoder features that fire on known versus unknown entities causally steer refusal and hallucination behavior ( Ferrando et al. , 2025 ) , and circuit-tracing analyses find “known answer” features whose misfiring produces hallucinations ( Lindsey et al. , 2025 ) . A global dispersion measure could easily average such a directional signal into noise.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This leaves a concrete, if narrower than it once was, empirical gap. Unsupervised single-pass dispersion statistics for hallucination detection have recently appeared: D 2 HScore computes intra-layer dispersion of token representations plus inter-layer drift ( Ding et al. , 2025 ) , EigenTrack tracks covariance-spectrum statistics of hidden states, including spectral entropy, during generation ( Ettori et al. , 2025 ) , and MIND builds an unsupervised internal-state detector from pseudo-labeled generations ( Su et al. , 2024 ) . All of these, however, score generated answers using token-span or covariance-level quantities. What remains untested, to our knowledge, is closed-form, per-neuron dispersion of a single activation vector (IPR and per-neuron spectral entropy - the physics-flavored localization statistics) read at the prompt point, before any answer token , as a detector of knowledge presence rather than answer-level hallucination - benchmarked simultaneously against the supervised ceiling (a linear probe), a selection-aware permutation floor, a token-count lexical baseline, and the standard multi-sample baseline (semantic entropy), across a scale sweep. Nor has any internal-state hallucination study been carried out on a Polish model family: the multilingual shared task Mu-SHROOM spans 14 languages without Polish ( Vázquez et al. , 2025 ) , and existing multilingual internal-state probing covers French, Bangla, and Amharic ( Alvi et al. , 2026 ) - while tokenization of Polish named entities and knowledge of Polish domains are exactly the conditions under which a Polish-first model like Bielik ( Ociepa et al. , 2025a , b ) is deployed.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We run this test across four Bielik v3.0 instruction-tuned models (1.5B, 4.5B, an 11B, and a 7B Minitron variant obtained by pruning and distillation), on a three-condition dataset spanning four entity domains: Polish athletes, cities, writers, and musicians. Each condition is holding famous ( known ), real but obscure ( unknown-real ), and fabricated but morphologically plausible ( fabricated ) entities, with fabricated names approximately length-matched to famous ones (the residual mismatch is measured and reported as an explicit per-domain lexical baseline). The four domains let us test whether the familiarity signal is a property of a single entity type or generalizes across them (cf. Ferrando et al. , 2025 ) . Beyond detection, we measure (on the athletes domain) the model’s behavioral reliability on the same entities: how often it actually answers correctly about entities it plausibly knows, which lets us put the representational signal and the behavioral phenotype on the same scale axis.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Contributions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A systematic test of closed-form, per-neuron activation dispersion at the prompt point as an unsupervised knowledge-presence detector, with full uncertainty quantification. One-pass IPR and spectral entropy over post-SwiGLU MLP activations separate known from fabricated athletes with Area Under the Receiver Operating Characteristic (AUROC) 0.94–0.98 on every model in the sweep; a supervised linear probe reaches 0.99–1.00. All values clear selection-aware permutation floors of ≈ \\\\approx 0.70–0.74 (empirical p ≤ 10 − 3 p\\\\leq 10^{-3} ) and survive held-out layer selection at 0.93–0.99 (Section 4.1 ). On the fabricated-string-free control contrast ( known vs. unknown-real ), dispersion reaches 0.97–1.00 (Section 4.2 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Generalization across four entity types. The same one-pass detector separates known from fabricated across athletes, cities, writers, and musicians (best dispersion 0.95–1.00, probe 0.99–1.00), and probes transfer across domains at mean off-diagonal AUROC 0.987/0.959/0.919/0.931 (1.5B/4.5B/7B/11B). The only large transfer drops involve cities, the one domain whose prompt template differs; a matched-template counterfactual shows the target-side drop is template-caused. Under a shared neutral stem, transfer into cities recovers to 0.999–1.000 at every scale, with a residual cities-as-source asymmetry (0.82–0.98) consistent with a tighter places manifold. A per-head analysis shows the attention-entropy signal is diffuse, with only 1–2 heads shared across all four domains’ top-20 (Section 4.8 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Two dissociated scaling curves, measured on one dataset. The representational familiarity signal (best of the metric pair) is at ceiling on this contrast already at 1.5B, while behavioral factual reliability rises sharply with scale: 0 → \\\\rightarrow 2 → \\\\rightarrow 10 → \\\\rightarrow 19 of 42 known athletes answered fully correctly under a strict judge, 6 → \\\\rightarrow 16 → \\\\rightarrow 24 → \\\\rightarrow 33 under a soft key-facts rubric (Section 4.3 ). This ordering is what frequency-based accounts of factual recall predict ( Mallen et al. , 2023 ; Kandpal et al. , 2023 ) ; the contribution is the clean same-dataset, same-entities measurement of both axes across scale.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An honest mostly-negative result on the behavioral axis. Within known athletes, correct-vs-hallucination is far harder than known-vs-fabricated, and the full model × \\\\times threshold grid is inconsistent: dispersion sometimes beats the probe, the probe once falls below its own permutation floor, and the strongest cell (11B probe 0.929) is suggestive rather than established. No metric we tested reads factual reliability the way dispersion reads familiarity (Section 4.6 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A complementarity analysis against semantic entropy. Discrete semantic entropy over five samples reaches AUROC 0.71–0.83 on known-vs-fabricated - below the one-pass dispersion signal with non-overlapping CIs - but wins the correctness contrast (up to 0.87) where dispersion fails; the two consume different information (prompt-side familiarity vs. generation-side consistency) (Section 4.7 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A near-zero-refusal regime, quantified by an answer-level LLM audit. Across 2,520 sampled athlete answers (42 entities × \\\\times 3 conditions × \\\\times 5 samples × \\\\times 4 models), the models decline twice and hedge once (99.88% direct assertions). Only the largest model ever abstains - despite carrying a near-perfectly decodable internal familiarity signal at every scale. This is a concrete, measurable alignment gap for Polish LLM deployments (Section 5 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Related work\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"White-box detectors on internal states.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Azaria and Mitchell ( 2023 ) first showed that a classifier on hidden activations predicts statement truthfulness, establishing the supervised-probe paradigm; Orgad et al. ( 2025 ) sharpened it, finding truthfulness information concentrated in exact-answer tokens, and notably probes that fail to generalize across datasets. Unsupervised and dispersion-flavored detectors are an active line. INSIDE ( Chen et al. , 2024 ) performs eigen-analysis of the covariance of hidden states across multiple sampled responses (EigenScore); LLM-Check ( Sriramanan et al. , 2024 ) makes the covariance line single-pass - its Hidden Score is the log-determinant of the Gram matrix of hidden states over response tokens, alongside logit-entropy and attention-kernel scores; D 2 HScore ( Ding et al. , 2025 ) scores hallucination from training-free intra-layer dispersion of token representations plus inter-layer drift; EigenTrack ( Ettori et al. , 2025 ) tracks covariance-spectrum statistics, including spectral entropy, during generation; MIND ( Su et al. , 2024 ) trains an internal-state detector without human labels from pseudo-labeled generations; HaloScope ( Du et al. , 2024 ) finds a truth subspace in activation space from unlabeled generations. Semantic Entropy Probes ( Kossen et al. , 2024 ) distill the multi-sample semantic-entropy signal into a cheap linear probe on hidden states; Lookback Lens ( Chuang et al. , 2024 ) detects contextual hallucination from attention-allocation ratios alone; Binkowski et al. ( 2025 ) use spectral features (Laplacian eigenvalues) of attention maps; TSV ( Park et al. , 2025 ) steers latents with learned truthful directions. All of the unsupervised detectors above score generated answers from token-span or covariance-level statistics. Our contribution is deliberately minimalist and differently aimed: two closed-form per-neuron dispersion statistics of a single activation vector, read at the prompt point before generation, as a detector of knowledge presence - no training, one forward pass - evaluated against the probe that upper-bounds what the activations contain.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Black-box baselines.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Semantic entropy ( Kuhn et al. , 2023 ; Farquhar et al. , 2024 ) clusters multiple sampled answers by bidirectional entailment and measures entropy over meaning classes; SelfCheckGPT ( Manakul et al. , 2023 ) checks sampled answers for mutual consistency. These methods are model-agnostic but require several forward passes plus auxiliary inference; they are our cost reference point (Section 4.7 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Self-knowledge, long-tail knowledge, and mechanistic accounts.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Kadavath et al. ( 2022 ) showed that models can largely predict whether they know the answer to a question ( P ( IK ) P(\\\\mathrm{IK}) ), and Yin et al. ( 2023 ) benchmarked whether models know what they do not know at the behavioral level - both foreshadowing the familiarity/reliability split we measure internally and externally on one dataset. Popularity and pretraining-frequency effects bear directly on that split: factual recall tracks entity popularity ( Mallen et al. , 2023 ) and models struggle to learn long-tail knowledge without many supporting documents ( Kandpal et al. , 2023 ) , which predicts exactly the pattern we observe: entity recognition is arriving long before reliable facts about the entity. Mechanistically, Ferrando et al. ( 2025 ) identified causal known-entity/unknown-entity SAE directions on the entity token, which modulate attribute-extraction attention heads; Obeso et al. ( 2025 ) carried that entity-familiarity signal into real-time detection of hallucinated entities in long-form generation; Lindsey et al. ( 2025 ) traced known-answer circuits whose failure modes produce hallucination; and Wu et al. ( 2025 ) showed that long-context factuality is carried by a sparse ( < < 5%), causally necessary set of retrieval heads. Closest to our probe, Moreno Cencerrado et al. ( 2025 ) predict answer accuracy from question-only linear probes - supervised pre-generation prediction, evaluated against answer-level correctness across 7B–70B models; our supervised probe replicates that pre-generation readability on a new family and language, while our headline detector needs no labels at all. Our dispersion results are a coarse-grained, label-free shadow of this machinery.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Non-English and multilingual hallucination detection.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The multilingual state of the art bounds our language claim. Mu-SHROOM, the SemEval-2025 shared task on multilingual hallucination detection, spans 14 languages and 43 teams - without Polish - and operates at the output-span level ( Vázquez et al. , 2025 ) . MultiHaluDet probes hidden states multilingually, covering French, Bangla, and Amharic ( Alvi et al. , 2026 ) ; HalluVerse25 offers a fine-grained multilingual benchmark (English, Arabic, Turkish) with no internal-state analysis ( Abdaljalil et al. , 2025 ) . We find no internal-state hallucination study on a Polish-native model family; to our knowledge this work is the first.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Pitfalls for global measures.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Massive activations - a handful of neurons with values orders of magnitude larger than the rest, largely input-independent ( Sun et al. , 2024 ) - can dominate any participation-ratio statistic; we winsorize activations before computing dispersion (Section 3.5 ). The effective rank of a representation matrix ( Roy and Vetterli , 2007 ) extends dispersion from single vectors to token spans and is a one-pass relative of EigenScore; we include it in the extended metric set. Contextual-sparsity results ( Liu et al. , 2023 ; Li et al. , 2023 ) calibrate expectations: sparsity is ubiquitous and input-dependent, so only relative dispersion differences between conditions are meaningful.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Method\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 Dataset: three conditions across four entity domains\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We construct the dataset in four entity domains: athletes, cities, writers, and musicians, each holding three conditions of 42 entities, 1 1 1 The count of 42 per condition is a deliberate nod to Adams ( 1979 ) . for 126 entities per domain and 504 per model:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"known : famous entities the models can be presumed to have seen abundantly in pretraining (e.g., Robert Lewandowski and Adam Małysz for athletes, Warszawa for cities, Adam Mickiewicz for writers, Fryderyk Chopin for musicians);\"\n },\n {\n \"type\": \"para\",\n \"text\": \"unknown-real : real but obscure entities (lower-league athletes, small villages, minor authors and performers) that the models almost certainly cannot describe;\"\n },\n {\n \"type\": \"para\",\n \"text\": \"fabricated : invented names, morphologically well-formed in Polish (in the spirit of “Roman Lewandowicz”), constructed to be approximately length-matched to the known set of the same domain.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The athletes domain is our primary domain: it alone carries behavioral labels, semantic-entropy scores, and the refusal audit (Sections 3.2 – 3.7 ); the other three domains carry condition-based detection labels only. The athletes set extends the original 41-per-condition list of the round-1 study by one entity per condition.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Two Polish prompt templates are used, differing only in the question stem so that a person is asked who and a place what :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Kim jest { entity } ? Odpowiedz jednym zdaniem. (“Who is {entity}? Answer in one sentence.”) - athletes, writers, musicians; Czym jest { entity } ? Odpowiedz jednym zdaniem. (“What is {entity}? Answer in one sentence.”) - cities.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Within a domain the template is uniform, so all variation between prompts is confined to the entity span; the stem difference between the cities template and the three people templates is a controlled confound we return to in Section 4.8 . For the template-causality control reported there, the cities and writers domains were additionally re-extracted under a third, neutral stem grammatical for both people and places - Co wiesz o { entity } ? Odpowiedz jednym zdaniem. (“What do you know about {entity}? Answer in one sentence.”), on all four models.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The length matching is only approximate, and the model family uses two tokenizers, not one: 1.5B and 4.5B share a tokenizer under which athlete names span 2–8 tokens ( known mean 3.19, unknown-real 3.98, fabricated 3.90), while Minitron-7B and 11B share a different one under which the same names span 4–11 tokens (means 6.17, 6.02, 6.76). Entity token count alone therefore separates known from fabricated athletes at AUROC 0.681 (small-model tokenizer) and 0.612 (large-model tokenizer), a weak but nonzero lexical baseline that we report explicitly and control for in Section 4.2 . It is however far below the dispersion signal. The three new domains were matched at least as tightly as the athletes baseline: token-count-only K-vs-F AUROC is 0.637/0.494 (cities), 0.500/0.500 (writers), and 0.509/0.505 (musicians) under the small/large tokenizers - writers and musicians at chance, cities saturating at 0.637 under the small tokenizer because a handful of single-token major cities cannot be matched by any invented multi-token toponym. Full per-domain screening, replacement logs, and web-verification coverage are documented with the release. The experiments were orchestrated with AI coding assistance. All results were produced and verified by the author.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2 Behavioral labeling (athletes)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Behavioral labeling is carried out on the athletes domain only. For each entity and model we sample five answers (chat template, at most 64 new tokens each). Answers to known entities are graded by an LLM judge (Claude Opus 4.8) in two regimes:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Strict : a deliberately minimal judge - a bare one-line prompt asking whether the answer is factually correct about the real world, with no rubric, no tolerance for answers truncated by the 64-token generation cap, and unparseable judge outputs defaulting to incorrect . An entity is fully correct if all 5 of 5 samples pass. Small models fail this bar mostly by decorating true identities with invented achievements (e.g., a world-championship title Lewandowski never won), but a part of the strict failure count reflects truncation and parse defaults rather than confabulation - one motivation for the soft rubric below.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Soft : a key-facts rubric - identity and discipline must be correct; fabricated significant achievements still count as errors; harmless omissions and truncations are tolerated. We report thresholds of 5/5, ≥ \\\\geq 4/5, and ≥ \\\\geq 3/5 correct samples.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The main detection contrast ( known vs. fabricated ) is condition-based and therefore independent of the judge; judges enter only through the behavioral axis (Section 4.6 ). In manual spot checks of 615 soft-judge verdicts (5 samples × \\\\times 41 known entities × \\\\times three dense models, on the round-1 athletes list) we found 2 anomalous verdicts ( ≈ \\\\approx 0.3%); see Section 6 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Refusal and hedging are audited at the answer level over all three athlete conditions: an LLM classifier (Claude Haiku 4.5) with randomized human control labels each of the 2,520 stored answers (42 entities × \\\\times 3 conditions × \\\\times 5 samples × \\\\times 4 models) as a direct assertion, a refusal, or a hedged non-answer (Section 5 ). An earlier six-substring marker heuristic proved brittle in both directions: its single hit was a false positive on a song title, and it missed both refusals the audit found, so we report only the LLM-audited counts.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.3 Models\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We sweep four instruction-tuned Bielik v3.0 checkpoints ( Ociepa et al. , 2025a , b ) : 1.5B (32 transformer layers, d = 1536 d=1536 ), 4.5B (60 layers, d = 2048 d=2048 ), 11B (50 layers, d = 4096 d=4096 ), and Minitron-7B (40 layers, d = 4096 d=4096 ) ( Kinas et al. , 2026 ) , the last obtained by Minitron-style pruning and knowledge distillation ( Muralidharan et al. , 2024 ) from a larger Bielik model rather than trained densely from scratch. The Minitron point therefore extends the scale curve between 4.5B and 11B but is architecturally not a pure member of the dense family; we flag it explicitly wherever it behaves differently. All models share a Mistral-like decoder architecture with SwiGLU MLPs.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.4 Signal extraction and measurement points\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For every prompt we run a single forward pass and record, at each layer: (i) the post-SwiGLU gate activations of the MLP (via forward hooks on act_fn ); (ii) residual-stream hidden states; (iii) attention maps; and (iv) logit-lens distributions (hidden state decoded through the output head). Signals are read at two token positions:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"the entity point : the last token of the entity name, where mechanistic work locates entity-recognition features ( Ferrando et al. , 2025 ) ;\"\n },\n {\n \"type\": \"para\",\n \"text\": \"the prompt point : the last token of the prompt, several tokens downstream of the entity.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The entity point yields nominally higher separability, but it is also maximally exposed to a lexical confound: a rare fabricated string can produce atypical activation statistics for reasons of token frequency alone, independent of knowledge. The prompt point, where the model has integrated the question and lexical identity is no longer the input, is the more conservative measurement, and all headline numbers in this paper are prompt-point numbers . Entity-token indices are computed from the very encoding fed to the model, avoiding Beginning of Sequence (BOS) off-by-one misalignment (verified empirically).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.5 Dispersion metrics\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let a ∈ ℝ d mlp a\\\\in\\\\mathbb{R}^{d_{\\\\mathrm{mlp}}} be the vector of post-SwiGLU MLP activations of layer ℓ \\\\ell at the measurement token. Define the normalized weights\"\n },\n {\n \"type\": \"para\",\n \"text\": \"and from them\"\n },\n {\n \"type\": \"para\",\n \"text\": \"1 / IPR 1/\\\\mathrm{IPR} is the effective number of active neurons; S S is the Shannon entropy of the activation distribution (labelled spectral entropy in the figures). Localized (concentrated) activity means high Inverse Participation Ration (IPR) and low S S ; delocalized activity the reverse. Because a few massive-activation outliers can dominate these statistics irrespective of content ( Sun et al. , 2024 ) , activations are winsorized (nearest-value clipping above the q = 0.99 q=0.99 quantile) before computing p p . Since SwiGLU activations have no exact zeros ( Li et al. , 2023 ) , both measures are continuous rather than combinatorial, which is irrelevant for IPR and entropy.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The winsorization quantile is a researcher degree of freedom sitting inside the headline metric, so we probed it directly on the 1.5B model by re-extracting raw activations and sweeping q ∈ { 0.95 , 0.99 , 0.999 , 1.0 } q\\\\in\\\\{0.95,0.99,0.999,1.0\\\\} (the last effectively skips winsorization). A strong signal exists at every setting - best-layer AUROC 0.899/0.933/0.836/0.889 for IPR and 0.910/0.944/0.890/0.905 for entropy - but the paper’s setting q = 0.99 q=0.99 is the most favorable of the four, the best layer moves with q q , and the per-layer value at the layer reported in Table 1 degrades substantially at other settings (entropy at layer 22: 0.509 at q = 0.95 q=0.95 , 0.782 at q = 0.999 q=0.999 , 0.731 without winsorization). Dispersion-as-familiarity is robust to the choice, while specific layer-level values are not. Only the 1.5B model was swept.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The extended metric set comprises of: the effective rank ( Roy and Vetterli , 2007 ) of the matrix of hidden states over the prompt tokens (per layer) - the exponential of the entropy of the normalized singular-value spectrum, a one-pass relative of EigenScore ( Chen et al. , 2024 ) ; attention entropy per head, averaged per layer, in raw and length-normalized variants; and logit-lens entropy ( nostalgebraist , 2020 ) per layer, measuring how early in depth the model commits to an output distribution. As a strictly cheaper output-side baseline we use the entropy of the first answer-token logprobs .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.6 Probe, controls, and evaluation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For each residual layer we train an ℓ 2 \\\\ell_{2} -regularized logistic-regression probe on the raw hidden state (5-fold cross-validated), giving a per-layer supervised upper bound on the linearly decodable signal.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Because every headline number is a maximum over layers (and, for dispersion, over the {IPR, entropy} pair), we calibrate selection optimism explicitly with three complementary tools. Permutation null with selection: we permute condition labels and rerun the entire pipeline, including the layer (and metric) re-selection, per permutation - 10,000 permutations for dispersion, 1,000 for the probe - and take the 95th percentile of the null maxima as the floor. These selection-aware floors land at 0.701–0.715 (dispersion) and 0.721–0.742 (probe) across the four models, with null means around 0.65; a single label permutation, by contrast, yields 0.62–0.64 and understates the null’s upper range. Any metric must clear these floors, not 0.5. Bootstrap confidence intervals (CI): 95% CIs from 10,000 stratified resamples with the layer fixed at the observed best layer; these quantify sampling noise, not selection - and because they condition on the observed best layer, they are mildly optimistic post-selection (the held-out analysis below is the out-of-sample complement). Held-out layer selection: 200 stratified half-splits in which the layer (and metric) is selected on one half of the entities and the AUROC is evaluated on the other half.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All discrimination results are AUROC, reported as separability max ( AUC , 1 − AUC ) \\\\max(\\\\mathrm{AUC},\\\\,1-\\\\mathrm{AUC}) ; the primary contrast is known (0) vs. fabricated (1) with n = 84 n=84 per contrast (42 per class; likewise 84 per contrast in each of the three new domains).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.7 Semantic-entropy baseline\"\n },\n {\n \"type\": \"para\",\n \"text\": \"From the five stored samples per athlete entity we compute discrete semantic entropy ( Kuhn et al. , 2023 ; Farquhar et al. , 2024 ) . The answers are clustered into meaning-equivalence classes by an LLM clustering judge (Claude Sonnet 5, bidirectional-entailment criterion), and the entropy of the resulting cluster distribution is the score. Clustering succeeded for every row (zero fallback rows in all 4 × 126 4\\\\times 126 athlete entities). This baseline costs five forward passes of the subject model plus clustering calls, versus one forward pass and no auxiliary model for dispersion.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Results\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.1 Dispersion separates known from fabricated at every scale\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 1 and Figure 1 (left) give the headline result for athletes. On every model in the sweep, at least one unsupervised dispersion metric separates known from fabricated with AUROC between 0.946 [0.887, 0.988] (1.5B) and 0.984 [0.952, 1.000] (4.5B), read from a single forward pass at the prompt point with no labels. The supervised probe confirms that the information content of the activations is near-ceiling everywhere (0.990–1.000).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Neither number is an artifact of layer-maximum selection (Table 2 ). In 10,000 label permutations that re-select the best layer and metric, the null maximum never reached the observed dispersion values (95th percentiles 0.701–0.715; empirical p ≤ 10 − 4 p\\\\leq 10^{-4} ); the probe behaves identically against its own null (95th percentiles 0.721–0.742; p ≤ 10 − 3 p\\\\leq 10^{-3} at 1,000 permutations). The signal also survives held-out selection: choosing the layer (and metric) on a random half of the entities and evaluating on the other half yields out-of-sample dispersion AUROCs of 0.928–0.970 and probe AUROCs of 0.977–0.993. The unsupervised metrics recover most of the gap between the selection floor and the probe ceiling, so the hypothesis in its volumetric form holds: the global dispersion measures work, not only low-dimensional directions. Section 4.8 shows the same detector holds across three further entity domains.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.2 Control contrasts: the signal is familiarity, with a measurable lexical residue\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The three-condition design supports two control contrasts that bound how much of the headline could be lexical rather than epistemic (Table 3 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Known vs. unknown-real (real names only). Dispersion separates famous from obscure real athletes at 0.971–1.000 - as strong as or stronger than the headline contrast (probe: 0.998–1.000) - on names with natural Polish subword statistics, where the token-count baseline is 0.694 under the small-model tokenizer and near chance (0.518) under the large-model one. This is the strongest single piece of evidence that the signal tracks familiarity rather than fabricated-string weirdness.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Unknown-real vs. fabricated (both unfamiliar). If dispersion were a pure familiarity readout, this contrast should sit near 0.5. But it does not, the dispersion reaches 0.699–0.824 (probe 0.819–0.913), while token count alone is at chance for the small-model tokenizer (0.515) and 0.638 for the large-model one. Some residual property of the fabricated strings (lexical statistics beyond raw length, or morphological plausibility) is detectable in the activations. We treat 0.70–0.82 as an upper bound on the lexical/plausibility component of the headline contrast (it may also reflect genuine partial familiarity with real name-forms): large enough that the headline should not be read as purely epistemic, far too small to account for 0.94–0.98, and cleanly excluded from the known -vs- unknown-real control above.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.3 Two scaling curves: signal at ceiling, rising reliability\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 1 juxtaposes the two axes for athletes. The activation signal (panel a) is flat and at or near ceiling for the best-of-pair statistic max ( IPR , S ) \\\\max(\\\\mathrm{IPR},S) : it peaks at 4.5B (0.984 [0.952, 1.000]) and dips at 11B (0.972 [0.939, 0.994]); with bootstrap CIs spanning roughly ± \\\\pm 0.03–0.06, the dip is uninterpretable as a trend, so we read the statistic as at ceiling on this contrast at every scale - noting that famous-vs-fabricated is an easy contrast, and a ceiling here does not rule out further growth of familiarity itself on a graded-popularity axis. The individual metrics are less stable than the pair: the better metric flips from entropy (1.5B, 4.5B, 7B) to IPR (11B), and 11B entropy falls to 0.896 [0.827, 0.954], whose CI barely touches the 4.5B entropy CI [0.952, 1.000]; held-out selection shows the same picture (11B entropy 0.836 ± 0.056 0.836\\\\pm 0.056 out-of-sample vs. IPR 0.972 ± 0.016 0.972\\\\pm 0.016 ). The claim that survives scrutiny is that some member of the dispersion pair is near ceiling on this contrast at every scale - not that either metric alone is scale-stable. Behavioral reliability (panel b) tells the opposite story: fully correct known answers go 0 → \\\\rightarrow 2 → \\\\rightarrow 10 → \\\\rightarrow 19 of 42 (strict) and 6 → \\\\rightarrow 16 → \\\\rightarrow 24 → \\\\rightarrow 33 (soft 5/5) from 1.5B to 11B. Small models know which entities are familiar - their activations say so almost perfectly - yet routinely decorate answers about those entities with fabricated details.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The Minitron-7B point fits the behavioral curve cleanly between 4.5B and 11B on the strict and soft-5/5 criteria (10 and 24 of 42), which is itself noteworthy: pruning and distillation preserved both the familiarity geometry (dispersion 0.957) and a scale-appropriate level of factual reliability. The one exception is the most lenient threshold, where Minitron is non-monotonic (soft ≥ \\\\geq 3/5: 34 vs. 36 for the 4.5B model) - a two-entity difference we do not over-interpret here.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.4 Depth structure of the signal\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 2 shows the per-layer picture on a shared relative-depth axis (layer index divided by the deepest layer of its family, which sidesteps the one-index offset between MLP and residual layer numbering). The probe saturates from roughly 20% depth onward in every model: familiarity is linearly decodable through most of the network. Dispersion is choppier - individual layers alternate between strongly expressing and not expressing the signal - but its envelope forms a recognizable mid-depth band.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A band analysis (band = layers within 90% of the per-model maximum AUROC) quantifies which metrics occupy a consistent relative depth across all four models: spectral entropy shares a common band at relative depth [ 0.48 , 0.77 ] [0.48,0.77] , the probe at [ 0.34 , 1.00 ] [0.34,1.00] , logit-lens entropy at [ 0.58 , 0.68 ] [0.58,0.68] , and mean attention entropy overlaps only at the single depth 0.74 0.74 . IPR and effective rank have no common band across the four models. The mid-to-late-depth “knowledge band” is therefore a real, scale-stable feature for entropy-flavored measures, but not a universal property of every dispersion statistic - consistent with the picture that different layers of different models express the underlying directional signal ( Ferrando et al. , 2025 ) to different volumetric degrees.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.5 Extended metrics: not all global measures are equal\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 4 and Figure 3 extend the comparison beyond MLP dispersion. The following three findings stand out. Attention entropy is robust across scale (0.839–0.991), consistent with the mechanistic account in which unknown entities suppress attribute-extraction attention, diffusing attention distributions ( Ferrando et al. , 2025 ; Chuang et al. , 2024 ; Binkowski et al. , 2025 ) . Logit-lens entropy grows with scale (0.722 at 1.5B, 0.937 at 4.5B, then ≈ \\\\approx 0.98 at 7B and 11B): in larger models, how early the network commits to an answer becomes an increasingly reliable familiarity readout. Effective rank is unstable (0.673–0.942 with no scale trend and no common depth band): extending dispersion from a single activation vector to the token-span covariance does not survive as a reliable one-pass statistic in this setting, despite its kinship to multi-sample EigenScore ( Chen et al. , 2024 ) . Global measures are not interchangeable; the physically motivated single-vector measures and attention entropy transfer across scale, effective rank does not.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.6 The behavioral axis is much harder\"\n },\n {\n \"type\": \"para\",\n \"text\": \"If dispersion measured factual reliability rather than mere familiarity, it should also separate correct answers from hallucinations within the known condition. Table 5 shows the full model × \\\\times threshold grid, and its eight adequately powered cells (both classes ≥ 8 \\\\geq 8 ) are inconsistent in every direction that matters. At the Minitron-7B soft-5/5 cell, dispersion (0.748) nominally beats the probe (0.706, which is itself below the single-permutation floor of 0.819); at the 4.5B soft-5/5 cell the ordering reverses (dispersion 0.748 vs. probe 0.762), and its dispersion maximum sits at layer 0. At the 1.5B ≥ \\\\geq 3/5 cell (the best-balanced cell in the grid, 23/19), the probe (0.584) falls below even its own single-permutation floor (0.677): at 1.5B no reliable behavioral signal is readable at all. At the 11B soft-5/5 cell, the probe reaches 0.929 against a single-permutation floor of 0.744, while dispersion (0.681) is indistinguishable from the first-token-entropy baseline (0.701) and its best layer is layer 1 - embedding-adjacent, a flag that the number is noise rather than signal, which itself supports the negative reading. Given that selection-aware nulls reach 0.72–0.74 at n = 84 n=84 on the detection contrast (Section 3.6 ) and n n is only 42 here, we read the 11B probe cell as suggestive of some weak, linearly decodable correctness signal, not as an established one; the remaining high values in the grid live in underpowered cells (minority class ≤ 7 \\\\leq 7 ) and should not be read as signal.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We frame this as a mostly-negative result and, we believe, the paper’s most useful distinction: activation dispersion detects entity familiarity, not factual reliability . Putting this into words, a one-pass dispersion gate can tell you the model has never heard of “Roman Lewandowicz”, while it cannot tell you whether the model is about to attach a fabricated world championship to Robert Lewandowski.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.7 Semantic-entropy baseline: a complementary information source\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 6 and Figure 4 compare against the standard multi-sample baseline. On the familiarity contrast, semantic entropy reaches 0.71–0.83 - squarely in the range reported for SE-style methods ( Kuhn et al. , 2023 ; Farquhar et al. , 2024 ) - while one-pass dispersion reaches 0.94–0.98; the 95% CIs of the two do not overlap for any model (e.g., 11B: SE 0.832 [0.743, 0.911] vs. IPR 0.972 [0.939, 0.994]), and SE additionally costs five forward passes of the subject model plus clustering inference.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The comparison is, however, not apples-to-apples, and the two methods are better read as complementary information sources than as competitors. Dispersion consumes the deterministic prefill and measures prompt-side familiarity. SE on the other hand, consumes the generation distribution (five samples at temperature 0.7) and measures answer-side consistency. On SE’s home turf - the judge-labeled correctness contrast over all 126 athlete entities - SE reaches 0.78–0.87 (0.872 on 11B), precisely where dispersion fails (Section 4.6 ). SE is also strong on known vs. unknown-real (0.83–0.93). And five samples is the low end for SE (ten or more are typical), so the SE numbers here should be read as a floor. Dispersion wins the familiarity contrast; SE wins the correctness contrast; a deployment stack may well want both.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The per-condition means expose a clean scaling effect: mean SE on known falls from 0.91 (1.5B) through 0.59 and 0.67 to 0.26 (11B), while remaining high on fabricated and unknown-real . Larger models become increasingly self-consistent about entities they know - even when the consistent answer still contains a fabricated detail, which is precisely why SE and factual correctness are not the same axis either.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"4.8 Generalization across entity types\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The results so far are all on athletes. To test whether the familiarity signal is a property of one entity type or of entity familiarity in general, we replicate the detection contrast on three further domains - cities, writers, and musicians - each with 42 known / unknown-real / fabricated entities (condition-based labels only). Figure 5 summarizes; Table 7 gives the per-domain, per-model separability.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"The signal transfers across domains.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Training the probe on one domain and evaluating on another (Figure 5 , right) preserves most of the separability: the mean off-diagonal transfer AUROC is 0.987 (1.5B), 0.959 (4.5B), 0.919 (7B), and 0.931 (11B). Transfer among the three people domains (athletes, writers, musicians) is near-perfect at every scale (0.985–0.998). This parallels the cross-entity-type generalization reported by Ferrando et al. ( 2025 ) for their known-entity direction, while extending it to a global, unsupervised statistic on a different model family and language - and it is the opposite of the cross-dataset generalization failures reported for supervised truthfulness probes ( Orgad et al. , 2025 ) , consistent with familiarity being a more portable representation-level quantity than answer truthfulness.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"The transfer drops are template-caused: a matched-template counterfactual.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Every large off-diagonal drop involves the cities domain, which is the only domain using the Czym jest (“what is”) stem rather than Kim jest (“who is”); the worst cells are cities → \\\\rightarrow athletes at 0.637 (Minitron) and cities → \\\\rightarrow musicians at 0.652 (11B), and the people-to-people versus cities-involving gap is + 0.078 +0.078 (4.5B), + 0.133 +0.133 (Minitron), and + 0.129 +0.129 (11B), but only − 0.005 -0.005 at 1.5B. Template and entity type are confounded for cities in the main matrix, so we ran a counterfactual control: cities and writers re-extracted under the shared neutral stem Co wiesz o { entity } ? (Section 3.1 ), which matches the template while varying entity type, on all four models. The neutral extractions are internally valid (within-domain K-vs-F probe 0.99–1.00, dispersion 0.96–1.00), and the decisive entity-type-varying cell recovers to ceiling: writers → \\\\rightarrow cities under the matched stem reaches probe transfer 0.999/1.000/1.000/1.000 (1.5B/4.5B/7B/11B), indistinguishable from the people → \\\\rightarrow people reference (0.98–1.00). The target-side drop into cities is therefore template-caused, not an entity-type effect . One asymmetry survives the matched template: cities as source transfers outward less well (cities → \\\\rightarrow writers probe 0.978/0.918/0.825/0.816), consistent with a tighter, lower-variance places manifold whose fitted probes generalize outward less well - not with places being harder to represent (within-cities separability remains the highest of any domain, Table 7 ) - and matching the original worst cells, which all have cities as the training domain. The template caveat is one Ferrando et al. ( 2025 ) did not encounter, since their cross-entity-type transfer held the prompt frame fixed.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"The knowledge band sits at a common relative depth across domains.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A layer-band analysis (band = relative depths within 90% of the per-domain maximum AUROC) shows the detector occupies a shared mid-depth band across all four domains: for the probe, the cross-domain intersection is relative depth [ 0.16 , 1.00 ] [0.16,1.00] – [ 0.38 , 1.00 ] [0.38,1.00] with overlap/union rising from 0.74 (1.5B) to 0.91 (11B); for dispersion the common band is roughly [ 0.4 , 0.85 ] [0.4,0.85] , loosest on the 11B dispersion metric (overlap/union 0.28, best-depth spread 0.45). The band is thus a scale-stable, largely domain-independent feature for the probe and for entropy-flavored dispersion, and noisiest exactly where the individual dispersion metric was already least stable (11B, Section 4.3 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"The signal is diffuse across attention heads.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Ranking every head by the known -vs- fabricated separability of its prompt-point attention entropy (Figure 6 ), only 1–2 heads sit in all four domains’ top-20 at any scale, and the mean pairwise cross-domain Spearman correlation of the per-head AUROC vectors rises only modestly with scale: 0.27 (1.5B, 384 heads), 0.50 (4.5B, 960), 0.52 (Minitron, 1280), 0.56 (11B, 1600). The best cross-domain heads sit at relative depth 0.63–0.80 (e.g., 11B L31H0, mean 0.986). The attention-entropy familiarity signal is therefore largely diffuse - spread across many heads that overlap only partially between domains - with the overlap strengthening but staying modest as models grow. This does not contradict the concentrated circuits reported in causal work - the attribute-extraction heads modulated by entity-recognition directions ( Ferrando et al. , 2025 ) , or the sparse, causally necessary retrieval heads of Wu et al. ( 2025 ) : those claims are mediational, ours is marginal correlational separability, and a concentrated causal circuit is fully compatible with diffuse marginal separability once its effects propagate downstream. What the diffuseness does establish is that our global statistics read a broadly distributed correlate of familiarity, not the circuit itself. As on athletes, effective rank remains the weakest metric across domains.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Discussion\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Two phenomena on different scaling curves.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The central empirical claim is a dissociation. Entity familiarity at the representation level is already at ceiling on our contrast at 1.5B: cheap, label-free dispersion statistics read it out at 0.94+, and a linear probe at ≈ \\\\approx 0.99 (whether familiarity as a graded quantity still improves with scale is not measurable on a famous-vs-fabricated contrast that every model aces). Behavioral factual reliability - answering about familiar entities without invented decorations - is nowhere near solved at 1.5B and improves steeply through 11B. These curves cannot be two views of one underlying quantity; a detector of the first is not a detector of the second, as Section 4.6 shows directly. This refines the localization intuition that motivated the study: the delocalized-activation signature marks missing entities , not missing facts . It also gives a coarse-grained, purely volumetric confirmation of the mechanistic picture in which entity-recognition circuitry ( Ferrando et al. , 2025 ; Lindsey et al. , 2025 ) is present and functional even in small models, while the knowledge those circuits gate remains scale-limited - consistent with frequency-based accounts of factual recall, in which long-tail facts arrive only with capacity and data repetition ( Mallen et al. , 2023 ; Kandpal et al. , 2023 ) while entity recognition arrives much earlier. The familiarity axis is moreover not an athletes-specific curiosity: it reproduces across cities, writers, and musicians and transfers between them (Section 4.8 ), so the detector reads a general property of entity familiarity rather than a domain-specific lexical quirk. Though, being diffuse across heads, it is a coarse volumetric shadow of the circuitry, not a localization of it. Nor is it a Bielik-specific quirk. A compact control on a non-Polish-centric family (Gemma-4 at two sizes, identical dataset and pipeline; Appendix A ) replicates both the familiarity signal and the dissociation qualitatively: weaker, and still rising with scale, weakest exactly in the domains whose famous tier is most Polish-local, which is what a signal that tracks actual model familiarity should do.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"The near-zero-refusal regime.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An answer-level LLM audit (Claude Haiku 4.5) of all 2,520 sampled athlete answers - including 42 entities that do not exist and 42 the models demonstrably know nothing about - finds exactly 2 refusals and 1 hedged answer: 99.88% of answers are direct assertions. The models fluently biografize fabricated athletes while their own activations carry a near-perfectly decodable “I don’t know this entity” signal. Strikingly, both refusals come from the 11B model (one unknown-real , one fabricated entity; the single hedge is from Minitron-7B): only the largest model ever declines , and even it at a per-mille rate. This sharpens rather than weakens the abstention-gap point - the capacity to refuse emerges with scale, exactly as the steering results of Ferrando et al. ( 2025 ) (where activating the unknown-entity direction causes refusal) and the default-refusal circuits of Lindsey et al. ( 2025 ) would lead one to expect, yet it remains behaviorally negligible at 11B while the internal familiarity signal is already near-ceiling at 1.5B. One qualifier is owed: the prompt commands a one-sentence answer ( “Odpowiedz jednym zdaniem.” ), which plausibly pushes an instruction-following model away from abstaining, so the regime is partly a property of the prompt (Section 6 ). The Gemma-4 control bounds that qualifier: under the identical prompts, Gemma-4-12B abstains condition-selectively - 24/42 fabricated and 28/42 unknown-real athletes draw all-five-sample refusals, versus 7/42 known (Appendix A ) - so the prompt does not preclude abstention, and a model of this scale range can surface the familiarity representation behaviorally. That Bielik does not is therefore a tuning/alignment property , not an inevitability of the representational signal or the scale range. The representational precondition for calibrated abstention exists at every scale; the behavior barely exists at any Bielik scale. That is an actionable alignment gap for Bielik family rather than a capability gap .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Practical implication: one-pass familiarity gating.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For Polish LLM deployments, the immediate application is a cheap gate. Compute IPR/spectral entropy at the prompt’s last token during the prefill pass (essentially free), and route high-dispersion queries to retrieval, abstention, or escalation. Unlike SelfCheckGPT or semantic entropy ( Manakul et al. , 2023 ; Farquhar et al. , 2024 ) , this requires no extra samples and no auxiliary model, and unlike probe-based detectors ( Azaria and Mitchell , 2023 ; Kossen et al. , 2024 ) it needs no labeled training data - though where labels are available, the probe’s near-ceiling performance argues for using both. Two caveats apply. The gate reads familiarity, not existence: it scores real-but-obscure and fabricated entities similarly ( unknown-real vs. fabricated 0.65–0.82 across domains), which is acceptable for routing - both belong in retrieval or abstention - but rules out using it to tell “obscure but real” from “nonexistent.” And, per Section 4.6 , the gate flags unfamiliar entities ; it must not be sold as a hallucination detector for content about familiar ones.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6 Limitations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Lexical confound (C1), quantified. Fabricated names are rare strings, and rare tokens alone perturb activation statistics. The length matching between fabricated and known is only approximate: for athletes, token count alone reaches K-vs-F AUROC 0.681/0.612 (small/large tokenizer), and the unknown-real -vs- fabricated dispersion separation of 0.70–0.82 upper-bounds the lexical-plus-plausibility component (Section 4.2 ). The known -vs- unknown-real control (0.97–1.00), free of fabricated strings, shows familiarity dominates; residual lexical contributions to the headline cannot be excluded, but they cannot be the story. The three new domains are matched at least as tightly (token-count K-vs-F AUROC 0.49–0.64; Section 3.1 ), so the generalization result is not carried by a lexical artifact either. Sample size. Each detection contrast has n = 84 n=84 (42 per class; 42 per class on the behavioral axis); bootstrap CIs on headline cells span roughly ± \\\\pm 0.03–0.06 and selection-aware permutation nulls reach 0.70–0.74, so single-model AUROC differences below ≈ \\\\approx 0.05 should not be interpreted. Behavioral, SE, and refusal claims are athletes-only. The behavioral axis, semantic-entropy comparison, and refusal audit are measured on athletes; the cities, writers, and musicians domains carry condition-based detection labels only, so the dissociation and abstention-gap claims are demonstrated on one domain even though the detection signal itself is shown to generalize. Template effects in cross-domain transfer. The cities domain uses a different prompt stem ( Czym jest rather than Kim jest ), confounding template and entity type in the main transfer matrix. The matched-template control (Section 4.8 ) resolves the target-side attribution causally - transfer into cities recovers to ceiling under a shared neutral stem - but two caveats remain: the control re-extracted only cities and writers under a single alternative stem, and the residual cities-as-source asymmetry (probe 0.82–0.98 under matched templates, widening with scale) is characterized, not explained; we read it as a manifold-tightness effect, which a targeted geometry analysis could test directly. Language and model family. The main study is one Polish family. A compact control on a non-Polish-centric family (Gemma-4 at two sizes, same Polish dataset, pipeline, and judges; Appendix A ) qualitatively replicates the familiarity signal and the dissociation, which removes the sharpest version of this concern - but it is two checkpoints of one additional family on one language; the full multi-family, multi-language treatment (including an English translation of the dataset) is future work (already in progress). Per-head diffuseness. The attention-entropy signal is spread across many partially overlapping heads (only 1–2 shared across all four domains’ top-20; cross-domain Spearman 0.27–0.56), which qualifies any mechanistic reading: our global statistics are a coarse volumetric shadow of the entity-recognition circuitry ( Ferrando et al. , 2025 ) , not a localization of a small head set. Fabricated non-existence. The 168 invented names (42 per domain) were screened. Automated pool-stage filtering, manual morphology/confusability/notability checks, and, for the three new domains, per-name search-engine verification with a logged replacement trail - but not exhaustively verified against all real persons and places; rare collisions with obscure real individuals are possible, though the models would, with high likelihood, not know such individuals either, so labels are robust to this failure mode. Judge stack and circularity. One model family occupies every rung of the evaluation ladder: strict and soft judging (Claude Opus 4.8), semantic-entropy clustering (Claude Sonnet 5), and the refusal audit (Claude Haiku 4.5). A family-level blind spot about Polish sports biography would contaminate the behavioral counts, the SE correctness column, and the refusal counts simultaneously; no non-Claude judge was run (but is planned for future work). This exposure extends to the behavioral scaling claim itself: the 0 → \\\\rightarrow 2 → \\\\rightarrow 10 → \\\\rightarrow 19 strict and 6 → \\\\rightarrow 16 → \\\\rightarrow 24 → \\\\rightarrow 33 soft curves rest entirely on one judge family: the monotone trend is corroborated by two independent rubrics and is too large to plausibly be judge noise, but it has not been confirmed by a second judge family. The strict judge is additionally a bare one-line prompt whose parse failures default to incorrect (Section 3.2 ), so strict counts conflate confabulation with truncation and parsing artifacts - the soft rubric mitigates but does not eliminate this. Manual inspection of 615 soft-judge verdicts (the known answers of the three dense models) surfaced 2 anomalous cases ( ≈ \\\\approx 0.3%): one flag of a 4.5B answer crediting Ewa Swoboda with a 2018 European indoor title, where the flag was arguably correct (no European championships were held in 2018) and the strict judge’s pass was the error; and one false flag of an 11B answer about Agnieszka Radwańska that was truncated mid-sentence by the 64-token cap. We report this as a manual-inspection finding, not an on-disk artifact. Refusal operationalization and prompt pressure. Refusal counts are answer-level LLM classifications, not human annotations; and the one-sentence instruction in the prompt plausibly discourages abstention, so the near-zero refusal rate is partly a property of the prompt. The Gemma-4 control bounds this concern - under the identical one-sentence prompt, Gemma-4-12B refuses condition-selectively (Appendix A ), so the prompt does not preclude abstention - but a prompt-variant ablation on Bielik itself (without the instruction, or with an explicit “say so if you are unsure”) was still not run. Winsorization sensitivity. The dispersion quantile q = 0.99 q=0.99 is the most favorable of the four settings swept, the sweep covered only the 1.5B model on the round-1 athletes list (41 per condition), and per-layer values shift substantially with q q (Section 3.5 ); the familiarity signal itself persists at every setting. Underpowered behavioral cells. Four of the twelve grid cells have a minority class below 8 (1.5B 5/5; 4.5B ≥ \\\\geq 3/5; 11B ≥ \\\\geq 4/5 and ≥ \\\\geq 3/5); conclusions rest on the eight powered cells, which are themselves mutually inconsistent (Section 4.6 ) - the honest summary is that no reliable behavioral signal has been demonstrated, not that one was ruled out. Minitron. The 7B point is a pruned/distilled variant ( Muralidharan et al. , 2024 ) , not a dense family member; it tracks the dense scale curve on strict and soft-5/5 criteria but is non-monotonic at soft ≥ \\\\geq 3/5 (34 vs. 36 for 4.5B), and its layer count and depth profile are not directly comparable.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"7 Reproducibility\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The full pipeline is included in the aforementioned public code repository: dataset construction, answer sampling and judging, activation extraction, and analysis, is a single command per model. Experiments ran on one Apple-silicon workstation (MPS backend, bf16); no GPU cluster is required at these scales. With Python tooling via uv :\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Hugging Face access to the gated Bielik v3.0 checkpoints and an Anthropic API key (judging, semantic-entropy clustering) are configured via environment variables . Per-model artifacts are written to results/<model>/ ( signals.parquet , hidden_states.npz , extended_signals.parquet , semantic_entropy.parquet , per-layer AUROC figures) and data/<model>/ ( labeled.parquet with all five sampled answers per entity, plus soft-judge labels); the three additional domains live under data/<model>/domains/<domain>/ and results/<model>/domains/<domain>/ with condition-based signals only. Judged with Claude Opus 4.8 (strict and soft rubrics); semantic-entropy clustering with Claude Sonnet 5; refusal audit with Claude Haiku 4.5. One reproducibility asymmetry should be stated plainly: answer sampling was unseeded (temperature 0.7), so the behavioral counts, SE values, and refusal audit are properties of the frozen answer artifacts shipped with the repository rather than of a fresh run; the activation/dispersion side is deterministic and exactly re-derivable. All numbers in this paper are recomputed from the raw artifacts by included consolidation scripts.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"8 Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Bielik knows what it doesn’t know. At every scale we tested, and legibly enough that two closed-form, unsupervised dispersion statistics read the distinction from a single forward pass at 0.94–0.98 AUROC. What Bielik does not know, below 11B, is the facts themselves: behavioral reliability about familiar athletes climbs from 0/42 to 19/42 across the same sweep, and no volumetric statistic we tested tracks it. Between these two curves sits the deployment problem: a model family that says “I don’t know” twice in 2,520 answers while its activations say exactly that, almost perfectly, at every scale and in every entity domain we tested. Closing that gap - by steering, fine-tuning, or simple prefill-time gating - looks less like a research puzzle and more like unfinished engineering.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To test whether the findings are Bielik-specific, we ran the unchanged Polish pipeline - identical entity lists, prompts, judges (strict: Claude Opus 4.8), measurement point, and statistics on two checkpoints of a non-Polish-centric family: google/gemma-4-E4B-it (MatFormer-heritage, 42 layers; paired with Bielik-4.5B) and google/gemma-4-12b-it (48 layers; paired with Bielik-11B), identified here by their Hugging Face model IDs. The only adaptation is architectural: the decoder lives under model.language_model and dispersion hooks attach to the per-layer mlp.act_fn , mirroring the Bielik hook path. One interpretive caveat is pre-registered in the analysis: the known tier was designed to be famous for Bielik , and Gemma’s strict known correctness is only 7–17% (Table 8 ), so some “known” entities are genuinely borderline for Gemma; the fabricated-string-free known -vs- unknown-real contrast is therefore the primary familiarity claim for this family, with known -vs- fabricated reported alongside. Refusal counts here are entity-level, marker-based all-five-sample refusals per condition (not the LLM audit used for Bielik).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The familiarity signal replicates qualitatively - weaker, patchier, and, unlike Bielik, still rising with scale (mean K-vs-F dispersion 0.83 → \\\\rightarrow 0.91 from E4B to 12B; on this family the signal is not at ceiling at the smaller scale). The per-domain structure is the informative part: cities and writers are Bielik-grade (K-vs-F dispersion 0.92–1.00), while athletes and musicians (the domains whose famous tier is most Polish-local) drop to 0.66–0.87, exactly where Gemma’s actual familiarity with the known tier is demonstrably low (strict correctness 3/42 and 7/42). The signal dilutes where genuine familiarity is low, as a frequency-based account predicts: it tracks what the model knows, not what the dataset labels as famous. Cross-domain transfer is correspondingly weaker (mean off-diagonal probe 0.74/0.77 vs. Bielik’s 0.92–0.99), bounded by the weak source signal in the people domains rather than by the cities template. The familiarity-vs-reliability dissociation replicates (K-vs-UR probe 0.94 at both scales against strict correctness of 7–17%) - with the abstention twist reported in Section 5 : Gemma-4-12B, unlike every Bielik model and unlike Gemma-4-E4B, refuses condition-selectively in exactly the direction its internal familiarity signal points. The full multi-family treatment is deliberately out of scope here and reserved for follow-up work.\"\n }\n]","isConversational":false}}},{"rowIdx":98,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.07665"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"Guidance Breaks the Fitted Operator: A Terminal-Fitted Repair for Classifier-Free Guidance"},"authors":{"kind":"list like","value":["Shiheng Zhang"],"string":"[\n \"Shiheng Zhang\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.07665v1"},"published":{"kind":"timestamp","value":"2026-07-08T00:00:00","string":"2026-07-08T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Diffusion and flow-matching samplers integrate a probability-flow ODE whose velocity field stiffens as the noise scale σ → σ min \\sigma\\to\\sigma_{\\min} : on a coordinate normal to the data manifold the exact flow contracts at exponent one, by σ n + 1 / σ n \\sigma_{n+1}/\\sigma_{n} per step. A recent fitted-operator analysis of unguided samplers [Zhang, 2026 ] shows that this terminal layer admits a unique fitted (layer-exact) one-step operator: a frozen-field Euler step—Euler with the velocity held at its start-of-step value—reproduces that exact contraction if and only if its integration variable is affine in σ \\sigma . This singles out the σ \\sigma -clock step, algebraically identical to deterministic DDIM [Song et al. , 2021 ] , and makes it asymptotic-preserving (AP) on the exactly solvable calibration models studied there [Zhang, 2026 ] —uniformly accurate as σ min → 0 \\sigma_{\\min}\\to 0 . This paper asks one question of that lens: what survives of DDIM’s fitted-operator protection once we turn on guidance?"},{"type":"heading","level":4,"text":"Guidance re-stiffens the discriminative subspace."},{"type":"para","text":"Classifier-free guidance (CFG) [Ho and Salimans, 2022 ] runs the sampler on an extrapolated denoiser D w = ( 1 + w ) D c − w D u D_{w}=(1+w)D_{\\mathrm{c}}-wD_{\\mathrm{u}} , that is, on a second velocity field η w = η c + w ( η c − η u ) \\eta_{w}=\\eta_{\\mathrm{c}}+w(\\eta_{\\mathrm{c}}-\\eta_{\\mathrm{u}}) (the η \\eta ’s are noise-prediction fields; Section 2 ) with guidance scale g = 1 + w g=1+w . This is not a uniform stiffening. On the commuting Gaussian model of Section 2 , the guided terminal exponent μ w ( σ ) \\mu_{w}(\\sigma) splits by layer (a layer here is a regime of state-space directions, not a network layer; Figure 1 ): it stays exactly one on directions shared by the class and the marginal, freezes to zero on tangential directions, and rises to 1 + w 1+w on the discriminative directions—the subspace that separates the class from the marginal (Proposition 1 ). Guidance re-stiffens precisely this discriminative subspace, and it does so with a second singular parameter: w w enters the terminal structure on the same footing as σ min \\sigma_{\\min} . Because D w D_{w} is not any distribution’s posterior mean, DDIM’s fitted-operator protection lapses there: on coarse meshes, vanilla DDIM + + CFG leaves the class of fitted operators and its terminal residual diverges as σ min → 0 \\sigma_{\\min}\\to 0 ."},{"type":"heading","level":4,"text":"Two orthogonal bills."},{"type":"para","text":"This is a statement about the solver . A parallel line studies how the continuous guided law itself departs from the tilted conditional it is meant to sample [Bradley and Nakkiran, 2025 ; Chidambaram et al. , 2024 ] : that charges the model ; we charge the discretization —our reference solution is always the exact guided flow’s own pushforward. The two bills are orthogonal, and conflating them has obscured how much of guidance’s terminal pathology—the oversaturation and norm inflation seen at large w w [Sadat et al. , 2025 ] —has a solver-induced component that appears even when the exact guided flow is the reference, from stepping guidance with an operator it has quietly un-fitted."},{"type":"heading","level":4,"text":"The repair is one coefficient."},{"type":"para","text":"Rectifying the anomalous exponent 1 + w 1+w by a gauge change and pushing the σ \\sigma -clock step through it yields a one-coefficient modification of CFG that touches nothing else in the sampler—only the coefficient on the guidance direction D u − D c D_{\\mathrm{u}}-D_{\\mathrm{c}} changes:"},{"type":"para","text":"with r = σ n + 1 / σ n r=\\sigma_{n+1}/\\sigma_{n} . It uses the same two denoiser evaluations per step—the implementation delta is a single line, at zero additional NFE (network function evaluations)—agrees with CFG to first order, and reduces to DDIM at w = 0 w=0 . We call it the guided fitted step, or fitted CFG . It is the unique scalar coefficient that is exact on the discriminative terminal layer using no spectral information (Lemma 2 ); it never reflects a coordinate across the class manifold (Proposition 3 ); and its one-step r → 0 r\\to 0 limit lands on the conditional denoiser D c D_{\\mathrm{c}} (the terminal-limit class projection), where vanilla CFG lands on the overshot D w D_{w} , the discrete mechanism of oversaturation."},{"type":"heading","level":4,"text":"Contributions."},{"type":"para","text":"Structure and a barrier. A layer classification of the guided terminal exponent (Proposition 1 ) and a guided clock barrier (Theorem 1 ) with three ordered step-size thresholds—reflection, residual-amplification failure (Section 3 ), absolute stability—and the resulting two-tier guidance tax on step count (Corollary 1 )."},{"type":"para","text":"Oversaturation as an endpoint, and the guidance interval. A model-free identity reading one-step DDIM + + CFG’s overshoot to D w D_{w} as the barrier’s r → 0 r\\to 0 endpoint (Remark 2 ), and a parameter-free prediction of the terminal edge of limited-interval guidance [Kynkäänniemi et al. , 2024 ] (Remark 3 )."},{"type":"para","text":"A spectrum-free fitted repair. The guided fitted step (Definition 1 ): first-order consistent, sign-preserving and terminal-exact on the model, unique (Lemma 2 ), and—on the discriminative crossover—free of the σ min \\sigma_{\\min} -divergent blow-up and first-order accurate as σ min → 0 \\sigma_{\\min}\\to 0 (Proposition 4 , Theorem 2 ): asymptotic-preserving in σ min \\sigma_{\\min} , though not uniform in w w ."},{"type":"para","text":"A scoped empirical program. On learned CIFAR-10 edm [Karras et al. , 2022 ] checkpoints, fitted CFG is a zero-extra-NFE high-guidance stabilizer: the residual and clipping certificates improve in both w ∈ { 6.5 , 8 } w\\in\\{6.5,8\\} cells; the hard-cell image evaluations (two 5k blocks, a 50k replication) give it the best FID with target accuracy preserved, and a DINOv2 backbone-swap keeps the same fitted-favoring ordering; a nine-cell grid gives 9 / 9 9/9 FID wins over CFG. We report the limits alongside—the CFG-favoring KID split and threshold-derived interval baselines that win FID/KID at lower conditionality (Section 4 )."},{"type":"para","text":"Section 6 collects the claim boundaries."},{"type":"heading","level":1,"text":"2 The guided flow and its terminal layers"},{"type":"heading","level":4,"text":"The unguided flow."},{"type":"para","text":"We work throughout in the variance-exploding family: p σ = p data ∗ 𝒩 ( 0 , σ 2 I ) p_{\\sigma}=p_{\\rm data}*\\mathcal{N}(0,\\sigma^{2}I) for σ ∈ [ σ min , σ max ] \\sigma\\in[\\sigma_{\\min},\\sigma_{\\max}] , with denoiser D ( x , σ ) = 𝔼 [ x 0 ∣ x σ = x ] D(x,\\sigma)=\\mathbb{E}[x_{0}\\mid x_{\\sigma}=x] and normalized residual η = ( x − D ) / σ \\eta=(x-D)/\\sigma —the noise prediction ε ^ \\hat{\\varepsilon} of practice ( − σ ∇ x log p σ -\\sigma\\nabla_{x}\\log p_{\\sigma} ); not DDIM’s stochasticity parameter, which is 0 throughout. The probability flow ODE is d X / d σ = η ( X , σ ) \\mathrm{d}X/\\mathrm{d}\\sigma=\\eta(X,\\sigma) , and the decomposition x = D + σ η x=D+\\sigma\\eta splits the state into a slow manifold component and a stiff normal component—normal to the data manifold, contracting fastest—whose exponent is 1 1 : on a normal coordinate the exact flow contracts by ( σ n + 1 / σ n ) 1 (\\sigma_{n+1}/\\sigma_{n})^{1} per step. A fitted-operator analysis of the unguided terminal layer [Zhang, 2026 ] shows that a frozen-field Euler step is layer-exact if and only if its clock is affine in σ \\sigma , making the σ \\sigma -clock step—algebraically the deterministic DDIM update [Song et al. , 2021 ] —the unique fitted operator up to affine reparameterization, with rectified flow its flow-matching counterpart; this fitted-operator property is what makes it asymptotic-preserving as σ min → 0 \\sigma_{\\min}\\to 0 . The present paper asks what survives of that protection under guidance."},{"type":"heading","level":4,"text":"Guidance as a second velocity field."},{"type":"para","text":"Classifier-free guidance [Ho and Salimans, 2022 ] replaces the conditional denoiser D c ( x , σ ) D_{\\mathrm{c}}(x,\\sigma) by the extrapolation"},{"type":"para","text":"where D u D_{\\mathrm{u}} is the unconditional (marginal) denoiser and g = 1 + w g=1+w is the guidance scale of practice, so w = 6.5 w=6.5 is CFG scale g = 7.5 g=7.5 . Writing η c = ( x − D c ) / σ \\eta_{\\mathrm{c}}=(x-D_{\\mathrm{c}})/\\sigma and η u = ( x − D u ) / σ \\eta_{\\mathrm{u}}=(x-D_{\\mathrm{u}})/\\sigma , the sampled dynamics is the guided flow"},{"type":"para","text":"with γ \\gamma the normalized guidance direction. Equation ( 2 ) is the dynamical statement of guidance: the sampler follows the conditional velocity plus w w times the discriminative correction. Two facts follow. First, D w D_{w} is in general not a VE posterior mean with a positive-semidefinite Tweedie covariance, so η w \\eta_{w} is not the residual of such a denoiser; the rigidity theorem that singles out DDIM as layer-exact inside the unguided denoiser class [Zhang, 2026 ] no longer applies directly. The exit is quantitative, not merely definitional: for any true denoiser, J D = ∇ x D = Cov ( X 0 ∣ x ) / σ 2 ⪰ 0 J_{D}=\\nabla_{x}D=\\mathrm{Cov}(X_{0}\\mid x)/\\sigma^{2}\\succeq 0 [Efron, 2011 ] caps the exponent of the residual field at 1 1 (on the model, the residual exponent is A ≤ 1 A\\leq 1 in ( 3 )), while on discriminative directions the guided exponent exceeds 1 1 at every σ \\sigma (Proposition 1 )—no repackaging of any denoiser, affine or otherwise, reproduces the guided field. Second, the guided flow carries a second parameter: any AP statement must state its w w -dependence, and the fitted step below removes the σ min \\sigma_{\\min} -divergent terminal barrier for bounded w w , though not uniformly in w w ."},{"type":"heading","level":4,"text":"The model problem."},{"type":"para","text":"Our analysis is carried out on the guided analog of the unguided model problem [Zhang, 2026 ] ."},{"type":"para","text":"p c = 𝒩 ( 0 , C c ) p_{\\mathrm{c}}=\\mathcal{N}(0,C_{\\mathrm{c}}) and p u = 𝒩 ( 0 , C u ) p_{\\mathrm{u}}=\\mathcal{N}(0,C_{\\mathrm{u}}) with [ C c , C u ] = 0 [C_{\\mathrm{c}},C_{\\mathrm{u}}]=0 , and on the shared eigenbasis the spectra satisfy a i ≤ b i a_{i}\\leq b_{i} ."},{"type":"para","text":"The ordering a i ≤ b i a_{i}\\leq b_{i} is the class-subset hypothesis : the class is narrower than the marginal in every probed direction (as when the marginal is a mixture whose components share the class covariance). Real learned checkpoints need not obey it; the reverse case is a genuine model boundary, discussed in Remark 4 . On an eigen-coordinate ξ \\xi with class variance a a and marginal variance b b , the two denoisers are linear and the residuals are"},{"type":"para","text":"The guided flow ( 2 ) therefore closes coordinate-wise,"},{"type":"para","text":"and every question about clocks, stability, and fitted operators reduces to the behavior of the exponent function μ w \\mu_{w} ."},{"type":"para","text":"Under Hypothesis 1 , as σ → 0 \\sigma\\to 0 the exponent function ( 4 ) satisfies: (i) on shared normal directions ( a = b = 0 a=b=0 ), μ w ≡ 1 \\mu_{w}\\equiv 1 : the guidance weight cancels and the layer is the unguided exponent- 1 1 layer; (ii) on tangential directions ( 0 < a ≤ b 0<a\\leq b ), μ w → 0 \\mu_{w}\\to 0 : the coordinate freezes; (iii) on discriminative normal directions ( a = 0 < b a=0<b ),"},{"type":"para","text":"monotonically as σ \\sigma decreases, with crossover midpoint at σ 2 = b \\sigma^{2}=b ."},{"type":"para","text":"(Proofs for Sections 2 and 3 are deferred to Appendix A .)"},{"type":"para","text":"Proposition 1 is the structural fact of the paper. Guidance does not stiffen the flow uniformly; it re-stiffens exactly the discriminative subspace —the directions collapsed within the class but present in the marginal ( a = 0 < b a=0<b )—and ( 5 ) locates where the re-stiffening begins: the anomalous exponent switches on as σ \\sigma descends through the class-separation scale b \\sqrt{b} . Above that scale the guided flow is, to leading order, the unguided flow; below it, the terminal layer runs at exponent 1 + w 1+w , and w w enters the singular structure of the problem on the same footing as σ min \\sigma_{\\min} ."},{"type":"para","text":"Under Hypothesis 1 , the exact solution of ( 2 ) on an eigen-coordinate over one step σ n → σ n + 1 \\sigma_{n}\\to\\sigma_{n+1} is ξ n + 1 = Φ w ξ n \\xi_{n+1}=\\Phi_{w}\\,\\xi_{n} with"},{"type":"para","text":"In particular, on a discriminative coordinate ( a = 0 a=0 ), writing r = σ n + 1 / σ n r=\\sigma_{n+1}/\\sigma_{n} ,"},{"type":"para","text":"which tends to r 1 + w r^{1+w} once σ n 2 ≪ b \\sigma_{n}^{2}\\ll b ."},{"type":"para","text":"The singular limit now has two knobs. As σ min → 0 \\sigma_{\\min}\\to 0 the discriminative layer contracts by the anomalous power σ 1 + w \\sigma^{1+w} ; as w w grows at fixed σ min \\sigma_{\\min} the same layer stiffens without bound. We ask two questions: which part of DDIM + + CFG’s failure is the pure terminal-layer solver barrier, and can that barrier be removed by a one-coefficient fitted repair? Uniform accuracy through the finite- σ \\sigma crossover is separate, needing either mesh resolution of the crossover or the spectrum-aware Gaussian factor."},{"type":"heading","level":1,"text":"3 The guided barrier and the fitted operator"},{"type":"para","text":"This section establishes one claim: classifier-free guidance introduces a second singular parameter w w , and on discriminative terminal directions the σ \\sigma -clock frozen-field step—DDIM applied to the guided denoiser—is no longer fitted. The program has three parts. First, a barrier: the guided analog of the unguided clock dichotomy [Zhang, 2026 ] , with sharp thresholds in the log-step h h and weight w w (Theorem 1 ). Second, an operator: a fitted step that removes the pure discriminative terminal-layer reflection and σ min \\sigma_{\\min} -divergent residual blow-up at zero additional cost (Definition 1 ). Third, accuracy: a first-order certificate on the discriminative crossover (Theorem 2 ); the whole-model theory remains future work. Throughout, the reference solution is the exact guided flow’s own pushforward: we charge the discretization of ( 2 ), not the distortion of the continuous guided law away from the tilted conditional (Section 1 )."},{"type":"heading","level":4,"text":"The frozen step."},{"type":"para","text":"DDIM applied to D w D_{w} —equivalently, σ \\sigma -clock Euler with the guided field frozen at the left endpoint—is"},{"type":"para","text":"which on an eigen-coordinate of the model contracts by"},{"type":"para","text":"On the pure discriminative layer ( A , B ) = ( 1 , 0 ) (A,B)=(1,0) this is"},{"type":"para","text":"to be compared with the exact factor e − ( 1 + w ) h \\mathrm{e}^{-(1+w)h} of Lemma 1 . Here h = log ( σ n / σ n + 1 ) h=\\log(\\sigma_{n}/\\sigma_{n+1}) is the step in λ = − log σ \\lambda=-\\log\\sigma (half the VE log-SNR), so a uniform λ \\lambda -mesh has constant h h . Along a trajectory, call max n ‖ η w ( x n , σ n ) ‖ / ‖ η w ( x 0 , σ 0 ) ‖ \\max_{n}\\|\\eta_{w}(x_{n},\\sigma_{n})\\|/\\|\\eta_{w}(x_{0},\\sigma_{0})\\| the residual-amplification certificate , written amp \\mathrm{amp} ; the sampler already forms every quantity in it, and a scheme is residual-stable if the certificate stays bounded as σ min → 0 \\sigma_{\\min}\\to 0 . Everything in Theorem 1 is a statement about the elementary function ( 10 )."},{"type":"para","text":"Consider the frozen step ( 8 ) on the pure discriminative layer, η w = ( 1 + w ) ξ / σ \\eta_{w}=(1+w)\\,\\xi/\\sigma with w > 0 w>0 , on a uniform λ \\lambda -mesh of step h h . Let"},{"type":"para","text":"Then h ♭ < h ♯ < h ∞ h_{\\flat}<h_{\\sharp}<h_{\\infty} , and:"},{"type":"para","text":"Sign. G w ( h ) > 0 G_{w}(h)>0 iff h < h ♭ h<h_{\\flat} ; equivalently, at fixed h h , sign preservation fails once w > 1 / ( e h − 1 ) w>1/(\\mathrm{e}^{h}-1) . For h > h ♭ h>h_{\\flat} every step reflects the coordinate across the class manifold."},{"type":"para","text":"Residual amplification. The per-step residual amplification is | G w ( h ) | e h |G_{w}(h)|\\,\\mathrm{e}^{h} , which is ≤ 1 \\leq 1 iff h ≤ h ♯ h\\leq h_{\\sharp} ; equivalently, raw amplification begins once w > 2 / ( e h − 1 ) w>2/(\\mathrm{e}^{h}-1) . For h > h ♯ h>h_{\\sharp} the amplification factor is w e h − ( 1 + w ) > 1 w\\mathrm{e}^{h}-(1+w)>1 per step. On a terminal subwindow σ ∈ [ σ min , ε b ] \\sigma\\in[\\sigma_{\\min},\\varepsilon\\sqrt{b}\\,] , where B ≤ ε 2 / ( 1 + ε 2 ) B\\leq\\varepsilon^{2}/(1+\\varepsilon^{2}) makes the exact per-step factor pure-layer up to O ( ε 2 ) O(\\varepsilon^{2}) , of λ \\lambda -length Λ ε = log ( ε b / σ min ) \\Lambda_{\\varepsilon}=\\log(\\varepsilon\\sqrt{b}/\\sigma_{\\min}) the guided residual grows by"},{"type":"para","text":"so for fixed ε > 0 \\varepsilon>0 and h > h ♯ h>h_{\\sharp} the scheme fails the amplification certificate as σ min → 0 \\sigma_{\\min}\\to 0 ."},{"type":"para","text":"Absolute stability. | G w ( h ) | ≤ 1 |G_{w}(h)|\\leq 1 for all h h when w ≤ 1 w\\leq 1 ; for w > 1 w>1 , iff h ≤ h ∞ h\\leq h_{\\infty} ."},{"type":"para","text":"All three thresholds decay like c / w c/w as w → ∞ w\\to\\infty , with c = 1 , 2 , 2 c=1,2,2 respectively."},{"type":"para","text":"In words: sign preservation fails first (a, above h ♭ h_{\\flat} : every step reflects across the class manifold), the certificate second (b, above h ♯ h_{\\sharp} : the residual compounds, diverging as σ min → 0 \\sigma_{\\min}\\to 0 ), absolute stability last (c, only for w > 1 w>1 )."},{"type":"para","text":"On a uniform λ \\lambda -mesh covering the discriminative window (of λ \\lambda -length Λ b = log ( b / σ min ) \\Lambda_{b}=\\log(\\sqrt{b}/\\sigma_{\\min}) ), avoiding reflection requires"},{"type":"para","text":"while merely keeping the residual nonexpansive requires N ≥ Λ b / log ( 1 + 2 / w ) ∼ w 2 Λ b N\\geq\\Lambda_{b}/\\log(1+2/w)\\sim\\tfrac{w}{2}\\Lambda_{b} . A uniform global mesh must meet this bound throughout the window, so Λ b \\Lambda_{b} is replaced by the full horizon Λ = log ( σ max / σ min ) \\Lambda=\\log(\\sigma_{\\max}/\\sigma_{\\min}) : at working scale g = 7.5 g=7.5 ( Λ = log ( 80 / 0.002 ) ≈ 10.6 \\Lambda=\\log(80/0.002)\\approx 10.6 ), reflection-free sampling already demands N ≳ 74 N\\gtrsim 74 steps—an order of magnitude above unguided-DDIM-accurate budgets [Karras et al. , 2022 ] ."},{"type":"para","text":"In the window h ∈ ( h ♭ , h ♯ ] h\\in(h_{\\flat},h_{\\sharp}] the scheme is reflecting but nonexpansive : the flip does not grow the residual. The flip is the mechanism of visible artifacts—the iterate lands on the wrong side of the class manifold each step—while the amplification crossing at h ♯ h_{\\sharp} is where the certificate, and any uniform accuracy claim, actually fails. Practitioners tune w w down at low σ \\sigma well before the certificate diverges: part (a), not part (b), is the threshold their tuning discovers."},{"type":"para","text":"The following identity is model-free. A single frozen step ( 8 ) from σ n \\sigma_{n} to σ n + 1 = r σ n \\sigma_{n+1}=r\\sigma_{n} with r → 0 r\\to 0 gives"},{"type":"para","text":"one-step DDIM + + CFG lands on the extrapolated denoiser, displaced from the class manifold by − w ( D u − D c ) -w(D_{\\mathrm{u}}-D_{\\mathrm{c}}) . On the model this is the reflection ξ ↦ − w ξ \\xi\\mapsto-w\\,\\xi of Theorem 1 (a) pushed to its endpoint. We read this as the discrete mechanism of the oversaturation and norm-inflation phenomenology reported for large guidance scales [Sadat et al. , 2025 ] : it is a property of the solver at coarse steps, not of the continuous guided flow—on the calibration model the exact flow contracts to D c D_{\\mathrm{c}} as r 1 + w r^{1+w} while the frozen step overshoots to D w D_{w} ."},{"type":"heading","level":4,"text":"The guided fitted step."},{"type":"para","text":"The repair is dictated by the same logic that produced DDIM in the unguided setting [Zhang, 2026 ] : integrate the stiff layer exactly and freeze what is slow. The gauge ξ ~ = σ − w ξ \\tilde{\\xi}=\\sigma^{-w}\\xi rectifies the anomalous exponent 1 + w 1+w to exponent 1 1 ; transporting the σ \\sigma -clock Euler step through the gauge and back yields, on the whole state,"},{"type":"para","text":"With r = σ n + 1 / σ n r=\\sigma_{n+1}/\\sigma_{n} and all fields evaluated at ( x n , σ n ) (x_{n},\\sigma_{n}) ,"},{"type":"para","text":"equivalently, in residual form,"},{"type":"para","text":"The scheme touches exactly one coefficient of classifier-free guidance: vanilla DDIM + + CFG is ( 15 ) with the coefficient w ( r − 1 ) w(r-1) in place of r 1 + w − r r^{1+w}-r on the guidance direction D u − D c D_{\\mathrm{u}}-D_{\\mathrm{c}} . It uses the same two denoiser evaluations per step, so the cost—and the implementation delta—is one line. Proposition 3 verifies the factor coordinate-wise; Lemma 2 pins the coefficient from terminal exactness alone."},{"type":"para","text":"r 1 + w − r = w ( r − 1 ) + 1 2 w ( w + 1 ) h 2 + O ( h 3 ) r^{1+w}-r=w(r-1)+\\tfrac{1}{2}w(w+1)\\,h^{2}+O(h^{3}) as h → 0 h\\to 0 , so ( 15 ) is a first-order-consistent discretization of the guided flow ( 2 ), agreeing with DDIM + + CFG through O ( h ) O(h) ; and at w = 0 w=0 both coefficients vanish identically, collapsing the scheme to DDIM."},{"type":"para","text":"Under Hypothesis 1 , the step ( 15 ) contracts an eigen-coordinate by"},{"type":"para","text":"a sum of nonnegative terms; hence G fit > 0 G_{\\rm fit}>0 on every coordinate, for every mesh, and every w ≥ 0 w\\geq 0 : the fitted step preserves the sign of every analyzed coordinate. It is exact in the three terminal regimes— G fit = r G_{\\rm fit}=r on shared normals ( A = B = 1 ) (A=B=1) , G fit = r 1 + w G_{\\rm fit}=r^{1+w} on pure discriminative normals ( A = 1 , B = 0 ) (A=1,B=0) , and G fit = 1 G_{\\rm fit}=1 on frozen tangential directions ( A = B = 0 ) (A=B=0) —and its one-jump limit is x n + 1 → D c x_{n+1}\\to D_{\\mathrm{c}} , the conditional denoiser (the terminal-limit class projection), in contrast with ( 14 )."},{"type":"para","text":"Consider scalar coefficient updates of the form"},{"type":"para","text":"where α \\alpha depends only on ( r , w ) (r,w) . The unique such coefficient that is exact on the pure discriminative terminal regime ( A , B ) = ( 1 , 0 ) (A,B)=(1,0) is"},{"type":"para","text":"Consequently, terminal exactness, DDIM recovery at w = 0 w=0 , and exactness on the shared-normal and tangent terminal regimes determine the fitted coefficient without any spectrum information."},{"type":"para","text":"On a single discriminative Gaussian coordinate with a = 0 < b a=0<b , define"},{"type":"para","text":"Let σ 0 > σ 1 > ⋯ > σ K \\sigma_{0}>\\sigma_{1}>\\cdots>\\sigma_{K} be any decreasing mesh, r n = σ n + 1 / σ n r_{n}=\\sigma_{n+1}/\\sigma_{n} , q n = q ( σ n ) q_{n}=q(\\sigma_{n}) , and μ n = μ ( σ n ) \\mu_{n}=\\mu(\\sigma_{n}) . For the fitted step ( 15 ), the guided residual ratio over one step is"},{"type":"para","text":"Hence for every prefix k k ,"},{"type":"para","text":"Thus the fitted step removes the σ min \\sigma_{\\min} -divergent terminal blow-up of Theorem 1 on the discriminative crossover, but it may still pay a finite O ( 1 + w ) O(1+w) crossover tax on a coarse mesh."},{"type":"para","text":"Bounded amplification in fact upgrades to accuracy: on this crossover the accumulated log-defect (uniform mesh, h ≤ 1 h\\leq 1 ) obeys 0 ≤ log Θ K ≤ 1 + h 4 w ( w + 2 ) h 0\\leq\\log\\Theta_{K}\\leq\\tfrac{1+h}{4}\\,w(w+2)\\,h , so the fitted step is first-order uniformly accurate against the exact guided flow’s own pushforward, with a W 2 W_{2} bound whose constant is independent of b b , K K , and σ min \\sigma_{\\min} (the O ( w 2 ) O(w^{2}) factor is not uniform in w w ). We defer the formal statement and proof to Theorem 2 in Appendix A ."},{"type":"para","text":"Extending this crossover accuracy to the tangential layers and learned geometry needs the spectrum-aware Gaussian factor ( 6 ) and is future work."},{"type":"para","text":"Limited-interval guidance [Kynkäänniemi et al. , 2024 ] disables guidance outside a tuned band ( σ lo , σ hi ) (\\sigma_{\\rm lo},\\sigma_{\\rm hi}) and leaves open whether the band can be derived rather than tuned. Theorem 1 supplies the terminal side without free parameters: on schedules whose local step h n h_{n} grows toward low σ \\sigma (e.g. Karras power meshes; on a uniform λ \\lambda -mesh h n h_{n} is constant and there is no distinct crossing), the level at which h n h_{n} crosses h ♭ ( w ) = log ( 1 + 1 / w ) h_{\\flat}(w)=\\log(1+1/w) predicts σ lo \\sigma_{\\rm lo} : below it, frozen-field guidance at weight w w is unaffordable and the interval or the fitted step must take over. The high -noise cutoff σ hi \\sigma_{\\rm hi} is not explained by the barrier: there ‖ D u − D c ‖ \\|D_{\\mathrm{u}}-D_{\\mathrm{c}}\\| is small against the model’s own error, a signal-level criterion outside this paper’s ledger."},{"type":"para","text":"If a direction has b = 0 < a b=0<a , the exponent ( 4 ) tends to − w -w : the continuous guided flow expands as σ → 0 \\sigma\\to 0 , an instability of the guidance target that no solver can repair; the class-subset hypothesis excludes it by assumption (Appendix B )."},{"type":"heading","level":4,"text":"Certificates."},{"type":"para","text":"The certificate is removal of the σ min \\sigma_{\\min} -divergent pure-layer mechanism (Proposition 4 ) plus first-order accuracy on the discriminative crossover (Theorem 2 ), not global residual nonexpansion through every finite- σ \\sigma crossover."},{"type":"heading","level":1,"text":"4 Empirical diagnostics on learned checkpoints"},{"type":"heading","level":4,"text":"Synthetic crossover."},{"type":"para","text":"Our primary residual and image-quality comparisons fix their schemes, cells, and metrics before each run; the feature-space, latent-diffusion, and dense-reference results are exploratory diagnostics. The single-coordinate Gaussian diagnostic isolates the theorem’s singular mechanism: on uniform log- σ \\sigma meshes with a = 0 < b a=0<b , CFG develops the coarse-mesh branch of Theorem 1 while the fitted coefficient stays below Proposition 4 ’s finite-tax bound (Figure 2 ). In the default w w – N N sweep, worst-case amplification reaches 10 3 10^{3} – 10 9 10^{9} for CFG but stays ≤ 4.66 \\leq 4.66 for the fitted step, below each 1 + w 1+w bound—a theorem figure, not a learned-checkpoint quality metric."},{"type":"heading","level":4,"text":"Real-checkpoint residual diagnostics."},{"type":"para","text":"We then test the coefficient as a drop-in replacement on NVIDIA edm CIFAR-10 VP checkpoints [Karras et al. , 2022 ] (the ‘VP’ label is network preconditioning only; sampling uses the σ \\sigma -parameterization): D c D_{\\mathrm{c}} is the class-conditional checkpoint, D u D_{\\mathrm{u}} the separately trained unconditional one, and both schemes use the same Karras mesh, seeds, labels, and NFE. The grid is w ∈ { 4 , 6.5 , 8 } w\\in\\{4,6.5,8\\} and N ∈ { 8 , 16 , 32 } N\\in\\{8,16,32\\} , with two 64-seed blocks (class 0 , seeds from 1000 1000 ; class 1 1 , from 2000 2000 ). Table 1 reports paired fitted/CFG ratios for the amplification median and tail, and paired clipping differences; lower is better."},{"type":"para","text":"In both blocks, for every tested N ∈ { 8 , 16 , 32 } N\\in\\{8,16,32\\} at w = 6.5 w=6.5 and w = 8 w=8 , fitted CFG improves the amp median, amp p95, and clipping median. The hardest cell is the clearest: at w = 8 , N = 8 w=8,N=8 , the class- 0 amp median/p95 falls from 4.57 / 13.35 4.57/13.35 under CFG to 1.41 / 1.71 1.41/1.71 under fitted CFG, while the clipping median drops from 0.379 0.379 to 0 ; the class- 1 1 replicate matches (amp p95 ratio 0.116 0.116 , clipping delta − 0.23 -0.23 ). At w = 4 w=4 , where the theory predicts a much weaker tax, the median amplification is mixed, but the amp tail and saturation proxy still improve in most cells. We use this as evidence for high-guidance stabilization, not uniform superiority at weak guidance."},{"type":"heading","level":4,"text":"Image-quality evaluation."},{"type":"para","text":"We next ran two 5000-image balanced-class evaluations on the hard cell w = 8 , N = 8 w=8,N=8 , plus a 50k replication, each including a terminal-interval baseline (vanilla CFG correction off when h > h ♭ h>h_{\\flat} , after Kynkäänniemi et al. [ 2024 ] ; on this coarse mesh every step crosses h ♭ h_{\\flat} , so it reduces to unguided conditional DDIM). We score with FID and KID (feature-space quality metrics; lower is better). Table 2 collects the results. Fitted CFG has the best FID in all three blocks at zero extra NFE, and the certificates replicate (amp p95 falls from 13 + 13{+} to ≈ 1.6 {\\approx}1.6 , clipping p95 from ≈ 0.57 {\\approx}0.57 to 0 ). The metrics disagree, however: KID is lowest for CFG in every block. We therefore read this as support for high-guidance stabilization and FID improvement, not an unqualified image-quality win; feature-manifold and sharpness diagnostics are consistent with CFG’s oversaturation contributing to the split (Figure 3 ), and a DINOv2 backbone-swapped audit gives the same fitted-favoring ordering, so the split is not Inception-specific (Appendices C and D )."},{"type":"para","text":"Finally, the target-accuracy column of Table 2 is a classifier-conditionality diagnostic (classifier at 94.37 % 94.37\\% on the CIFAR-10 test set): in every block fitted CFG stays within 0.6 0.6 pp of CFG yet ∼ 5 {\\sim}5 pp above interval, keeping CFG’s conditional pull without its clipping pathology."},{"type":"heading","level":4,"text":"Robustness grid."},{"type":"para","text":"The 50k replication (Table 2 ) reduces the 5k-sampling-artifact concern for the FID ordering, and the KID split persists at scale. A nine-cell grid ( w ∈ { 4 , 6.5 , 8 } w\\in\\{4,6.5,8\\} , N ∈ { 8 , 16 , 32 } N\\in\\{8,16,32\\} , 5k images each) shows it is not cell-specific: fitted CFG improves FID in 9 / 9 9/9 cells, clipping p95 in 9 / 9 9/9 , and amp p95 in 8 / 9 8/9 (exception w = 4 , N = 32 w{=}4,N{=}32 , both near 1 1 ). The terminal-interval baseline wins FID at N ∈ { 16 , 32 } N\\in\\{16,32\\} and KID at N = 32 N=32 but loses conditionality everywhere ( 3.24 3.24 – 5.72 5.72 pp)."},{"type":"heading","level":4,"text":"Interval-baseline separation."},{"type":"para","text":"Is the fitted step just guidance shutdown? We compared it against a tuned conditionality-preserving limited interval (selected per cell) and the parameter-free terminal interval (Appendix E ). In the tested cells, the tuned interval preserves target accuracy but does not beat the fitted step on FID; the terminal interval can win FID (e.g. 12.63 12.63 vs 17.12 17.12 at w = 8 , N = 16 w{=}8,N{=}16 ) but drops target accuracy by 5 5 – 6 6 pp. So the fitted step gives interval-like residual stabilization while keeping the guided update active—it is not explained away by turning guidance off."},{"type":"heading","level":4,"text":"All-class audit."},{"type":"para","text":"Extending the residual diagnostic from two classes to all ten CIFAR-10 classes on the high-guidance cells (60 class–cell pairs), clipping/saturation is non-increased in 60 / 60 60/60 pairs and amp p95 improves in 54 / 60 54/60 : global saturation robustness and near-global residual-tail improvement (Appendix G )."},{"type":"heading","level":4,"text":"Latent-diffusion smoke."},{"type":"para","text":"The mechanism is not specific to pixel-space CIFAR. On Stable Diffusion 1.5 DDIM at high guidance, the same coefficient sharply reduces pixel saturation (p95 0.134 → 0.009 0.134\\to 0.009 at guidance 12 12 , over 256 images) while preserving CLIP image–text alignment ( 0.320 → 0.314 0.320\\to 0.314 ); terminal guidance shutdown reaches lower saturation but collapses CLIP to 0.271 0.271 (Appendix F ). This is a cross-domain smoke test, not a Stable Diffusion benchmark."},{"type":"heading","level":1,"text":"5 Related work"},{"type":"para","text":"The CFG-repair family is crowded, and methods differ mainly in their mathematical object : a manifold projection, a guidance-vector rescaling, an interval schedule, a dynamic weight, or—ours—a terminal fitted-operator coefficient. In brief: CFG++ [Chung et al. , 2025 ] constrains the update to the data manifold; limited-interval guidance [Kynkäänniemi et al. , 2024 ] switches guidance off outside a σ \\sigma band (Remark 3 derives its terminal edge parameter-free); APG [Sadat et al. , 2025 ] down-weights the update’s parallel component; CFG-Zero ⋆ [Fan et al. , 2025 ] optimizes the scale and zero-inits early steps; and C 2 FG [Gao et al. , 2026 ] decays the weight in time. None derives the terminal-fitted coefficient r 1 + w − r r^{1+w}-r from the guided exponent—where the barrier lives. An orthogonal line analyses what the continuous guided law samples—CFG is a predictor–corrector, not a sampler of its tilted target [Bradley and Nakkiran, 2025 ; Chidambaram et al. , 2024 ] —the “model” bill complementary to ours."},{"type":"heading","level":1,"text":"6 Discussion and limitations"},{"type":"heading","level":4,"text":"What is proved."},{"type":"para","text":"On the commuting class-subset Gaussian model, the barrier (Theorem 1 ) is sharp; the fitted step is consistent, sign-preserving, terminal-exact, pays only a finite O ( 1 + w ) O(1+w) crossover tax (Prop. 4 ), and is first-order accurate on the crossover (Theorem 2 )—all about the discretization against the guided flow’s own pushforward, AP in σ min \\sigma_{\\min} but not uniform in w w ."},{"type":"heading","level":4,"text":"What is evidence."},{"type":"para","text":"On learned CIFAR-10 edm checkpoints, fitted CFG is a drop-in, zero-extra-NFE high-guidance stabilizer: the certificates improve residual amplification and clipping across every w ∈ { 6.5 , 8 } w\\in\\{6.5,8\\} cell; the hard-cell evaluations (5k twice, 50k once) give it the best FID with target accuracy preserved, and a DINOv2 backbone-swap the same ordering; the nine-cell grid gives 9 / 9 9/9 FID wins over CFG. It separates from tuned and terminal interval baselines (which either lose target accuracy or fail to beat it on FID), holds across all ten classes ( 60 / 60 60/60 saturation non-increase, 54 / 60 54/60 amp-tail), and transfers to Stable Diffusion 1.5 DDIM as a saturation-reducing coefficient that keeps CLIP alignment far better than guidance shutdown (Section 4 , Appendices E – G )."},{"type":"heading","level":4,"text":"What is neither."},{"type":"para","text":"We do not claim (i) an unqualified image-quality win—KID stays lowest for CFG at 5k and 50k, a split tied to CFG’s oversaturation without proven causation; (ii) that CIFAR verifies the sharp threshold law—the learned diagnostics are only consistent with the model ordering; or (iii) a 50k KID win or many-cell large-sample result—the 50k run is one hard cell. Nor is the fitted step a uniformly better integrator of the vanilla CFG field on learned checkpoints: against a 512 512 -step CFG reference its residual tails shrink but its final-image L 2 L_{2} does not (Appendix H ), an interval can win FID in some cells by weakening guidance, and the Stable Diffusion result is a smoke test, not a benchmark. The class-subset hypothesis is itself a boundary: where the class is wider than the marginal, the continuous guided flow is unstable (exponent − w -w ), which no solver can repair (Remark 4 )."},{"type":"heading","level":4,"text":"Scope and outlook."},{"type":"para","text":"Every certificate uses only sampler-computed quantities from two separate checkpoints, so shared-head correlations cannot inflate them. ‘Asymptotic-preserving’ is used in the calibration-model sense: on learned checkpoints we audit AP-relevant residuals rather than certify AP. Next: shared-dropout CFG, broader latent-diffusion benchmarks, and a whole-model accuracy theory."},{"type":"heading","level":1,"text":"Reproducibility Statement"},{"type":"para","text":"All theoretical results are stated with explicit constants on the model of Hypothesis 1 , and complete proofs appear in Appendix A . Every empirical diagnostic in Section 4 is produced by a self-contained script operating on the public NVIDIA edm CIFAR-10 VP checkpoints with stated seeds, meshes, sample counts, schemes, and metrics; the synthetic-crossover and checkpoint-certificate scripts additionally ship CPU --self-test modes, and the synthetic crossover figure requires no checkpoints. The supplementary package contains the scripts, the exact commands, the run manifests, and all summary artifacts (CSV/JSON) behind every number reported in the paper, including the 50k and nine-cell grid diagnostics."},{"type":"para","text":"For a = 0 a=0 one has A ≡ 1 A\\equiv 1 , so μ w = ( 1 + w ) − w B = 1 + w ( 1 − B ) = 1 + w b / ( b + σ 2 ) \\mu_{w}=(1+w)-wB=1+w(1-B)=1+wb/(b+\\sigma^{2}) ; the limits and monotonicity are immediate. Cases (i) and (ii) follow from A , B → 1 A,B\\to 1 and A , B → 0 A,B\\to 0 respectively. ∎"},{"type":"para","text":"Integrate ( 4 ): ∫ μ w d log σ = 1 + w 2 log ( a + σ 2 ) − w 2 log ( b + σ 2 ) \\int\\mu_{w}\\,\\mathrm{d}\\log\\sigma=\\tfrac{1+w}{2}\\log(a+\\sigma^{2})-\\tfrac{w}{2}\\log(b+\\sigma^{2}) up to a constant, since d log ( c + σ 2 ) = 2 A c d log σ \\mathrm{d}\\log(c+\\sigma^{2})=2A_{c}\\,\\mathrm{d}\\log\\sigma with A c = σ 2 / ( c + σ 2 ) A_{c}=\\sigma^{2}/(c+\\sigma^{2}) . Exponentiate the increment. ∎"},{"type":"para","text":"From ( 10 ): G w > 0 ⇔ e − h > w / ( 1 + w ) ⇔ h < log ( 1 + 1 / w ) G_{w}>0\\iff\\mathrm{e}^{-h}>w/(1+w)\\iff h<\\log(1+1/w) , which is (a). For (b), on the pure layer η w ∝ ξ / σ \\eta_{w}\\propto\\xi/\\sigma , so the residual ratio per step is | ξ n + 1 / ξ n | ⋅ σ n / σ n + 1 = | G w | e h |\\xi_{n+1}/\\xi_{n}|\\cdot\\sigma_{n}/\\sigma_{n+1}=|G_{w}|\\mathrm{e}^{h} . On the branch G w ≤ 0 G_{w}\\leq 0 this is ( w − ( 1 + w ) e − h ) e h = w e h − ( 1 + w ) (w-(1+w)\\mathrm{e}^{-h})\\mathrm{e}^{h}=w\\mathrm{e}^{h}-(1+w) , which exceeds 1 1 iff e h > 1 + 2 / w \\mathrm{e}^{h}>1+2/w ; on the branch G w ≥ 0 G_{w}\\geq 0 it equals ( 1 + w ) − w e h ≤ 1 (1+w)-w\\mathrm{e}^{h}\\leq 1 always. Compounding over Λ ε / h \\Lambda_{\\varepsilon}/h steps gives ( 12 ). For (c), G w ≤ 1 G_{w}\\leq 1 always, and G w ≥ − 1 ⇔ ( 1 + w ) e − h ≥ w − 1 G_{w}\\geq-1\\iff(1+w)\\mathrm{e}^{-h}\\geq w-1 , vacuous for w ≤ 1 w\\leq 1 and equivalent to h ≤ h ∞ h\\leq h_{\\infty} otherwise. The ordering h ♯ < h ∞ h_{\\sharp}<h_{\\infty} follows from 1 + w w − 1 − ( 1 + 2 w ) = 2 w ( w − 1 ) > 0 \\frac{1+w}{w-1}-\\bigl(1+\\frac{2}{w}\\bigr)=\\frac{2}{w(w-1)}>0 . ∎"},{"type":"para","text":"r 1 + w − r = e − ( 1 + w ) h − e − h r^{1+w}-r=\\mathrm{e}^{-(1+w)h}-\\mathrm{e}^{-h} and w ( r − 1 ) = w ( e − h − 1 ) w(r-1)=w(\\mathrm{e}^{-h}-1) ; both are − w h + O ( h 2 ) -wh+O(h^{2}) , and expanding to second order, r 1 + w − r − w ( r − 1 ) = e − ( 1 + w ) h − ( 1 + w ) e − h + w = 1 2 w ( w + 1 ) h 2 + O ( h 3 ) r^{1+w}-r-w(r-1)=\\mathrm{e}^{-(1+w)h}-(1+w)\\mathrm{e}^{-h}+w=\\tfrac{1}{2}w(w+1)h^{2}+O(h^{3}) . ∎"},{"type":"para","text":"On an eigen-coordinate, D c = ( 1 − A ) ξ D_{\\mathrm{c}}=(1-A)\\xi , x − D c = A ξ x-D_{\\mathrm{c}}=A\\xi , and D u − D c = ( A − B ) ξ D_{\\mathrm{u}}-D_{\\mathrm{c}}=(A-B)\\xi ; substituting into ( 15 ) gives the first expression in ( 17 ), and regrouping 1 − A + r A − r ( A − B ) + r 1 + w ( A − B ) 1-A+rA-r(A-B)+r^{1+w}(A-B) gives the second. Nonnegativity of each term is 0 ≤ B ≤ A ≤ 1 0\\leq B\\leq A\\leq 1 and 0 < r < 1 0<r<1 . The regime values and the r → 0 r\\to 0 limit are read off termwise. ∎"},{"type":"para","text":"On shared normals ( A , B ) = ( 1 , 1 ) (A,B)=(1,1) one has D u − D c = 0 D_{\\mathrm{u}}-D_{\\mathrm{c}}=0 , so every scalar coefficient gives the DDIM factor r r . On tangent directions ( A , B ) = ( 0 , 0 ) (A,B)=(0,0) , both denoisers equal the state and every coefficient gives factor 1 1 . On a pure discriminative normal ( A , B ) = ( 1 , 0 ) (A,B)=(1,0) , the factor of the scalar update is r + α ( r , w ) r+\\alpha(r,w) . Exactness requires r + α ( r , w ) = r 1 + w r+\\alpha(r,w)=r^{1+w} , hence the displayed coefficient, uniquely. ∎"},{"type":"para","text":"For a = 0 a=0 , A ≡ 1 A\\equiv 1 and q = 1 − B q=1-B . The decomposition ( 17 ) gives the coordinate factor"},{"type":"para","text":"Since η w = μ ( σ ) ξ / σ \\eta_{w}=\\mu(\\sigma)\\xi/\\sigma , the residual ratio is ( μ n + 1 / μ n ) ( G fit / r n ) (\\mu_{n+1}/\\mu_{n})(G_{\\rm fit}/r_{n}) , which is ( 18 ). The bracket is at most 1 1 because 0 ≤ q n ≤ 1 0\\leq q_{n}\\leq 1 and 0 < r n w ≤ 1 0<r_{n}^{w}\\leq 1 , so products telescope:"},{"type":"para","text":"∎"},{"type":"para","text":"On a discriminative Gaussian coordinate ( a = 0 < b a=0<b ), for w ≥ 0 w\\geq 0 and a uniform λ \\lambda -mesh of step 0 < h ≤ 1 0<h\\leq 1 , the fitted step ( 15 ) and the exact guided flow satisfy ξ K fit = Θ K ξ K ex \\xi_{K}^{\\rm fit}=\\Theta_{K}\\,\\xi_{K}^{\\rm ex} with 0 ≤ log Θ K ≤ 1 + h 4 w ( w + 2 ) h 0\\leq\\log\\Theta_{K}\\leq\\tfrac{1+h}{4}\\,w(w+2)\\,h , hence W 2 ( Law ( ξ K fit ) , Law ( ξ K ex ) ) ≤ ( e w ( w + 2 ) h / 2 − 1 ) ( 𝔼 | ξ 0 | 2 ) 1 / 2 W_{2}\\bigl(\\mathrm{Law}(\\xi_{K}^{\\rm fit}),\\mathrm{Law}(\\xi_{K}^{\\rm ex})\\bigr)\\leq\\bigl(\\mathrm{e}^{w(w+2)h/2}-1\\bigr)(\\mathbb{E}|\\xi_{0}|^{2})^{1/2} . The multiplier bound and its prefactor are independent of b b , K K , and σ min \\sigma_{\\min} (the W 2 W_{2} scale carries the initial second moment ( 𝔼 | ξ 0 | 2 ) 1 / 2 (\\mathbb{E}|\\xi_{0}|^{2})^{1/2} ): on this crossover the fitted step is first-order uniformly accurate against the exact flow’s own pushforward, though the O ( w 2 ) O(w^{2}) prefactor is not uniform in w w ."},{"type":"para","text":"On a = 0 a=0 the exact one-step factor (Lemma 1 ) is Φ n = r 1 + w ( ( 1 + u n ) / ( 1 + r 2 u n ) ) w / 2 \\Phi_{n}=r^{1+w}\\bigl((1+u_{n})/(1+r^{2}u_{n})\\bigr)^{w/2} , with u n = σ n 2 / b u_{n}=\\sigma_{n}^{2}/b , r = e − h r=\\mathrm{e}^{-h} ; the fitted factor (Proposition 3 , A = 1 A=1 ) is G n = e − h ( u n + e − w h ) / ( 1 + u n ) G_{n}=\\mathrm{e}^{-h}(u_{n}+\\mathrm{e}^{-wh})/(1+u_{n}) . Writing y n = log u n y_{n}=\\log u_{n} , δ ( h , y ) := log ( G n / Φ n ) = w h + log ( e y + e − w h ) − ( 1 + w 2 ) log ( 1 + e y ) + w 2 log ( 1 + e y − 2 h ) \\delta(h,y):=\\log(G_{n}/\\Phi_{n})=wh+\\log(\\mathrm{e}^{y}+\\mathrm{e}^{-wh})-\\bigl(1+\\tfrac{w}{2}\\bigr)\\log(1+\\mathrm{e}^{y})+\\tfrac{w}{2}\\log(1+\\mathrm{e}^{y-2h}) . Then δ ( 0 , y ) = ∂ h δ ( 0 , y ) = 0 \\delta(0,y)=\\partial_{h}\\delta(0,y)=0 (first-order consistency, Proposition 2 ), and with F ( y ) = e y / ( 1 + e y ) 2 F(y)=\\mathrm{e}^{y}/(1+\\mathrm{e}^{y})^{2} one has ∂ h 2 δ = w 2 F ( y + w h ) + 2 w F ( y − 2 h ) ≥ 0 \\partial_{h}^{2}\\delta=w^{2}F(y+wh)+2wF(y-2h)\\geq 0 ; hence δ ≥ 0 \\delta\\geq 0 , so Θ K = ∏ n e δ ( h , y n ) ≥ 1 \\Theta_{K}=\\prod_{n}\\mathrm{e}^{\\delta(h,y_{n})}\\geq 1 , and Taylor’s formula gives δ ( h , y n ) = ∫ 0 h ( h − s ) [ w 2 F ( y n + w s ) + 2 w F ( y n − 2 s ) ] 𝑑 s \\delta(h,y_{n})=\\int_{0}^{h}(h-s)\\bigl[w^{2}F(y_{n}+ws)+2wF(y_{n}-2s)\\bigr]\\,ds . On the uniform mesh y n + 1 = y n − 2 h y_{n+1}=y_{n}-2h ; since ∫ F = 1 \\int F=1 and TV ( F ) = 1 2 \\mathrm{TV}(F)=\\tfrac{1}{2} , the rectangle bound gives 2 h ∑ n F ( y n + c ) ≤ 1 + h 2h\\sum_{n}F(y_{n}+c)\\leq 1+h for every shift c c , so log Θ K ≤ ∫ 0 h ( h − s ) 1 + h 2 h ( w 2 + 2 w ) 𝑑 s = 1 + h 4 w ( w + 2 ) h \\log\\Theta_{K}\\leq\\int_{0}^{h}(h-s)\\tfrac{1+h}{2h}(w^{2}+2w)\\,ds=\\tfrac{1+h}{4}w(w+2)h . Synchronous coupling with | ξ K ex | ≤ | ξ 0 | |\\xi_{K}^{\\rm ex}|\\leq|\\xi_{0}| (the exact discriminative factor contracts) gives the W 2 W_{2} bound. ∎"},{"type":"para","text":"This appendix expands Remark 4 ."},{"type":"para","text":"If a reversed singular direction has b = 0 < a b=0<a —the marginal collapsed where the class has not—then the exponent ( 4 ) tends to − w -w , negative for every w > 0 w>0 : the guided flow itself expands the coordinate as σ → 0 \\sigma\\to 0 , so no discretization can be uniformly stable because the continuous target is not. More generally, reverse ordering with 0 < b < a 0<b<a can create finite- σ \\sigma expansion windows, but only the reversed singular case creates this terminal instability. This is a property of the guidance target, not of the solver; the class-subset hypothesis excludes it by assumption, and autoguidance-style constructions aim to enforce the same directionality."},{"type":"para","text":"To interpret the FID/KID split at the hard cell, we computed post-hoc feature-manifold and sharpness diagnostics on the same images. Fitted CFG improves Inception precision/recall over CFG in both 5k blocks (e.g. 0.799 / 0.474 0.799/0.474 versus 0.704 / 0.360 0.704/0.360 ); interval has higher recall but lower precision. The pixel-saturation median is 0.16 0.16 for CFG and 0 for fitted CFG and interval, and the median Laplacian energy drops by an order of magnitude—consistent with the fitted step removing oversaturated high-frequency artifacts (we use the Laplacian only as an oversaturation proxy, not as evidence of sharpness). The DINOv2 audit below is a backbone-swapped control on the same split."},{"type":"para","text":"To test whether the FID/KID split is Inception-specific, we recompute feature-space distances with a DINOv2 backbone ( dinov2_vits14 , CLS-token features; images bilinearly resized to 224 224 ) against 50 50 k CIFAR-10 training images. We report the Fréchet distance FD-DINOv2 and an unbiased RBF-kernel MMD (median-heuristic bandwidth, 100 100 subsets of size 1000 1000 ). On the two 5k hard-cell blocks, FD-DINOv2 for CFG/fitted/interval is 469.73 / 230.00 / 257.88 469.73/230.00/257.88 and 469.10 / 227.86 / 254.25 469.10/227.86/254.25 , and the DINO-MMD is 0.0258 / 0.0126 / 0.0168 0.0258/0.0126/0.0168 and 0.0257 / 0.0125 / 0.0164 0.0257/0.0125/0.0164 ; both favor fitted CFG in both blocks. FD-DINOv2 is the clean backbone-only control; the MMD changes both backbone and kernel relative to Inception KID, so it is corroborating, not a direct KID analogue."},{"type":"para","text":"The Section 4 interval baseline is the parameter-free terminal cutoff, which on the hard mesh reduces to unguided conditional DDIM. To test whether a tuned interval that preserves conditionality can match the fitted step, we selected per cell the conditionality-viable guidance interval [ σ lo , σ hi ] [\\sigma_{\\rm lo},\\sigma_{\\rm hi}] from a nine-candidate grid (the one whose target accuracy stays closest to CFG), then scored it on a fresh 5k block. Table 3 reports CFG, the fitted step, the tuned interval, and the terminal interval. In the tested cells, a conditionality-preserving tuned interval does not beat the fitted step on FID; the terminal interval can win FID in a cell ( w = 8 , N = 16 w{=}8,N{=}16 : 12.63 12.63 vs 17.12 17.12 ) but pays a clear target-accuracy cost. The fitted step gives interval-like residual stabilization while keeping the guided update active. (Target accuracy is a classifier proxy; the high fitted value at N = 16 N{=}16 should not be read as fuller conditional fidelity, as it may partly reflect reduced diversity.)"},{"type":"para","text":"As a cross-domain check that the coefficient is not specific to pixel-space CIFAR edm , we ran a small Stable Diffusion 1.5 DDIM smoke test at high guidance, swapping only the terminal guidance coefficient. We report pixel saturation (95th percentile of the clipped-pixel fraction) and CLIP image–text alignment; the interval baseline is terminal guidance shutdown. Table 4 shows the fitted step sharply reduces saturation while preserving CLIP alignment far better than shutdown—the same saturation-repair pattern as on CIFAR. This is a smoke/deepen test, not a Stable Diffusion benchmark."},{"type":"para","text":"Setup. We use runwayml/stable-diffusion-v1-5 at 512 × 512 512\\times 512 with deterministic DDIM ( η = 0 \\eta=0 , ϵ \\epsilon -prediction) and float16, over a fixed list of everyday-scene captions (repeated to 256 256 and 128 128 prompts) with no negative prompt and seeds matched across schemes. The fitted step swaps only the terminal guidance coefficient; the interval baseline disables the guidance correction on the terminal steps below the same h ♭ h_{\\flat} -derived cutoff. Saturation p95 is the 95th percentile over images of the fraction of final RGB pixels at the valid-range extremes; CLIP alignment uses openai/clip-vit-base-patch32 ."},{"type":"para","text":"The real-checkpoint residual diagnostic (Table 1 ) uses class 0 and class 1 blocks. To rule out class cherry-picking, we extended it to all ten CIFAR-10 classes on the high-guidance cells ( w ∈ { 6.5 , 8 } w\\in\\{6.5,8\\} , N ∈ { 8 , 16 , 32 } N\\in\\{8,16,32\\} ; 60 class–cell pairs, all fixed before the run). Clipping/saturation is non-increased in 60 / 60 60/60 pairs, and amp p95 improves in 54 / 60 54/60 ; the other six are modestly worse on the amp tail (ratios 1.02 1.02 – 1.25 1.25 , largest class 4 at w = 6.5 , N = 8 w{=}6.5,N{=}8 ), with clipping still non-increased there. We describe this as global saturation robustness and near-global residual-tail improvement , and leave the raw counts for the reader to weigh."},{"type":"para","text":"Theorem 2 is an accuracy statement against the exact guided flow’s own pushforward on the calibration model. On learned checkpoints this does not extend to final-image accuracy: measured against a 512 512 -step vanilla-CFG reference trajectory, the fitted step is not uniformly closer in final denoised-image L 2 L_{2} (Table 5 , ratios > 1 >1 ), even though its residual and amp tails are far smaller (ratios ≪ 1 \\ll 1 ). So the learned-checkpoint evidence supports terminal residual/saturation repair, not a claim that the fitted step is a uniformly better coarse integrator of the vanilla CFG field."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Diffusion and flow-matching samplers integrate a probability-flow ODE whose velocity field stiffens as the noise scale σ → σ min \\\\sigma\\\\to\\\\sigma_{\\\\min} : on a coordinate normal to the data manifold the exact flow contracts at exponent one, by σ n + 1 / σ n \\\\sigma_{n+1}/\\\\sigma_{n} per step. A recent fitted-operator analysis of unguided samplers [Zhang, 2026 ] shows that this terminal layer admits a unique fitted (layer-exact) one-step operator: a frozen-field Euler step—Euler with the velocity held at its start-of-step value—reproduces that exact contraction if and only if its integration variable is affine in σ \\\\sigma . This singles out the σ \\\\sigma -clock step, algebraically identical to deterministic DDIM [Song et al. , 2021 ] , and makes it asymptotic-preserving (AP) on the exactly solvable calibration models studied there [Zhang, 2026 ] —uniformly accurate as σ min → 0 \\\\sigma_{\\\\min}\\\\to 0 . This paper asks one question of that lens: what survives of DDIM’s fitted-operator protection once we turn on guidance?\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Guidance re-stiffens the discriminative subspace.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Classifier-free guidance (CFG) [Ho and Salimans, 2022 ] runs the sampler on an extrapolated denoiser D w = ( 1 + w ) D c − w D u D_{w}=(1+w)D_{\\\\mathrm{c}}-wD_{\\\\mathrm{u}} , that is, on a second velocity field η w = η c + w ( η c − η u ) \\\\eta_{w}=\\\\eta_{\\\\mathrm{c}}+w(\\\\eta_{\\\\mathrm{c}}-\\\\eta_{\\\\mathrm{u}}) (the η \\\\eta ’s are noise-prediction fields; Section 2 ) with guidance scale g = 1 + w g=1+w . This is not a uniform stiffening. On the commuting Gaussian model of Section 2 , the guided terminal exponent μ w ( σ ) \\\\mu_{w}(\\\\sigma) splits by layer (a layer here is a regime of state-space directions, not a network layer; Figure 1 ): it stays exactly one on directions shared by the class and the marginal, freezes to zero on tangential directions, and rises to 1 + w 1+w on the discriminative directions—the subspace that separates the class from the marginal (Proposition 1 ). Guidance re-stiffens precisely this discriminative subspace, and it does so with a second singular parameter: w w enters the terminal structure on the same footing as σ min \\\\sigma_{\\\\min} . Because D w D_{w} is not any distribution’s posterior mean, DDIM’s fitted-operator protection lapses there: on coarse meshes, vanilla DDIM + + CFG leaves the class of fitted operators and its terminal residual diverges as σ min → 0 \\\\sigma_{\\\\min}\\\\to 0 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Two orthogonal bills.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This is a statement about the solver . A parallel line studies how the continuous guided law itself departs from the tilted conditional it is meant to sample [Bradley and Nakkiran, 2025 ; Chidambaram et al. , 2024 ] : that charges the model ; we charge the discretization —our reference solution is always the exact guided flow’s own pushforward. The two bills are orthogonal, and conflating them has obscured how much of guidance’s terminal pathology—the oversaturation and norm inflation seen at large w w [Sadat et al. , 2025 ] —has a solver-induced component that appears even when the exact guided flow is the reference, from stepping guidance with an operator it has quietly un-fitted.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"The repair is one coefficient.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Rectifying the anomalous exponent 1 + w 1+w by a gauge change and pushing the σ \\\\sigma -clock step through it yields a one-coefficient modification of CFG that touches nothing else in the sampler—only the coefficient on the guidance direction D u − D c D_{\\\\mathrm{u}}-D_{\\\\mathrm{c}} changes:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"with r = σ n + 1 / σ n r=\\\\sigma_{n+1}/\\\\sigma_{n} . It uses the same two denoiser evaluations per step—the implementation delta is a single line, at zero additional NFE (network function evaluations)—agrees with CFG to first order, and reduces to DDIM at w = 0 w=0 . We call it the guided fitted step, or fitted CFG . It is the unique scalar coefficient that is exact on the discriminative terminal layer using no spectral information (Lemma 2 ); it never reflects a coordinate across the class manifold (Proposition 3 ); and its one-step r → 0 r\\\\to 0 limit lands on the conditional denoiser D c D_{\\\\mathrm{c}} (the terminal-limit class projection), where vanilla CFG lands on the overshot D w D_{w} , the discrete mechanism of oversaturation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Contributions.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Structure and a barrier. A layer classification of the guided terminal exponent (Proposition 1 ) and a guided clock barrier (Theorem 1 ) with three ordered step-size thresholds—reflection, residual-amplification failure (Section 3 ), absolute stability—and the resulting two-tier guidance tax on step count (Corollary 1 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Oversaturation as an endpoint, and the guidance interval. A model-free identity reading one-step DDIM + + CFG’s overshoot to D w D_{w} as the barrier’s r → 0 r\\\\to 0 endpoint (Remark 2 ), and a parameter-free prediction of the terminal edge of limited-interval guidance [Kynkäänniemi et al. , 2024 ] (Remark 3 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A spectrum-free fitted repair. The guided fitted step (Definition 1 ): first-order consistent, sign-preserving and terminal-exact on the model, unique (Lemma 2 ), and—on the discriminative crossover—free of the σ min \\\\sigma_{\\\\min} -divergent blow-up and first-order accurate as σ min → 0 \\\\sigma_{\\\\min}\\\\to 0 (Proposition 4 , Theorem 2 ): asymptotic-preserving in σ min \\\\sigma_{\\\\min} , though not uniform in w w .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A scoped empirical program. On learned CIFAR-10 edm [Karras et al. , 2022 ] checkpoints, fitted CFG is a zero-extra-NFE high-guidance stabilizer: the residual and clipping certificates improve in both w ∈ { 6.5 , 8 } w\\\\in\\\\{6.5,8\\\\} cells; the hard-cell image evaluations (two 5k blocks, a 50k replication) give it the best FID with target accuracy preserved, and a DINOv2 backbone-swap keeps the same fitted-favoring ordering; a nine-cell grid gives 9 / 9 9/9 FID wins over CFG. We report the limits alongside—the CFG-favoring KID split and threshold-derived interval baselines that win FID/KID at lower conditionality (Section 4 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Section 6 collects the claim boundaries.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 The guided flow and its terminal layers\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"The unguided flow.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We work throughout in the variance-exploding family: p σ = p data ∗ 𝒩 ( 0 , σ 2 I ) p_{\\\\sigma}=p_{\\\\rm data}*\\\\mathcal{N}(0,\\\\sigma^{2}I) for σ ∈ [ σ min , σ max ] \\\\sigma\\\\in[\\\\sigma_{\\\\min},\\\\sigma_{\\\\max}] , with denoiser D ( x , σ ) = 𝔼 [ x 0 ∣ x σ = x ] D(x,\\\\sigma)=\\\\mathbb{E}[x_{0}\\\\mid x_{\\\\sigma}=x] and normalized residual η = ( x − D ) / σ \\\\eta=(x-D)/\\\\sigma —the noise prediction ε ^ \\\\hat{\\\\varepsilon} of practice ( − σ ∇ x log p σ -\\\\sigma\\\\nabla_{x}\\\\log p_{\\\\sigma} ); not DDIM’s stochasticity parameter, which is 0 throughout. The probability flow ODE is d X / d σ = η ( X , σ ) \\\\mathrm{d}X/\\\\mathrm{d}\\\\sigma=\\\\eta(X,\\\\sigma) , and the decomposition x = D + σ η x=D+\\\\sigma\\\\eta splits the state into a slow manifold component and a stiff normal component—normal to the data manifold, contracting fastest—whose exponent is 1 1 : on a normal coordinate the exact flow contracts by ( σ n + 1 / σ n ) 1 (\\\\sigma_{n+1}/\\\\sigma_{n})^{1} per step. A fitted-operator analysis of the unguided terminal layer [Zhang, 2026 ] shows that a frozen-field Euler step is layer-exact if and only if its clock is affine in σ \\\\sigma , making the σ \\\\sigma -clock step—algebraically the deterministic DDIM update [Song et al. , 2021 ] —the unique fitted operator up to affine reparameterization, with rectified flow its flow-matching counterpart; this fitted-operator property is what makes it asymptotic-preserving as σ min → 0 \\\\sigma_{\\\\min}\\\\to 0 . The present paper asks what survives of that protection under guidance.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Guidance as a second velocity field.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Classifier-free guidance [Ho and Salimans, 2022 ] replaces the conditional denoiser D c ( x , σ ) D_{\\\\mathrm{c}}(x,\\\\sigma) by the extrapolation\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where D u D_{\\\\mathrm{u}} is the unconditional (marginal) denoiser and g = 1 + w g=1+w is the guidance scale of practice, so w = 6.5 w=6.5 is CFG scale g = 7.5 g=7.5 . Writing η c = ( x − D c ) / σ \\\\eta_{\\\\mathrm{c}}=(x-D_{\\\\mathrm{c}})/\\\\sigma and η u = ( x − D u ) / σ \\\\eta_{\\\\mathrm{u}}=(x-D_{\\\\mathrm{u}})/\\\\sigma , the sampled dynamics is the guided flow\"\n },\n {\n \"type\": \"para\",\n \"text\": \"with γ \\\\gamma the normalized guidance direction. Equation ( 2 ) is the dynamical statement of guidance: the sampler follows the conditional velocity plus w w times the discriminative correction. Two facts follow. First, D w D_{w} is in general not a VE posterior mean with a positive-semidefinite Tweedie covariance, so η w \\\\eta_{w} is not the residual of such a denoiser; the rigidity theorem that singles out DDIM as layer-exact inside the unguided denoiser class [Zhang, 2026 ] no longer applies directly. The exit is quantitative, not merely definitional: for any true denoiser, J D = ∇ x D = Cov ( X 0 ∣ x ) / σ 2 ⪰ 0 J_{D}=\\\\nabla_{x}D=\\\\mathrm{Cov}(X_{0}\\\\mid x)/\\\\sigma^{2}\\\\succeq 0 [Efron, 2011 ] caps the exponent of the residual field at 1 1 (on the model, the residual exponent is A ≤ 1 A\\\\leq 1 in ( 3 )), while on discriminative directions the guided exponent exceeds 1 1 at every σ \\\\sigma (Proposition 1 )—no repackaging of any denoiser, affine or otherwise, reproduces the guided field. Second, the guided flow carries a second parameter: any AP statement must state its w w -dependence, and the fitted step below removes the σ min \\\\sigma_{\\\\min} -divergent terminal barrier for bounded w w , though not uniformly in w w .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"The model problem.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our analysis is carried out on the guided analog of the unguided model problem [Zhang, 2026 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"p c = 𝒩 ( 0 , C c ) p_{\\\\mathrm{c}}=\\\\mathcal{N}(0,C_{\\\\mathrm{c}}) and p u = 𝒩 ( 0 , C u ) p_{\\\\mathrm{u}}=\\\\mathcal{N}(0,C_{\\\\mathrm{u}}) with [ C c , C u ] = 0 [C_{\\\\mathrm{c}},C_{\\\\mathrm{u}}]=0 , and on the shared eigenbasis the spectra satisfy a i ≤ b i a_{i}\\\\leq b_{i} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The ordering a i ≤ b i a_{i}\\\\leq b_{i} is the class-subset hypothesis : the class is narrower than the marginal in every probed direction (as when the marginal is a mixture whose components share the class covariance). Real learned checkpoints need not obey it; the reverse case is a genuine model boundary, discussed in Remark 4 . On an eigen-coordinate ξ \\\\xi with class variance a a and marginal variance b b , the two denoisers are linear and the residuals are\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The guided flow ( 2 ) therefore closes coordinate-wise,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"and every question about clocks, stability, and fitted operators reduces to the behavior of the exponent function μ w \\\\mu_{w} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Under Hypothesis 1 , as σ → 0 \\\\sigma\\\\to 0 the exponent function ( 4 ) satisfies: (i) on shared normal directions ( a = b = 0 a=b=0 ), μ w ≡ 1 \\\\mu_{w}\\\\equiv 1 : the guidance weight cancels and the layer is the unguided exponent- 1 1 layer; (ii) on tangential directions ( 0 < a ≤ b 0<a\\\\leq b ), μ w → 0 \\\\mu_{w}\\\\to 0 : the coordinate freezes; (iii) on discriminative normal directions ( a = 0 < b a=0<b ),\"\n },\n {\n \"type\": \"para\",\n \"text\": \"monotonically as σ \\\\sigma decreases, with crossover midpoint at σ 2 = b \\\\sigma^{2}=b .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"(Proofs for Sections 2 and 3 are deferred to Appendix A .)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Proposition 1 is the structural fact of the paper. Guidance does not stiffen the flow uniformly; it re-stiffens exactly the discriminative subspace —the directions collapsed within the class but present in the marginal ( a = 0 < b a=0<b )—and ( 5 ) locates where the re-stiffening begins: the anomalous exponent switches on as σ \\\\sigma descends through the class-separation scale b \\\\sqrt{b} . Above that scale the guided flow is, to leading order, the unguided flow; below it, the terminal layer runs at exponent 1 + w 1+w , and w w enters the singular structure of the problem on the same footing as σ min \\\\sigma_{\\\\min} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Under Hypothesis 1 , the exact solution of ( 2 ) on an eigen-coordinate over one step σ n → σ n + 1 \\\\sigma_{n}\\\\to\\\\sigma_{n+1} is ξ n + 1 = Φ w ξ n \\\\xi_{n+1}=\\\\Phi_{w}\\\\,\\\\xi_{n} with\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In particular, on a discriminative coordinate ( a = 0 a=0 ), writing r = σ n + 1 / σ n r=\\\\sigma_{n+1}/\\\\sigma_{n} ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"which tends to r 1 + w r^{1+w} once σ n 2 ≪ b \\\\sigma_{n}^{2}\\\\ll b .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The singular limit now has two knobs. As σ min → 0 \\\\sigma_{\\\\min}\\\\to 0 the discriminative layer contracts by the anomalous power σ 1 + w \\\\sigma^{1+w} ; as w w grows at fixed σ min \\\\sigma_{\\\\min} the same layer stiffens without bound. We ask two questions: which part of DDIM + + CFG’s failure is the pure terminal-layer solver barrier, and can that barrier be removed by a one-coefficient fitted repair? Uniform accuracy through the finite- σ \\\\sigma crossover is separate, needing either mesh resolution of the crossover or the spectrum-aware Gaussian factor.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 The guided barrier and the fitted operator\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This section establishes one claim: classifier-free guidance introduces a second singular parameter w w , and on discriminative terminal directions the σ \\\\sigma -clock frozen-field step—DDIM applied to the guided denoiser—is no longer fitted. The program has three parts. First, a barrier: the guided analog of the unguided clock dichotomy [Zhang, 2026 ] , with sharp thresholds in the log-step h h and weight w w (Theorem 1 ). Second, an operator: a fitted step that removes the pure discriminative terminal-layer reflection and σ min \\\\sigma_{\\\\min} -divergent residual blow-up at zero additional cost (Definition 1 ). Third, accuracy: a first-order certificate on the discriminative crossover (Theorem 2 ); the whole-model theory remains future work. Throughout, the reference solution is the exact guided flow’s own pushforward: we charge the discretization of ( 2 ), not the distortion of the continuous guided law away from the tilted conditional (Section 1 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"The frozen step.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"DDIM applied to D w D_{w} —equivalently, σ \\\\sigma -clock Euler with the guided field frozen at the left endpoint—is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"which on an eigen-coordinate of the model contracts by\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On the pure discriminative layer ( A , B ) = ( 1 , 0 ) (A,B)=(1,0) this is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"to be compared with the exact factor e − ( 1 + w ) h \\\\mathrm{e}^{-(1+w)h} of Lemma 1 . Here h = log ( σ n / σ n + 1 ) h=\\\\log(\\\\sigma_{n}/\\\\sigma_{n+1}) is the step in λ = − log σ \\\\lambda=-\\\\log\\\\sigma (half the VE log-SNR), so a uniform λ \\\\lambda -mesh has constant h h . Along a trajectory, call max n ‖ η w ( x n , σ n ) ‖ / ‖ η w ( x 0 , σ 0 ) ‖ \\\\max_{n}\\\\|\\\\eta_{w}(x_{n},\\\\sigma_{n})\\\\|/\\\\|\\\\eta_{w}(x_{0},\\\\sigma_{0})\\\\| the residual-amplification certificate , written amp \\\\mathrm{amp} ; the sampler already forms every quantity in it, and a scheme is residual-stable if the certificate stays bounded as σ min → 0 \\\\sigma_{\\\\min}\\\\to 0 . Everything in Theorem 1 is a statement about the elementary function ( 10 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Consider the frozen step ( 8 ) on the pure discriminative layer, η w = ( 1 + w ) ξ / σ \\\\eta_{w}=(1+w)\\\\,\\\\xi/\\\\sigma with w > 0 w>0 , on a uniform λ \\\\lambda -mesh of step h h . Let\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Then h ♭ < h ♯ < h ∞ h_{\\\\flat}<h_{\\\\sharp}<h_{\\\\infty} , and:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Sign. G w ( h ) > 0 G_{w}(h)>0 iff h < h ♭ h<h_{\\\\flat} ; equivalently, at fixed h h , sign preservation fails once w > 1 / ( e h − 1 ) w>1/(\\\\mathrm{e}^{h}-1) . For h > h ♭ h>h_{\\\\flat} every step reflects the coordinate across the class manifold.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Residual amplification. The per-step residual amplification is | G w ( h ) | e h |G_{w}(h)|\\\\,\\\\mathrm{e}^{h} , which is ≤ 1 \\\\leq 1 iff h ≤ h ♯ h\\\\leq h_{\\\\sharp} ; equivalently, raw amplification begins once w > 2 / ( e h − 1 ) w>2/(\\\\mathrm{e}^{h}-1) . For h > h ♯ h>h_{\\\\sharp} the amplification factor is w e h − ( 1 + w ) > 1 w\\\\mathrm{e}^{h}-(1+w)>1 per step. On a terminal subwindow σ ∈ [ σ min , ε b ] \\\\sigma\\\\in[\\\\sigma_{\\\\min},\\\\varepsilon\\\\sqrt{b}\\\\,] , where B ≤ ε 2 / ( 1 + ε 2 ) B\\\\leq\\\\varepsilon^{2}/(1+\\\\varepsilon^{2}) makes the exact per-step factor pure-layer up to O ( ε 2 ) O(\\\\varepsilon^{2}) , of λ \\\\lambda -length Λ ε = log ( ε b / σ min ) \\\\Lambda_{\\\\varepsilon}=\\\\log(\\\\varepsilon\\\\sqrt{b}/\\\\sigma_{\\\\min}) the guided residual grows by\"\n },\n {\n \"type\": \"para\",\n \"text\": \"so for fixed ε > 0 \\\\varepsilon>0 and h > h ♯ h>h_{\\\\sharp} the scheme fails the amplification certificate as σ min → 0 \\\\sigma_{\\\\min}\\\\to 0 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Absolute stability. | G w ( h ) | ≤ 1 |G_{w}(h)|\\\\leq 1 for all h h when w ≤ 1 w\\\\leq 1 ; for w > 1 w>1 , iff h ≤ h ∞ h\\\\leq h_{\\\\infty} .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All three thresholds decay like c / w c/w as w → ∞ w\\\\to\\\\infty , with c = 1 , 2 , 2 c=1,2,2 respectively.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In words: sign preservation fails first (a, above h ♭ h_{\\\\flat} : every step reflects across the class manifold), the certificate second (b, above h ♯ h_{\\\\sharp} : the residual compounds, diverging as σ min → 0 \\\\sigma_{\\\\min}\\\\to 0 ), absolute stability last (c, only for w > 1 w>1 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On a uniform λ \\\\lambda -mesh covering the discriminative window (of λ \\\\lambda -length Λ b = log ( b / σ min ) \\\\Lambda_{b}=\\\\log(\\\\sqrt{b}/\\\\sigma_{\\\\min}) ), avoiding reflection requires\"\n },\n {\n \"type\": \"para\",\n \"text\": \"while merely keeping the residual nonexpansive requires N ≥ Λ b / log ( 1 + 2 / w ) ∼ w 2 Λ b N\\\\geq\\\\Lambda_{b}/\\\\log(1+2/w)\\\\sim\\\\tfrac{w}{2}\\\\Lambda_{b} . A uniform global mesh must meet this bound throughout the window, so Λ b \\\\Lambda_{b} is replaced by the full horizon Λ = log ( σ max / σ min ) \\\\Lambda=\\\\log(\\\\sigma_{\\\\max}/\\\\sigma_{\\\\min}) : at working scale g = 7.5 g=7.5 ( Λ = log ( 80 / 0.002 ) ≈ 10.6 \\\\Lambda=\\\\log(80/0.002)\\\\approx 10.6 ), reflection-free sampling already demands N ≳ 74 N\\\\gtrsim 74 steps—an order of magnitude above unguided-DDIM-accurate budgets [Karras et al. , 2022 ] .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In the window h ∈ ( h ♭ , h ♯ ] h\\\\in(h_{\\\\flat},h_{\\\\sharp}] the scheme is reflecting but nonexpansive : the flip does not grow the residual. The flip is the mechanism of visible artifacts—the iterate lands on the wrong side of the class manifold each step—while the amplification crossing at h ♯ h_{\\\\sharp} is where the certificate, and any uniform accuracy claim, actually fails. Practitioners tune w w down at low σ \\\\sigma well before the certificate diverges: part (a), not part (b), is the threshold their tuning discovers.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The following identity is model-free. A single frozen step ( 8 ) from σ n \\\\sigma_{n} to σ n + 1 = r σ n \\\\sigma_{n+1}=r\\\\sigma_{n} with r → 0 r\\\\to 0 gives\"\n },\n {\n \"type\": \"para\",\n \"text\": \"one-step DDIM + + CFG lands on the extrapolated denoiser, displaced from the class manifold by − w ( D u − D c ) -w(D_{\\\\mathrm{u}}-D_{\\\\mathrm{c}}) . On the model this is the reflection ξ ↦ − w ξ \\\\xi\\\\mapsto-w\\\\,\\\\xi of Theorem 1 (a) pushed to its endpoint. We read this as the discrete mechanism of the oversaturation and norm-inflation phenomenology reported for large guidance scales [Sadat et al. , 2025 ] : it is a property of the solver at coarse steps, not of the continuous guided flow—on the calibration model the exact flow contracts to D c D_{\\\\mathrm{c}} as r 1 + w r^{1+w} while the frozen step overshoots to D w D_{w} .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"The guided fitted step.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The repair is dictated by the same logic that produced DDIM in the unguided setting [Zhang, 2026 ] : integrate the stiff layer exactly and freeze what is slow. The gauge ξ ~ = σ − w ξ \\\\tilde{\\\\xi}=\\\\sigma^{-w}\\\\xi rectifies the anomalous exponent 1 + w 1+w to exponent 1 1 ; transporting the σ \\\\sigma -clock Euler step through the gauge and back yields, on the whole state,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"With r = σ n + 1 / σ n r=\\\\sigma_{n+1}/\\\\sigma_{n} and all fields evaluated at ( x n , σ n ) (x_{n},\\\\sigma_{n}) ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"equivalently, in residual form,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The scheme touches exactly one coefficient of classifier-free guidance: vanilla DDIM + + CFG is ( 15 ) with the coefficient w ( r − 1 ) w(r-1) in place of r 1 + w − r r^{1+w}-r on the guidance direction D u − D c D_{\\\\mathrm{u}}-D_{\\\\mathrm{c}} . It uses the same two denoiser evaluations per step, so the cost—and the implementation delta—is one line. Proposition 3 verifies the factor coordinate-wise; Lemma 2 pins the coefficient from terminal exactness alone.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"r 1 + w − r = w ( r − 1 ) + 1 2 w ( w + 1 ) h 2 + O ( h 3 ) r^{1+w}-r=w(r-1)+\\\\tfrac{1}{2}w(w+1)\\\\,h^{2}+O(h^{3}) as h → 0 h\\\\to 0 , so ( 15 ) is a first-order-consistent discretization of the guided flow ( 2 ), agreeing with DDIM + + CFG through O ( h ) O(h) ; and at w = 0 w=0 both coefficients vanish identically, collapsing the scheme to DDIM.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Under Hypothesis 1 , the step ( 15 ) contracts an eigen-coordinate by\"\n },\n {\n \"type\": \"para\",\n \"text\": \"a sum of nonnegative terms; hence G fit > 0 G_{\\\\rm fit}>0 on every coordinate, for every mesh, and every w ≥ 0 w\\\\geq 0 : the fitted step preserves the sign of every analyzed coordinate. It is exact in the three terminal regimes— G fit = r G_{\\\\rm fit}=r on shared normals ( A = B = 1 ) (A=B=1) , G fit = r 1 + w G_{\\\\rm fit}=r^{1+w} on pure discriminative normals ( A = 1 , B = 0 ) (A=1,B=0) , and G fit = 1 G_{\\\\rm fit}=1 on frozen tangential directions ( A = B = 0 ) (A=B=0) —and its one-jump limit is x n + 1 → D c x_{n+1}\\\\to D_{\\\\mathrm{c}} , the conditional denoiser (the terminal-limit class projection), in contrast with ( 14 ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Consider scalar coefficient updates of the form\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where α \\\\alpha depends only on ( r , w ) (r,w) . The unique such coefficient that is exact on the pure discriminative terminal regime ( A , B ) = ( 1 , 0 ) (A,B)=(1,0) is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Consequently, terminal exactness, DDIM recovery at w = 0 w=0 , and exactness on the shared-normal and tangent terminal regimes determine the fitted coefficient without any spectrum information.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On a single discriminative Gaussian coordinate with a = 0 < b a=0<b , define\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Let σ 0 > σ 1 > ⋯ > σ K \\\\sigma_{0}>\\\\sigma_{1}>\\\\cdots>\\\\sigma_{K} be any decreasing mesh, r n = σ n + 1 / σ n r_{n}=\\\\sigma_{n+1}/\\\\sigma_{n} , q n = q ( σ n ) q_{n}=q(\\\\sigma_{n}) , and μ n = μ ( σ n ) \\\\mu_{n}=\\\\mu(\\\\sigma_{n}) . For the fitted step ( 15 ), the guided residual ratio over one step is\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Hence for every prefix k k ,\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Thus the fitted step removes the σ min \\\\sigma_{\\\\min} -divergent terminal blow-up of Theorem 1 on the discriminative crossover, but it may still pay a finite O ( 1 + w ) O(1+w) crossover tax on a coarse mesh.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Bounded amplification in fact upgrades to accuracy: on this crossover the accumulated log-defect (uniform mesh, h ≤ 1 h\\\\leq 1 ) obeys 0 ≤ log Θ K ≤ 1 + h 4 w ( w + 2 ) h 0\\\\leq\\\\log\\\\Theta_{K}\\\\leq\\\\tfrac{1+h}{4}\\\\,w(w+2)\\\\,h , so the fitted step is first-order uniformly accurate against the exact guided flow’s own pushforward, with a W 2 W_{2} bound whose constant is independent of b b , K K , and σ min \\\\sigma_{\\\\min} (the O ( w 2 ) O(w^{2}) factor is not uniform in w w ). We defer the formal statement and proof to Theorem 2 in Appendix A .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Extending this crossover accuracy to the tangential layers and learned geometry needs the spectrum-aware Gaussian factor ( 6 ) and is future work.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Limited-interval guidance [Kynkäänniemi et al. , 2024 ] disables guidance outside a tuned band ( σ lo , σ hi ) (\\\\sigma_{\\\\rm lo},\\\\sigma_{\\\\rm hi}) and leaves open whether the band can be derived rather than tuned. Theorem 1 supplies the terminal side without free parameters: on schedules whose local step h n h_{n} grows toward low σ \\\\sigma (e.g. Karras power meshes; on a uniform λ \\\\lambda -mesh h n h_{n} is constant and there is no distinct crossing), the level at which h n h_{n} crosses h ♭ ( w ) = log ( 1 + 1 / w ) h_{\\\\flat}(w)=\\\\log(1+1/w) predicts σ lo \\\\sigma_{\\\\rm lo} : below it, frozen-field guidance at weight w w is unaffordable and the interval or the fitted step must take over. The high -noise cutoff σ hi \\\\sigma_{\\\\rm hi} is not explained by the barrier: there ‖ D u − D c ‖ \\\\|D_{\\\\mathrm{u}}-D_{\\\\mathrm{c}}\\\\| is small against the model’s own error, a signal-level criterion outside this paper’s ledger.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"If a direction has b = 0 < a b=0<a , the exponent ( 4 ) tends to − w -w : the continuous guided flow expands as σ → 0 \\\\sigma\\\\to 0 , an instability of the guidance target that no solver can repair; the class-subset hypothesis excludes it by assumption (Appendix B ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Certificates.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The certificate is removal of the σ min \\\\sigma_{\\\\min} -divergent pure-layer mechanism (Proposition 4 ) plus first-order accuracy on the discriminative crossover (Theorem 2 ), not global residual nonexpansion through every finite- σ \\\\sigma crossover.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Empirical diagnostics on learned checkpoints\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Synthetic crossover.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our primary residual and image-quality comparisons fix their schemes, cells, and metrics before each run; the feature-space, latent-diffusion, and dense-reference results are exploratory diagnostics. The single-coordinate Gaussian diagnostic isolates the theorem’s singular mechanism: on uniform log- σ \\\\sigma meshes with a = 0 < b a=0<b , CFG develops the coarse-mesh branch of Theorem 1 while the fitted coefficient stays below Proposition 4 ’s finite-tax bound (Figure 2 ). In the default w w – N N sweep, worst-case amplification reaches 10 3 10^{3} – 10 9 10^{9} for CFG but stays ≤ 4.66 \\\\leq 4.66 for the fitted step, below each 1 + w 1+w bound—a theorem figure, not a learned-checkpoint quality metric.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Real-checkpoint residual diagnostics.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We then test the coefficient as a drop-in replacement on NVIDIA edm CIFAR-10 VP checkpoints [Karras et al. , 2022 ] (the ‘VP’ label is network preconditioning only; sampling uses the σ \\\\sigma -parameterization): D c D_{\\\\mathrm{c}} is the class-conditional checkpoint, D u D_{\\\\mathrm{u}} the separately trained unconditional one, and both schemes use the same Karras mesh, seeds, labels, and NFE. The grid is w ∈ { 4 , 6.5 , 8 } w\\\\in\\\\{4,6.5,8\\\\} and N ∈ { 8 , 16 , 32 } N\\\\in\\\\{8,16,32\\\\} , with two 64-seed blocks (class 0 , seeds from 1000 1000 ; class 1 1 , from 2000 2000 ). Table 1 reports paired fitted/CFG ratios for the amplification median and tail, and paired clipping differences; lower is better.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"In both blocks, for every tested N ∈ { 8 , 16 , 32 } N\\\\in\\\\{8,16,32\\\\} at w = 6.5 w=6.5 and w = 8 w=8 , fitted CFG improves the amp median, amp p95, and clipping median. The hardest cell is the clearest: at w = 8 , N = 8 w=8,N=8 , the class- 0 amp median/p95 falls from 4.57 / 13.35 4.57/13.35 under CFG to 1.41 / 1.71 1.41/1.71 under fitted CFG, while the clipping median drops from 0.379 0.379 to 0 ; the class- 1 1 replicate matches (amp p95 ratio 0.116 0.116 , clipping delta − 0.23 -0.23 ). At w = 4 w=4 , where the theory predicts a much weaker tax, the median amplification is mixed, but the amp tail and saturation proxy still improve in most cells. We use this as evidence for high-guidance stabilization, not uniform superiority at weak guidance.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Image-quality evaluation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We next ran two 5000-image balanced-class evaluations on the hard cell w = 8 , N = 8 w=8,N=8 , plus a 50k replication, each including a terminal-interval baseline (vanilla CFG correction off when h > h ♭ h>h_{\\\\flat} , after Kynkäänniemi et al. [ 2024 ] ; on this coarse mesh every step crosses h ♭ h_{\\\\flat} , so it reduces to unguided conditional DDIM). We score with FID and KID (feature-space quality metrics; lower is better). Table 2 collects the results. Fitted CFG has the best FID in all three blocks at zero extra NFE, and the certificates replicate (amp p95 falls from 13 + 13{+} to ≈ 1.6 {\\\\approx}1.6 , clipping p95 from ≈ 0.57 {\\\\approx}0.57 to 0 ). The metrics disagree, however: KID is lowest for CFG in every block. We therefore read this as support for high-guidance stabilization and FID improvement, not an unqualified image-quality win; feature-manifold and sharpness diagnostics are consistent with CFG’s oversaturation contributing to the split (Figure 3 ), and a DINOv2 backbone-swapped audit gives the same fitted-favoring ordering, so the split is not Inception-specific (Appendices C and D ).\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Finally, the target-accuracy column of Table 2 is a classifier-conditionality diagnostic (classifier at 94.37 % 94.37\\\\% on the CIFAR-10 test set): in every block fitted CFG stays within 0.6 0.6 pp of CFG yet ∼ 5 {\\\\sim}5 pp above interval, keeping CFG’s conditional pull without its clipping pathology.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Robustness grid.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The 50k replication (Table 2 ) reduces the 5k-sampling-artifact concern for the FID ordering, and the KID split persists at scale. A nine-cell grid ( w ∈ { 4 , 6.5 , 8 } w\\\\in\\\\{4,6.5,8\\\\} , N ∈ { 8 , 16 , 32 } N\\\\in\\\\{8,16,32\\\\} , 5k images each) shows it is not cell-specific: fitted CFG improves FID in 9 / 9 9/9 cells, clipping p95 in 9 / 9 9/9 , and amp p95 in 8 / 9 8/9 (exception w = 4 , N = 32 w{=}4,N{=}32 , both near 1 1 ). The terminal-interval baseline wins FID at N ∈ { 16 , 32 } N\\\\in\\\\{16,32\\\\} and KID at N = 32 N=32 but loses conditionality everywhere ( 3.24 3.24 – 5.72 5.72 pp).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Interval-baseline separation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Is the fitted step just guidance shutdown? We compared it against a tuned conditionality-preserving limited interval (selected per cell) and the parameter-free terminal interval (Appendix E ). In the tested cells, the tuned interval preserves target accuracy but does not beat the fitted step on FID; the terminal interval can win FID (e.g. 12.63 12.63 vs 17.12 17.12 at w = 8 , N = 16 w{=}8,N{=}16 ) but drops target accuracy by 5 5 – 6 6 pp. So the fitted step gives interval-like residual stabilization while keeping the guided update active—it is not explained away by turning guidance off.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"All-class audit.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Extending the residual diagnostic from two classes to all ten CIFAR-10 classes on the high-guidance cells (60 class–cell pairs), clipping/saturation is non-increased in 60 / 60 60/60 pairs and amp p95 improves in 54 / 60 54/60 : global saturation robustness and near-global residual-tail improvement (Appendix G ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Latent-diffusion smoke.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The mechanism is not specific to pixel-space CIFAR. On Stable Diffusion 1.5 DDIM at high guidance, the same coefficient sharply reduces pixel saturation (p95 0.134 → 0.009 0.134\\\\to 0.009 at guidance 12 12 , over 256 images) while preserving CLIP image–text alignment ( 0.320 → 0.314 0.320\\\\to 0.314 ); terminal guidance shutdown reaches lower saturation but collapses CLIP to 0.271 0.271 (Appendix F ). This is a cross-domain smoke test, not a Stable Diffusion benchmark.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Related work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The CFG-repair family is crowded, and methods differ mainly in their mathematical object : a manifold projection, a guidance-vector rescaling, an interval schedule, a dynamic weight, or—ours—a terminal fitted-operator coefficient. In brief: CFG++ [Chung et al. , 2025 ] constrains the update to the data manifold; limited-interval guidance [Kynkäänniemi et al. , 2024 ] switches guidance off outside a σ \\\\sigma band (Remark 3 derives its terminal edge parameter-free); APG [Sadat et al. , 2025 ] down-weights the update’s parallel component; CFG-Zero ⋆ [Fan et al. , 2025 ] optimizes the scale and zero-inits early steps; and C 2 FG [Gao et al. , 2026 ] decays the weight in time. None derives the terminal-fitted coefficient r 1 + w − r r^{1+w}-r from the guided exponent—where the barrier lives. An orthogonal line analyses what the continuous guided law samples—CFG is a predictor–corrector, not a sampler of its tilted target [Bradley and Nakkiran, 2025 ; Chidambaram et al. , 2024 ] —the “model” bill complementary to ours.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6 Discussion and limitations\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"What is proved.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On the commuting class-subset Gaussian model, the barrier (Theorem 1 ) is sharp; the fitted step is consistent, sign-preserving, terminal-exact, pays only a finite O ( 1 + w ) O(1+w) crossover tax (Prop. 4 ), and is first-order accurate on the crossover (Theorem 2 )—all about the discretization against the guided flow’s own pushforward, AP in σ min \\\\sigma_{\\\\min} but not uniform in w w .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"What is evidence.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On learned CIFAR-10 edm checkpoints, fitted CFG is a drop-in, zero-extra-NFE high-guidance stabilizer: the certificates improve residual amplification and clipping across every w ∈ { 6.5 , 8 } w\\\\in\\\\{6.5,8\\\\} cell; the hard-cell evaluations (5k twice, 50k once) give it the best FID with target accuracy preserved, and a DINOv2 backbone-swap the same ordering; the nine-cell grid gives 9 / 9 9/9 FID wins over CFG. It separates from tuned and terminal interval baselines (which either lose target accuracy or fail to beat it on FID), holds across all ten classes ( 60 / 60 60/60 saturation non-increase, 54 / 60 54/60 amp-tail), and transfers to Stable Diffusion 1.5 DDIM as a saturation-reducing coefficient that keeps CLIP alignment far better than guidance shutdown (Section 4 , Appendices E – G ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"What is neither.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We do not claim (i) an unqualified image-quality win—KID stays lowest for CFG at 5k and 50k, a split tied to CFG’s oversaturation without proven causation; (ii) that CIFAR verifies the sharp threshold law—the learned diagnostics are only consistent with the model ordering; or (iii) a 50k KID win or many-cell large-sample result—the 50k run is one hard cell. Nor is the fitted step a uniformly better integrator of the vanilla CFG field on learned checkpoints: against a 512 512 -step CFG reference its residual tails shrink but its final-image L 2 L_{2} does not (Appendix H ), an interval can win FID in some cells by weakening guidance, and the Stable Diffusion result is a smoke test, not a benchmark. The class-subset hypothesis is itself a boundary: where the class is wider than the marginal, the continuous guided flow is unstable (exponent − w -w ), which no solver can repair (Remark 4 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Scope and outlook.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Every certificate uses only sampler-computed quantities from two separate checkpoints, so shared-head correlations cannot inflate them. ‘Asymptotic-preserving’ is used in the calibration-model sense: on learned checkpoints we audit AP-relevant residuals rather than certify AP. Next: shared-dropout CFG, broader latent-diffusion benchmarks, and a whole-model accuracy theory.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"Reproducibility Statement\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All theoretical results are stated with explicit constants on the model of Hypothesis 1 , and complete proofs appear in Appendix A . Every empirical diagnostic in Section 4 is produced by a self-contained script operating on the public NVIDIA edm CIFAR-10 VP checkpoints with stated seeds, meshes, sample counts, schemes, and metrics; the synthetic-crossover and checkpoint-certificate scripts additionally ship CPU --self-test modes, and the synthetic crossover figure requires no checkpoints. The supplementary package contains the scripts, the exact commands, the run manifests, and all summary artifacts (CSV/JSON) behind every number reported in the paper, including the 50k and nine-cell grid diagnostics.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For a = 0 a=0 one has A ≡ 1 A\\\\equiv 1 , so μ w = ( 1 + w ) − w B = 1 + w ( 1 − B ) = 1 + w b / ( b + σ 2 ) \\\\mu_{w}=(1+w)-wB=1+w(1-B)=1+wb/(b+\\\\sigma^{2}) ; the limits and monotonicity are immediate. Cases (i) and (ii) follow from A , B → 1 A,B\\\\to 1 and A , B → 0 A,B\\\\to 0 respectively. ∎\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Integrate ( 4 ): ∫ μ w d log σ = 1 + w 2 log ( a + σ 2 ) − w 2 log ( b + σ 2 ) \\\\int\\\\mu_{w}\\\\,\\\\mathrm{d}\\\\log\\\\sigma=\\\\tfrac{1+w}{2}\\\\log(a+\\\\sigma^{2})-\\\\tfrac{w}{2}\\\\log(b+\\\\sigma^{2}) up to a constant, since d log ( c + σ 2 ) = 2 A c d log σ \\\\mathrm{d}\\\\log(c+\\\\sigma^{2})=2A_{c}\\\\,\\\\mathrm{d}\\\\log\\\\sigma with A c = σ 2 / ( c + σ 2 ) A_{c}=\\\\sigma^{2}/(c+\\\\sigma^{2}) . Exponentiate the increment. ∎\"\n },\n {\n \"type\": \"para\",\n \"text\": \"From ( 10 ): G w > 0 ⇔ e − h > w / ( 1 + w ) ⇔ h < log ( 1 + 1 / w ) G_{w}>0\\\\iff\\\\mathrm{e}^{-h}>w/(1+w)\\\\iff h<\\\\log(1+1/w) , which is (a). For (b), on the pure layer η w ∝ ξ / σ \\\\eta_{w}\\\\propto\\\\xi/\\\\sigma , so the residual ratio per step is | ξ n + 1 / ξ n | ⋅ σ n / σ n + 1 = | G w | e h |\\\\xi_{n+1}/\\\\xi_{n}|\\\\cdot\\\\sigma_{n}/\\\\sigma_{n+1}=|G_{w}|\\\\mathrm{e}^{h} . On the branch G w ≤ 0 G_{w}\\\\leq 0 this is ( w − ( 1 + w ) e − h ) e h = w e h − ( 1 + w ) (w-(1+w)\\\\mathrm{e}^{-h})\\\\mathrm{e}^{h}=w\\\\mathrm{e}^{h}-(1+w) , which exceeds 1 1 iff e h > 1 + 2 / w \\\\mathrm{e}^{h}>1+2/w ; on the branch G w ≥ 0 G_{w}\\\\geq 0 it equals ( 1 + w ) − w e h ≤ 1 (1+w)-w\\\\mathrm{e}^{h}\\\\leq 1 always. Compounding over Λ ε / h \\\\Lambda_{\\\\varepsilon}/h steps gives ( 12 ). For (c), G w ≤ 1 G_{w}\\\\leq 1 always, and G w ≥ − 1 ⇔ ( 1 + w ) e − h ≥ w − 1 G_{w}\\\\geq-1\\\\iff(1+w)\\\\mathrm{e}^{-h}\\\\geq w-1 , vacuous for w ≤ 1 w\\\\leq 1 and equivalent to h ≤ h ∞ h\\\\leq h_{\\\\infty} otherwise. The ordering h ♯ < h ∞ h_{\\\\sharp}<h_{\\\\infty} follows from 1 + w w − 1 − ( 1 + 2 w ) = 2 w ( w − 1 ) > 0 \\\\frac{1+w}{w-1}-\\\\bigl(1+\\\\frac{2}{w}\\\\bigr)=\\\\frac{2}{w(w-1)}>0 . ∎\"\n },\n {\n \"type\": \"para\",\n \"text\": \"r 1 + w − r = e − ( 1 + w ) h − e − h r^{1+w}-r=\\\\mathrm{e}^{-(1+w)h}-\\\\mathrm{e}^{-h} and w ( r − 1 ) = w ( e − h − 1 ) w(r-1)=w(\\\\mathrm{e}^{-h}-1) ; both are − w h + O ( h 2 ) -wh+O(h^{2}) , and expanding to second order, r 1 + w − r − w ( r − 1 ) = e − ( 1 + w ) h − ( 1 + w ) e − h + w = 1 2 w ( w + 1 ) h 2 + O ( h 3 ) r^{1+w}-r-w(r-1)=\\\\mathrm{e}^{-(1+w)h}-(1+w)\\\\mathrm{e}^{-h}+w=\\\\tfrac{1}{2}w(w+1)h^{2}+O(h^{3}) . ∎\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On an eigen-coordinate, D c = ( 1 − A ) ξ D_{\\\\mathrm{c}}=(1-A)\\\\xi , x − D c = A ξ x-D_{\\\\mathrm{c}}=A\\\\xi , and D u − D c = ( A − B ) ξ D_{\\\\mathrm{u}}-D_{\\\\mathrm{c}}=(A-B)\\\\xi ; substituting into ( 15 ) gives the first expression in ( 17 ), and regrouping 1 − A + r A − r ( A − B ) + r 1 + w ( A − B ) 1-A+rA-r(A-B)+r^{1+w}(A-B) gives the second. Nonnegativity of each term is 0 ≤ B ≤ A ≤ 1 0\\\\leq B\\\\leq A\\\\leq 1 and 0 < r < 1 0<r<1 . The regime values and the r → 0 r\\\\to 0 limit are read off termwise. ∎\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On shared normals ( A , B ) = ( 1 , 1 ) (A,B)=(1,1) one has D u − D c = 0 D_{\\\\mathrm{u}}-D_{\\\\mathrm{c}}=0 , so every scalar coefficient gives the DDIM factor r r . On tangent directions ( A , B ) = ( 0 , 0 ) (A,B)=(0,0) , both denoisers equal the state and every coefficient gives factor 1 1 . On a pure discriminative normal ( A , B ) = ( 1 , 0 ) (A,B)=(1,0) , the factor of the scalar update is r + α ( r , w ) r+\\\\alpha(r,w) . Exactness requires r + α ( r , w ) = r 1 + w r+\\\\alpha(r,w)=r^{1+w} , hence the displayed coefficient, uniquely. ∎\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For a = 0 a=0 , A ≡ 1 A\\\\equiv 1 and q = 1 − B q=1-B . The decomposition ( 17 ) gives the coordinate factor\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Since η w = μ ( σ ) ξ / σ \\\\eta_{w}=\\\\mu(\\\\sigma)\\\\xi/\\\\sigma , the residual ratio is ( μ n + 1 / μ n ) ( G fit / r n ) (\\\\mu_{n+1}/\\\\mu_{n})(G_{\\\\rm fit}/r_{n}) , which is ( 18 ). The bracket is at most 1 1 because 0 ≤ q n ≤ 1 0\\\\leq q_{n}\\\\leq 1 and 0 < r n w ≤ 1 0<r_{n}^{w}\\\\leq 1 , so products telescope:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"∎\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On a discriminative Gaussian coordinate ( a = 0 < b a=0<b ), for w ≥ 0 w\\\\geq 0 and a uniform λ \\\\lambda -mesh of step 0 < h ≤ 1 0<h\\\\leq 1 , the fitted step ( 15 ) and the exact guided flow satisfy ξ K fit = Θ K ξ K ex \\\\xi_{K}^{\\\\rm fit}=\\\\Theta_{K}\\\\,\\\\xi_{K}^{\\\\rm ex} with 0 ≤ log Θ K ≤ 1 + h 4 w ( w + 2 ) h 0\\\\leq\\\\log\\\\Theta_{K}\\\\leq\\\\tfrac{1+h}{4}\\\\,w(w+2)\\\\,h , hence W 2 ( Law ( ξ K fit ) , Law ( ξ K ex ) ) ≤ ( e w ( w + 2 ) h / 2 − 1 ) ( 𝔼 | ξ 0 | 2 ) 1 / 2 W_{2}\\\\bigl(\\\\mathrm{Law}(\\\\xi_{K}^{\\\\rm fit}),\\\\mathrm{Law}(\\\\xi_{K}^{\\\\rm ex})\\\\bigr)\\\\leq\\\\bigl(\\\\mathrm{e}^{w(w+2)h/2}-1\\\\bigr)(\\\\mathbb{E}|\\\\xi_{0}|^{2})^{1/2} . The multiplier bound and its prefactor are independent of b b , K K , and σ min \\\\sigma_{\\\\min} (the W 2 W_{2} scale carries the initial second moment ( 𝔼 | ξ 0 | 2 ) 1 / 2 (\\\\mathbb{E}|\\\\xi_{0}|^{2})^{1/2} ): on this crossover the fitted step is first-order uniformly accurate against the exact flow’s own pushforward, though the O ( w 2 ) O(w^{2}) prefactor is not uniform in w w .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On a = 0 a=0 the exact one-step factor (Lemma 1 ) is Φ n = r 1 + w ( ( 1 + u n ) / ( 1 + r 2 u n ) ) w / 2 \\\\Phi_{n}=r^{1+w}\\\\bigl((1+u_{n})/(1+r^{2}u_{n})\\\\bigr)^{w/2} , with u n = σ n 2 / b u_{n}=\\\\sigma_{n}^{2}/b , r = e − h r=\\\\mathrm{e}^{-h} ; the fitted factor (Proposition 3 , A = 1 A=1 ) is G n = e − h ( u n + e − w h ) / ( 1 + u n ) G_{n}=\\\\mathrm{e}^{-h}(u_{n}+\\\\mathrm{e}^{-wh})/(1+u_{n}) . Writing y n = log u n y_{n}=\\\\log u_{n} , δ ( h , y ) := log ( G n / Φ n ) = w h + log ( e y + e − w h ) − ( 1 + w 2 ) log ( 1 + e y ) + w 2 log ( 1 + e y − 2 h ) \\\\delta(h,y):=\\\\log(G_{n}/\\\\Phi_{n})=wh+\\\\log(\\\\mathrm{e}^{y}+\\\\mathrm{e}^{-wh})-\\\\bigl(1+\\\\tfrac{w}{2}\\\\bigr)\\\\log(1+\\\\mathrm{e}^{y})+\\\\tfrac{w}{2}\\\\log(1+\\\\mathrm{e}^{y-2h}) . Then δ ( 0 , y ) = ∂ h δ ( 0 , y ) = 0 \\\\delta(0,y)=\\\\partial_{h}\\\\delta(0,y)=0 (first-order consistency, Proposition 2 ), and with F ( y ) = e y / ( 1 + e y ) 2 F(y)=\\\\mathrm{e}^{y}/(1+\\\\mathrm{e}^{y})^{2} one has ∂ h 2 δ = w 2 F ( y + w h ) + 2 w F ( y − 2 h ) ≥ 0 \\\\partial_{h}^{2}\\\\delta=w^{2}F(y+wh)+2wF(y-2h)\\\\geq 0 ; hence δ ≥ 0 \\\\delta\\\\geq 0 , so Θ K = ∏ n e δ ( h , y n ) ≥ 1 \\\\Theta_{K}=\\\\prod_{n}\\\\mathrm{e}^{\\\\delta(h,y_{n})}\\\\geq 1 , and Taylor’s formula gives δ ( h , y n ) = ∫ 0 h ( h − s ) [ w 2 F ( y n + w s ) + 2 w F ( y n − 2 s ) ] 𝑑 s \\\\delta(h,y_{n})=\\\\int_{0}^{h}(h-s)\\\\bigl[w^{2}F(y_{n}+ws)+2wF(y_{n}-2s)\\\\bigr]\\\\,ds . On the uniform mesh y n + 1 = y n − 2 h y_{n+1}=y_{n}-2h ; since ∫ F = 1 \\\\int F=1 and TV ( F ) = 1 2 \\\\mathrm{TV}(F)=\\\\tfrac{1}{2} , the rectangle bound gives 2 h ∑ n F ( y n + c ) ≤ 1 + h 2h\\\\sum_{n}F(y_{n}+c)\\\\leq 1+h for every shift c c , so log Θ K ≤ ∫ 0 h ( h − s ) 1 + h 2 h ( w 2 + 2 w ) 𝑑 s = 1 + h 4 w ( w + 2 ) h \\\\log\\\\Theta_{K}\\\\leq\\\\int_{0}^{h}(h-s)\\\\tfrac{1+h}{2h}(w^{2}+2w)\\\\,ds=\\\\tfrac{1+h}{4}w(w+2)h . Synchronous coupling with | ξ K ex | ≤ | ξ 0 | |\\\\xi_{K}^{\\\\rm ex}|\\\\leq|\\\\xi_{0}| (the exact discriminative factor contracts) gives the W 2 W_{2} bound. ∎\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This appendix expands Remark 4 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"If a reversed singular direction has b = 0 < a b=0<a —the marginal collapsed where the class has not—then the exponent ( 4 ) tends to − w -w , negative for every w > 0 w>0 : the guided flow itself expands the coordinate as σ → 0 \\\\sigma\\\\to 0 , so no discretization can be uniformly stable because the continuous target is not. More generally, reverse ordering with 0 < b < a 0<b<a can create finite- σ \\\\sigma expansion windows, but only the reversed singular case creates this terminal instability. This is a property of the guidance target, not of the solver; the class-subset hypothesis excludes it by assumption, and autoguidance-style constructions aim to enforce the same directionality.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To interpret the FID/KID split at the hard cell, we computed post-hoc feature-manifold and sharpness diagnostics on the same images. Fitted CFG improves Inception precision/recall over CFG in both 5k blocks (e.g. 0.799 / 0.474 0.799/0.474 versus 0.704 / 0.360 0.704/0.360 ); interval has higher recall but lower precision. The pixel-saturation median is 0.16 0.16 for CFG and 0 for fitted CFG and interval, and the median Laplacian energy drops by an order of magnitude—consistent with the fitted step removing oversaturated high-frequency artifacts (we use the Laplacian only as an oversaturation proxy, not as evidence of sharpness). The DINOv2 audit below is a backbone-swapped control on the same split.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To test whether the FID/KID split is Inception-specific, we recompute feature-space distances with a DINOv2 backbone ( dinov2_vits14 , CLS-token features; images bilinearly resized to 224 224 ) against 50 50 k CIFAR-10 training images. We report the Fréchet distance FD-DINOv2 and an unbiased RBF-kernel MMD (median-heuristic bandwidth, 100 100 subsets of size 1000 1000 ). On the two 5k hard-cell blocks, FD-DINOv2 for CFG/fitted/interval is 469.73 / 230.00 / 257.88 469.73/230.00/257.88 and 469.10 / 227.86 / 254.25 469.10/227.86/254.25 , and the DINO-MMD is 0.0258 / 0.0126 / 0.0168 0.0258/0.0126/0.0168 and 0.0257 / 0.0125 / 0.0164 0.0257/0.0125/0.0164 ; both favor fitted CFG in both blocks. FD-DINOv2 is the clean backbone-only control; the MMD changes both backbone and kernel relative to Inception KID, so it is corroborating, not a direct KID analogue.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The Section 4 interval baseline is the parameter-free terminal cutoff, which on the hard mesh reduces to unguided conditional DDIM. To test whether a tuned interval that preserves conditionality can match the fitted step, we selected per cell the conditionality-viable guidance interval [ σ lo , σ hi ] [\\\\sigma_{\\\\rm lo},\\\\sigma_{\\\\rm hi}] from a nine-candidate grid (the one whose target accuracy stays closest to CFG), then scored it on a fresh 5k block. Table 3 reports CFG, the fitted step, the tuned interval, and the terminal interval. In the tested cells, a conditionality-preserving tuned interval does not beat the fitted step on FID; the terminal interval can win FID in a cell ( w = 8 , N = 16 w{=}8,N{=}16 : 12.63 12.63 vs 17.12 17.12 ) but pays a clear target-accuracy cost. The fitted step gives interval-like residual stabilization while keeping the guided update active. (Target accuracy is a classifier proxy; the high fitted value at N = 16 N{=}16 should not be read as fuller conditional fidelity, as it may partly reflect reduced diversity.)\"\n },\n {\n \"type\": \"para\",\n \"text\": \"As a cross-domain check that the coefficient is not specific to pixel-space CIFAR edm , we ran a small Stable Diffusion 1.5 DDIM smoke test at high guidance, swapping only the terminal guidance coefficient. We report pixel saturation (95th percentile of the clipped-pixel fraction) and CLIP image–text alignment; the interval baseline is terminal guidance shutdown. Table 4 shows the fitted step sharply reduces saturation while preserving CLIP alignment far better than shutdown—the same saturation-repair pattern as on CIFAR. This is a smoke/deepen test, not a Stable Diffusion benchmark.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Setup. We use runwayml/stable-diffusion-v1-5 at 512 × 512 512\\\\times 512 with deterministic DDIM ( η = 0 \\\\eta=0 , ϵ \\\\epsilon -prediction) and float16, over a fixed list of everyday-scene captions (repeated to 256 256 and 128 128 prompts) with no negative prompt and seeds matched across schemes. The fitted step swaps only the terminal guidance coefficient; the interval baseline disables the guidance correction on the terminal steps below the same h ♭ h_{\\\\flat} -derived cutoff. Saturation p95 is the 95th percentile over images of the fraction of final RGB pixels at the valid-range extremes; CLIP alignment uses openai/clip-vit-base-patch32 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The real-checkpoint residual diagnostic (Table 1 ) uses class 0 and class 1 blocks. To rule out class cherry-picking, we extended it to all ten CIFAR-10 classes on the high-guidance cells ( w ∈ { 6.5 , 8 } w\\\\in\\\\{6.5,8\\\\} , N ∈ { 8 , 16 , 32 } N\\\\in\\\\{8,16,32\\\\} ; 60 class–cell pairs, all fixed before the run). Clipping/saturation is non-increased in 60 / 60 60/60 pairs, and amp p95 improves in 54 / 60 54/60 ; the other six are modestly worse on the amp tail (ratios 1.02 1.02 – 1.25 1.25 , largest class 4 at w = 6.5 , N = 8 w{=}6.5,N{=}8 ), with clipping still non-increased there. We describe this as global saturation robustness and near-global residual-tail improvement , and leave the raw counts for the reader to weigh.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Theorem 2 is an accuracy statement against the exact guided flow’s own pushforward on the calibration model. On learned checkpoints this does not extend to final-image accuracy: measured against a 512 512 -step vanilla-CFG reference trajectory, the fitted step is not uniformly closer in final denoised-image L 2 L_{2} (Table 5 , ratios > 1 >1 ), even though its residual and amp tails are far smaller (ratios ≪ 1 \\\\ll 1 ). So the learned-checkpoint evidence supports terminal residual/saturation repair, not a claim that the fitted step is a uniformly better coarse integrator of the vanilla CFG field.\"\n }\n]","isConversational":false}}},{"rowIdx":99,"cells":{"source":{"kind":"string","value":"arxiv"},"paper_id":{"kind":"string","value":"2607.07557"},"doi":{"kind":"null"},"version":{"kind":"number","value":1,"string":"1"},"title":{"kind":"string","value":"PALS: Percentile-Aware Layerwise Sparsity for LLM Pruning"},"authors":{"kind":"list like","value":["Yazdan Jamshidi","Alexey Shvets"],"string":"[\n \"Yazdan Jamshidi\",\n \"Alexey Shvets\"\n]","isConversational":false},"license":{"kind":"string","value":"CC-BY-4.0"},"source_url":{"kind":"string","value":"https://arxiv.org/html/2607.07557v1"},"published":{"kind":"timestamp","value":"2026-07-08T00:00:00","string":"2026-07-08T00:00:00"},"blocks":{"kind":"list like","value":[{"type":"heading","level":1,"text":"1 Introduction"},{"type":"para","text":"Large language models routinely reach billions of parameters. A 7B-parameter model takes roughly 13 GB in half precision and processes billions of multiply-accumulate operations per token, which puts real-time inference out of reach on modest hardware. Post-training pruning offers a direct way to cut both memory and compute: remove a fraction of the weights and run inference on the sparser model, without retraining from scratch."},{"type":"para","text":"Recent one-shot pruning methods have made this practical for LLMs. Wanda Sun et al. ( 2024 ) scores each weight by the product of its magnitude and the mean absolute activation flowing through it ( | w i j | ⋅ mean ( | a i | ) |w_{ij}|\\cdot\\text{mean}(|a_{i}|) ), then drops the lowest-scoring fraction. SparseGPT Frantar and Alistarh ( 2023 ) solves a heavier problem—layer-wise sparse regression with approximate Hessian information—but in practice lands at similar perplexity on most benchmarks. Both methods have become standard baselines."},{"type":"para","text":"One thing both share is that they enforce uniform sparsity: if the target is 50%, every one of the 32 layers in LLaMA-2 is pruned to exactly 50%. This sits poorly with what we know about transformer layers. Attention heads vary widely in importance Voita et al. ( 2019 ); Michel et al. ( 2019 ) . Early layers tend to build generic representations; middle layers are often more redundant; late layers specialize for the output vocabulary. Pruning all of them equally seems like it should leave performance on the table—and it does."},{"type":"para","text":"We propose PALS (Percentile-Aware Layerwise Sparsity), which adjusts the sparsity ratio for each layer based on activation statistics gathered from a small calibration set. The idea is direct: compute the 99th percentile of activation magnitudes for each layer, normalize these scores, and use them to shift each layer’s sparsity target up or down. Layers with large activation outliers—a sign of important information flow—get pruned less; layers with uniformly small activations get pruned more. A ± 5 % \\pm 5\\% band around the target prevents any single layer from being starved or overstuffed. The procedure slots into Wanda’s pipeline with almost no extra cost, since Wanda already collects the activations."},{"type":"para","text":"Before settling on activations, we tried the seemingly more natural approach of using gradient magnitudes to estimate layer importance. This failed decisively. Gradient-based allocation produced 47 perplexity on LLaMA-2-7B, compared to 24 for random allocation and 11 for our activation-based scheme. We believe gradients reflect optimization dynamics—how much the loss would change under an infinitesimal perturbation—rather than the actual impact of removing a large fraction of weights. But the scale of the failure was not something we anticipated."},{"type":"para","text":"On LLaMA-2-7B at 50% sparsity, PALS brings WikiText-2 perplexity from 12.92 down to 10.96, averaged over nine independent runs with different random seeds (Welch’s t ( 14 ) = 8.1 t(14)=8.1 , p < 0.001 p<0.001 ). The picture is less clear on other architectures. LLaMA-3-8B benefits marginally (10.45 vs. 10.48), and Mistral-7B does not benefit at all (6.31 either way). We think this reflects how uniformly these newer models distribute computation across layers—when all layers are similarly important, non-uniform allocation has nothing to exploit. We present ablation studies covering the percentile choice, adaptation strength, and bound width, along with downstream evaluation on four benchmarks and practical speedup measurements."},{"type":"heading","level":1,"text":"2 Related Work"},{"type":"heading","level":2,"text":"2.1 One-Shot Pruning for LLMs"},{"type":"para","text":"Magnitude pruning Han et al. ( 2015 ) —removing weights with the smallest absolute values—is the oldest and simplest approach but ignores activation patterns, which limits it on modern LLMs. SparseGPT Frantar and Alistarh ( 2023 ) improved on this substantially by formulating each layer’s pruning as a sparse reconstruction problem, using an approximate inverse Hessian to choose which weights to remove and how to adjust the survivors. Wanda Sun et al. ( 2024 ) showed that a much simpler scoring rule— | w | ⋅ mean ( | a | ) |w|\\cdot\\text{mean}(|a|) , with no weight update—matches SparseGPT in most settings. Both methods use the same sparsity ratio for every layer. PALS keeps Wanda’s scoring but varies the fraction of weights removed per layer."},{"type":"heading","level":2,"text":"2.2 Structured Pruning"},{"type":"para","text":"An orthogonal line of work removes entire structural units for hardware-friendly sparsity. LLM-Pruner Ma et al. ( 2023 ) uses first-order Taylor expansion to score and remove attention heads and MLP neurons. SliceGPT Ashkboos et al. ( 2024 ) projects weight matrices into a lower-rank space and deletes entire rows and columns. Sheared LLaMA Zhang et al. ( 2024 ) combines structured pruning with continued pre-training. These approaches deliver better hardware utilization than unstructured pruning but typically require some form of fine-tuning. PALS stays within the unstructured, one-shot setting."},{"type":"heading","level":2,"text":"2.3 Non-Uniform Layer Treatment"},{"type":"para","text":"The idea that different layers should be treated differently is not new. NISP Yu et al. ( 2018 ) propagates importance scores between layers for vision models, but was designed for networks orders of magnitude smaller than current LLMs. For large language models, most work on non-uniform treatment has focused on depth pruning—removing entire transformer blocks Kurtic et al. ( 2022 ) —rather than adjusting within-layer sparsity ratios. To our knowledge, PALS is the first to report clear gains from varying per-layer sparsity in one-shot LLM pruning, and the first to document the failure of gradient-based allocation in this setting."},{"type":"heading","level":2,"text":"2.4 Activation Outliers in Transformers"},{"type":"para","text":"Dettmers et al. Dettmers et al. ( 2022 ) showed that large language models develop activation outliers—individual features with magnitudes 10–100 × \\times larger than the rest—concentrated in specific dimensions and layers. These outliers are essential for model quality: zeroing them out causes sharp accuracy drops. Our use of the 99th percentile as a layer importance signal connects directly to this observation. Layers with more extreme outliers are presumably processing more critical information, and PALS assigns them lower sparsity. This also explains why the 99th percentile outperforms the mean as an importance metric: the mean averages out exactly the outliers that matter."},{"type":"heading","level":1,"text":"3 Method"},{"type":"para","text":"Given a pretrained model with L L transformer layers and a target sparsity s target s_{\\text{target}} (e.g., 0.50 for 50% weight removal), PALS adjusts the per-layer sparsity based on activation statistics, then prunes each layer independently using Wanda’s scoring."},{"type":"heading","level":2,"text":"3.1 Layer Importance from Activation Percentiles"},{"type":"para","text":"For each layer l l , we collect activations A ( l ) ∈ ℝ n × d A^{(l)}\\in\\mathbb{R}^{n\\times d} on n n calibration tokens across d d hidden dimensions. The layer importance score is the 99th percentile of the absolute activations:"},{"type":"para","text":"The 99th percentile targets the outlier activations that Dettmers et al. Dettmers et al. ( 2022 ) identified as critical, while being less noisy than the maximum. To make scores comparable across layers, we standardize:"},{"type":"para","text":"where I ¯ = 1 L ∑ l = 1 L I ( l ) \\bar{I}=\\frac{1}{L}\\sum_{l=1}^{L}I^{(l)} and σ I \\sigma_{I} is the standard deviation of the raw scores."},{"type":"heading","level":2,"text":"3.2 Sparsity Allocation with Conservative Bounds"},{"type":"para","text":"The normalized importance I ^ ( l ) \\hat{I}^{(l)} determines how much each layer deviates from the global target:"},{"type":"para","text":"where α \\alpha controls adaptation strength. The clip enforces:"},{"type":"para","text":"The ± 5 % \\pm 5\\% bounds are important. In preliminary experiments, wider bounds ( ± 20 % \\pm 20\\% ) caused sharp degradation because heavily pruned layers created information bottlenecks—the rest of the network could not compensate for the lost capacity. Conservative bounds trade off some potential gain for robustness."},{"type":"heading","level":2,"text":"3.3 Weight Scoring and Pruning"},{"type":"para","text":"After determining s ( l ) s^{(l)} , we score and prune weights using Wanda’s criterion:"},{"type":"para","text":"For each layer l l , the bottom s ( l ) × 100 % s^{(l)}\\times 100\\% of weights by score are removed. No weight updates, no fine-tuning."},{"type":"heading","level":2,"text":"3.4 Complete Procedure"},{"type":"para","text":"Algorithm 1 summarizes the pipeline. Steps 1–5 take negligible time compared to Wanda’s scoring in steps 6–9, and the activation collection is shared with Wanda—so the only overhead is the percentile computation and sparsity reallocation."},{"type":"heading","level":1,"text":"4 Experimental Setup"},{"type":"heading","level":4,"text":"Models."},{"type":"para","text":"We evaluate on three architectures: LLaMA-2-7B (6.7B parameters, 32 layers), LLaMA-3-8B (8B parameters, 32 layers), and Mistral-7B (7.3B parameters, 32 layers). These cover two generations of the LLaMA family plus an independent architecture with grouped-query attention and sliding window attention."},{"type":"heading","level":4,"text":"Evaluation."},{"type":"para","text":"Perplexity is measured on the WikiText-2 Merity et al. ( 2016 ) test set using 2048-token sequences. For downstream evaluation, we report zero-shot accuracy on BoolQ Clark et al. ( 2019 ) , PIQA Bisk et al. ( 2020 ) , HellaSwag Zellers et al. ( 2019 ) , and MMLU, using the lm-evaluation-harness framework Gao et al. ( 2021 ) ."},{"type":"heading","level":4,"text":"Calibration."},{"type":"para","text":"All methods that require calibration data use 128 samples of 2048 tokens drawn from C4. Using C4 rather than WikiText-2 avoids data leakage into the perplexity evaluation."},{"type":"heading","level":4,"text":"Baselines."},{"type":"para","text":"We compare against: (1) Magnitude —remove weights with smallest | w | |w| , no calibration needed; (2) Wanda Sun et al. ( 2024 ) —activation-aware scoring with uniform per-layer sparsity; (3) SparseGPT Frantar and Alistarh ( 2023 ) —Hessian-based layer-wise optimization; and (4) PALS-Gradient —our allocation formula but substituting gradient norms ‖ ∇ W ( l ) ℒ ‖ F \\|\\nabla_{W^{(l)}}\\mathcal{L}\\|_{F} for activation percentiles."},{"type":"heading","level":4,"text":"Runs and seeds."},{"type":"para","text":"For the primary LLaMA-2-7B evaluation, we run PALS nine times and Wanda eight times with different random seeds controlling weight tie-breaking. This provides variance estimates and allows a two-sample test. Other configurations use single runs."},{"type":"heading","level":4,"text":"Implementation."},{"type":"para","text":"All experiments use PyTorch with HuggingFace Transformers on NVIDIA A100 80GB GPUs. PALS uses α = 0.05 \\alpha=0.05 and the 99th percentile, with ± 5 % \\pm 5\\% bounds. All pruning is one-shot with no fine-tuning or weight updates."},{"type":"heading","level":1,"text":"5 Results"},{"type":"heading","level":2,"text":"5.1 Main Comparison on LLaMA-2-7B"},{"type":"para","text":"Table 1 presents the primary result on LLaMA-2-7B at 50% unstructured sparsity."},{"type":"para","text":"PALS achieves 10.96 perplexity, down from Wanda’s 12.92—a reduction of 1.96 points. The standard deviation across nine runs is 0.59 and Wanda’s across eight runs is 0.40, so the distributions barely overlap. A Welch’s t-test gives t ( 14 ) = 8.1 t(14)=8.1 , p < 0.001 p<0.001 . 1 1 1 Degrees of freedom computed via the Welch-Satterthwaite approximation: df = ( 0.59 2 / 9 + 0.40 2 / 8 ) 2 / ( 0.59 4 / 9 2 ⋅ 8 − 1 + 0.40 4 / 8 2 ⋅ 7 − 1 ) ≈ 14 \\text{df}=(0.59^{2}/9+0.40^{2}/8)^{2}/(0.59^{4}/9^{2}\\cdot 8^{-1}+0.40^{4}/8^{2}\\cdot 7^{-1})\\approx 14 ."},{"type":"para","text":"SparseGPT lands at 13.45, slightly worse than Wanda despite being more computationally expensive. Both gradient-based methods—magnitude pruning (43.05) and PALS-Gradient (45.23)—fail badly, which we return to in Section 6.5 ."},{"type":"para","text":"Figure 1 shows the comparison visually."},{"type":"heading","level":2,"text":"5.2 Cross-Model Results"},{"type":"para","text":"Table 2 extends the evaluation to LLaMA-3-8B and Mistral-7B."},{"type":"para","text":"The pattern is clear: PALS helps LLaMA-2 substantially but does little for the other two. Mistral-7B is the most pruning-friendly model in absolute terms (6.31 at 50% sparsity, only 33% above its dense baseline of 4.74), but PALS offers no improvement over uniform allocation. LLaMA-3-8B falls between the two, with a negligible 0.03 perplexity reduction."},{"type":"para","text":"We think this reflects architectural differences. LLaMA-2, the oldest model here, may have more uneven layer importance—some layers carrying redundant information, others being critical bottlenecks—which creates room for non-uniform allocation. Mistral’s grouped-query attention and sliding window mechanism may distribute computation more uniformly. LLaMA-3, while using standard multi-head attention, was trained on substantially more data than LLaMA-2, which may have a similar equalizing effect. We discuss this further in Section 7.2 ."},{"type":"heading","level":2,"text":"5.3 Downstream Task Performance"},{"type":"para","text":"Perplexity measures language modeling quality, but deployment applications care about task accuracy. Table 3 reports zero-shot accuracy on four benchmarks."},{"type":"para","text":"PALS retains 93.3% of the dense model’s average accuracy, compared to 91.7% for Wanda. The 1.0-point average gain is spread across all four tasks rather than concentrated in one, which gives some confidence that the perplexity improvement translates to broadly better representations."},{"type":"para","text":"That said, the downstream gaps are modest. On MMLU, PALS gets 42.2 versus Wanda’s 41.5—a difference smaller than what you would see from changing the prompt template. Downstream benchmarks at this scale may lack the sensitivity to resolve the kind of gains PALS provides."},{"type":"heading","level":2,"text":"5.4 Efficiency"},{"type":"para","text":"Table 4 reports deployment metrics on an A100 GPU with batch size 1 and sequence length 512."},{"type":"para","text":"The 50% sparsity halves the stored model size from 13.2 GB to 6.6 GB. Throughput improves by 1.41 × \\times using sparse matrix kernels, while peak memory drops modestly (1.10 × \\times ) because activation memory is unchanged. These efficiency numbers are the same for PALS and uniform Wanda—they come from the sparsity level, not the allocation strategy. PALS gets better perplexity at the same sparsity budget."},{"type":"heading","level":2,"text":"5.5 Performance Across Sparsity Levels"},{"type":"para","text":"Figure 2 shows WikiText-2 perplexity across sparsity levels from 30% to 70%."},{"type":"para","text":"PALS outperforms uniform Wanda at every sparsity level tested, with the gap widening at higher sparsity. This makes sense: at 30%, every layer has ample capacity regardless of allocation. At 70%, capacity is scarce, and putting it where it counts matters more."},{"type":"heading","level":1,"text":"6 Analysis"},{"type":"heading","level":2,"text":"6.1 Learned Sparsity Patterns"},{"type":"para","text":"Figure 3 shows the per-layer sparsity budgets PALS assigns across our three models."},{"type":"para","text":"All three models show a roughly U-shaped pattern: early layers (0–8) and late layers (21–31) get sparsity below 50%, while middle layers (9–20) get sparsity above 50%. The range stays within the ± 5 % \\pm 5\\% bounds by construction, so every layer falls between 45% and 55%."},{"type":"para","text":"This aligns with what is known about transformer layer function. Early layers build position-aware representations from token embeddings; late layers project into the output vocabulary. Both are hard to replace with fewer weights. Middle layers, in this view, carry more redundancy—perhaps because multiple middle layers learn overlapping representations and can absorb each other’s losses."},{"type":"para","text":"One caveat: the pattern similarity across models is partly mechanical. Because PALS normalizes importance scores and clips to ± 5 % \\pm 5\\% , the output will look roughly U-shaped for any model whose early and late layers have larger activation outliers than the middle. Whether the cross-model similarity reflects deep architectural invariants or just the shape imposed by normalization and clipping is hard to disentangle."},{"type":"heading","level":2,"text":"6.2 Effect of Adaptation Strength α \\alpha"},{"type":"para","text":"Table 5 shows how α \\alpha affects perplexity on LLaMA-2-7B. These are single runs (except α = 0.05 \\alpha=0.05 , which is the 9-run mean), so differences within the run-to-run noise band ( ∼ \\sim 0.6 PPL) should not be over-interpreted."},{"type":"para","text":"There is an interesting detail here. For α ≥ 0.07 \\alpha\\geq 0.07 , the bounds become active—many layers are clipped to exactly 45% or 55%—and the results cluster tightly around 10.1–10.2. These single-run numbers are slightly better than our chosen α = 0.05 \\alpha=0.05 (10.96), but the difference of ∼ \\sim 0.8 PPL is within the run-to-run variance (std = 0.59 over the 9-run evaluation at α = 0.05 \\alpha=0.05 ). A single lucky seed at α = 0.07 \\alpha=0.07 could easily account for the gap."},{"type":"para","text":"We use α = 0.05 \\alpha=0.05 because it represents the transition point: below it, the allocation is too timid; above it, the bounds do most of the work and the specific value of α \\alpha stops mattering. The data is consistent with the interpretation that a coarser binary allocation (every layer at either 45% or 55%) works roughly as well as the smooth allocation PALS provides, though confirming this would require multi-run evaluation at each α \\alpha value."},{"type":"para","text":"The α = 0.01 \\alpha=0.01 entry (14.23 PPL) is worse than uniform Wanda (12.92)—a very small amount of adaptation apparently hurting. We suspect this is seed noise in a single run, not a real effect. Without multi-run data at each α \\alpha , we cannot say whether this non-monotonicity is genuine."},{"type":"heading","level":2,"text":"6.3 Effect of Bound Width"},{"type":"para","text":"Table 6 is the cleanest result in our ablations. Performance degrades sharply once the bounds exceed ± 10 % \\pm 10\\% , and unbounded allocation (12–88% range) is catastrophic. At ± 20 % \\pm 20\\% , the most aggressively pruned layers hit 70% sparsity, and the information loss there apparently cannot be compensated by gently pruned layers elsewhere. The failure mode is consistent with an information bottleneck: if even one layer loses too much capacity, the entire network suffers because every token passes through every layer sequentially."},{"type":"para","text":"The ± 5 % \\pm 5\\% setting sits in a good spot. Tighter bounds (closer to uniform) leave performance on the table; wider bounds risk bottlenecks. Whether ± 5 % \\pm 5\\% is universally best or happens to suit these three models is an open question."},{"type":"heading","level":2,"text":"6.4 Choice of Activation Percentile"},{"type":"para","text":"The mean is a poor importance signal (13.78 PPL, worse than uniform Wanda), confirming that outlier activations, not average activation level, drive layer importance for pruning. Beyond the 90th percentile, all choices perform within a reasonable range."},{"type":"para","text":"A careful reader will notice that the 99th percentile (10.96) is not the best in this table: both the 95th (10.67) and 99.5th (10.11) give lower perplexity. Since these are single runs and the run-to-run variance at α = 0.05 \\alpha=0.05 is 0.59, the differences are not statistically distinguishable. We chose the 99th percentile before running this ablation, based on its connection to the outlier phenomenon Dettmers et al. ( 2022 ) , and kept it rather than cherry-picking after the fact. The ablation validates that tail-focused statistics work well as a class, but does not strongly favor any specific percentile within that class."},{"type":"heading","level":2,"text":"6.5 Gradient-Based Allocation Fails"},{"type":"para","text":"Table 8 is, in our view, the most striking result in this paper. Gradient-based importance metrics—which have a long history of success in pruning smaller networks LeCun et al. ( 1989 ) —produce allocations that are worse than random . Gradient norm gives 47.32 PPL; first-order Taylor expansion ( ∑ | g i j w i j | \\sum|g_{ij}w_{ij}| ) does better at 38.61 but is still far worse than random at 24.37."},{"type":"para","text":"To be clear about what “random allocation” means here: each layer gets a sparsity ratio drawn uniformly from [ s target − 0.05 , s target + 0.05 ] [s_{\\text{target}}-0.05,\\;s_{\\text{target}}+0.05] , with no information about layer importance. It beats both gradient methods by a wide margin."},{"type":"para","text":"Why do gradients fail? Two possibilities, both speculative:"},{"type":"para","text":"Convergence confound. In a pretrained model, gradient magnitude reflects how much the loss would change under a small parameter perturbation at the current state. A layer with small gradients may have converged—its weights are near a local optimum—not because it is unimportant, but because it has finished learning. Gradient-based allocation would over-prune such layers."},{"type":"para","text":"Discrete–continuous mismatch. Pruning removes a large, discrete fraction of weights. Gradients measure infinitesimal, continuous sensitivity. At 50% sparsity, the “perturbation” is far from infinitesimal. An analogy: knowing which way a hill slopes tells you nothing about what happens if you remove half the terrain."},{"type":"para","text":"Neither explanation is fully satisfying, and we consider this an open problem. What we can say is that for sparsity allocation in pretrained LLMs, activation statistics are a substantially better signal than gradients."},{"type":"heading","level":2,"text":"6.6 Calibration Stability"},{"type":"para","text":"PALS relies on activation statistics from 128 calibration samples. A natural concern is whether the importance scores are stable across different calibration draws."},{"type":"para","text":"We did not run a formal calibration sensitivity experiment, but can offer indirect evidence. The 99th percentile is computed over roughly 128 × 2048 ≈ 262 128\\times 2048\\approx 262 K tokens per layer. Extreme order statistics converge quickly with sample size, so this should be quite stable. Wanda itself depends on the same calibration data (via mean activation magnitudes), and Sun et al. Sun et al. ( 2024 ) reported that Wanda’s results are stable across calibration subsets. Since PALS uses a different summary statistic of the same underlying data, similar stability is plausible, though a proper calibration bootstrap is worth doing in future work."},{"type":"para","text":"Our nine-run evaluation uses the same calibration set but different random seeds for tie-breaking. The resulting std of 0.59 bounds the contribution of tie-breaking noise. Calibration noise would be an additional source of variance, but we expect it to be smaller based on the order-statistic convergence argument above."},{"type":"heading","level":1,"text":"7 Discussion"},{"type":"heading","level":2,"text":"7.1 Why Activations Work"},{"type":"para","text":"The success of activation-based importance connects to the growing literature on activation outliers Dettmers et al. ( 2022 ) . Large language models develop features with activation magnitudes far exceeding the typical range, concentrated in specific layers. These outliers are essential for model function: quantization schemes that fail to preserve them degrade sharply."},{"type":"para","text":"PALS identifies layers with large outliers (via the 99th percentile) and assigns them lower sparsity. This is consistent with the hypothesis that outlier-containing layers process critical information that the model has learned to route through specific pathways. The connection also predicts that any importance metric sensitive to the tail of the activation distribution should work—and indeed, the 90th through max percentile all outperform the mean (Table 7 )."},{"type":"heading","level":2,"text":"7.2 Architecture Dependence"},{"type":"para","text":"The large improvement on LLaMA-2 (15%) versus zero on Mistral is worth understanding. Three hypotheses:"},{"type":"para","text":"Training data volume. LLaMA-2-7B was trained on 2T tokens; LLaMA-3-8B on roughly 15T tokens. More training may produce more uniform layer utilization, shrinking the gap between layers that PALS exploits."},{"type":"para","text":"Architecture. Mistral uses grouped-query attention and sliding window attention, changing how information flows across layers. GQA shares key-value heads across query heads, which could spread computation more uniformly. Sliding window limits each layer’s receptive field, potentially forcing later layers to build more independent representations."},{"type":"para","text":"Model maturity. Newer models may simply be better trained, with fewer “wasted” layers. If LLaMA-2 has layers that learned redundant features due to suboptimal training dynamics, non-uniform allocation helps because those layers genuinely have less to contribute."},{"type":"para","text":"We cannot distinguish between these with our current experiments. The practical upshot: if you run PALS and the sparsity pattern looks nearly flat (all layers close to 50%), the model probably will not benefit, and uniform allocation is fine."},{"type":"heading","level":2,"text":"7.3 Limitations"},{"type":"para","text":"Unstructured sparsity. The throughput gains in Table 4 require sparse matrix kernels or hardware support (e.g., NVIDIA’s 2:4 structured sparsity). Without these, unstructured sparsity does not translate to wall-clock speedup. Extending PALS to N:M or block sparsity patterns is a natural next step."},{"type":"para","text":"Single perplexity dataset. Our primary metric is WikiText-2 perplexity. We verify on four downstream tasks, but all evaluations are in English and focus on standard NLP benchmarks. Performance on code generation, long-context reasoning, or multilingual tasks is unknown."},{"type":"para","text":"No fine-tuning. We evaluate one-shot pruning only. Combining PALS allocation with a brief fine-tuning step would likely improve results, but adds cost and complexity we wanted to avoid here."},{"type":"para","text":"Model scale. All experiments are on 7–8B parameter models. Whether the gains hold at 13B, 70B, or larger is untested. Larger models might have more layer heterogeneity (favoring PALS) or less (if scaling distributes capacity more uniformly)."},{"type":"heading","level":1,"text":"8 Conclusion"},{"type":"para","text":"PALS adjusts per-layer sparsity in one-shot LLM pruning using activation percentiles, reducing LLaMA-2-7B perplexity from 12.92 to 10.96 at 50% sparsity with negligible extra cost. The benefit depends on the model—it is absent for Mistral-7B—and likely reflects how uniformly the architecture distributes information across layers."},{"type":"para","text":"The gradient failure finding is, to us, the more interesting result. Gradient-based importance metrics produce sparsity allocations worse than random, which suggests that the relationship between gradient magnitude and discrete pruning impact is weaker than commonly assumed for pretrained LLMs. Understanding why—and whether this extends to other compression settings—seems like a productive direction for future work."}],"string":"[\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"1 Introduction\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Large language models routinely reach billions of parameters. A 7B-parameter model takes roughly 13 GB in half precision and processes billions of multiply-accumulate operations per token, which puts real-time inference out of reach on modest hardware. Post-training pruning offers a direct way to cut both memory and compute: remove a fraction of the weights and run inference on the sparser model, without retraining from scratch.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Recent one-shot pruning methods have made this practical for LLMs. Wanda Sun et al. ( 2024 ) scores each weight by the product of its magnitude and the mean absolute activation flowing through it ( | w i j | ⋅ mean ( | a i | ) |w_{ij}|\\\\cdot\\\\text{mean}(|a_{i}|) ), then drops the lowest-scoring fraction. SparseGPT Frantar and Alistarh ( 2023 ) solves a heavier problem—layer-wise sparse regression with approximate Hessian information—but in practice lands at similar perplexity on most benchmarks. Both methods have become standard baselines.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"One thing both share is that they enforce uniform sparsity: if the target is 50%, every one of the 32 layers in LLaMA-2 is pruned to exactly 50%. This sits poorly with what we know about transformer layers. Attention heads vary widely in importance Voita et al. ( 2019 ); Michel et al. ( 2019 ) . Early layers tend to build generic representations; middle layers are often more redundant; late layers specialize for the output vocabulary. Pruning all of them equally seems like it should leave performance on the table—and it does.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We propose PALS (Percentile-Aware Layerwise Sparsity), which adjusts the sparsity ratio for each layer based on activation statistics gathered from a small calibration set. The idea is direct: compute the 99th percentile of activation magnitudes for each layer, normalize these scores, and use them to shift each layer’s sparsity target up or down. Layers with large activation outliers—a sign of important information flow—get pruned less; layers with uniformly small activations get pruned more. A ± 5 % \\\\pm 5\\\\% band around the target prevents any single layer from being starved or overstuffed. The procedure slots into Wanda’s pipeline with almost no extra cost, since Wanda already collects the activations.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Before settling on activations, we tried the seemingly more natural approach of using gradient magnitudes to estimate layer importance. This failed decisively. Gradient-based allocation produced 47 perplexity on LLaMA-2-7B, compared to 24 for random allocation and 11 for our activation-based scheme. We believe gradients reflect optimization dynamics—how much the loss would change under an infinitesimal perturbation—rather than the actual impact of removing a large fraction of weights. But the scale of the failure was not something we anticipated.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"On LLaMA-2-7B at 50% sparsity, PALS brings WikiText-2 perplexity from 12.92 down to 10.96, averaged over nine independent runs with different random seeds (Welch’s t ( 14 ) = 8.1 t(14)=8.1 , p < 0.001 p<0.001 ). The picture is less clear on other architectures. LLaMA-3-8B benefits marginally (10.45 vs. 10.48), and Mistral-7B does not benefit at all (6.31 either way). We think this reflects how uniformly these newer models distribute computation across layers—when all layers are similarly important, non-uniform allocation has nothing to exploit. We present ablation studies covering the percentile choice, adaptation strength, and bound width, along with downstream evaluation on four benchmarks and practical speedup measurements.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"2 Related Work\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.1 One-Shot Pruning for LLMs\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Magnitude pruning Han et al. ( 2015 ) —removing weights with the smallest absolute values—is the oldest and simplest approach but ignores activation patterns, which limits it on modern LLMs. SparseGPT Frantar and Alistarh ( 2023 ) improved on this substantially by formulating each layer’s pruning as a sparse reconstruction problem, using an approximate inverse Hessian to choose which weights to remove and how to adjust the survivors. Wanda Sun et al. ( 2024 ) showed that a much simpler scoring rule— | w | ⋅ mean ( | a | ) |w|\\\\cdot\\\\text{mean}(|a|) , with no weight update—matches SparseGPT in most settings. Both methods use the same sparsity ratio for every layer. PALS keeps Wanda’s scoring but varies the fraction of weights removed per layer.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.2 Structured Pruning\"\n },\n {\n \"type\": \"para\",\n \"text\": \"An orthogonal line of work removes entire structural units for hardware-friendly sparsity. LLM-Pruner Ma et al. ( 2023 ) uses first-order Taylor expansion to score and remove attention heads and MLP neurons. SliceGPT Ashkboos et al. ( 2024 ) projects weight matrices into a lower-rank space and deletes entire rows and columns. Sheared LLaMA Zhang et al. ( 2024 ) combines structured pruning with continued pre-training. These approaches deliver better hardware utilization than unstructured pruning but typically require some form of fine-tuning. PALS stays within the unstructured, one-shot setting.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.3 Non-Uniform Layer Treatment\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The idea that different layers should be treated differently is not new. NISP Yu et al. ( 2018 ) propagates importance scores between layers for vision models, but was designed for networks orders of magnitude smaller than current LLMs. For large language models, most work on non-uniform treatment has focused on depth pruning—removing entire transformer blocks Kurtic et al. ( 2022 ) —rather than adjusting within-layer sparsity ratios. To our knowledge, PALS is the first to report clear gains from varying per-layer sparsity in one-shot LLM pruning, and the first to document the failure of gradient-based allocation in this setting.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"2.4 Activation Outliers in Transformers\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Dettmers et al. Dettmers et al. ( 2022 ) showed that large language models develop activation outliers—individual features with magnitudes 10–100 × \\\\times larger than the rest—concentrated in specific dimensions and layers. These outliers are essential for model quality: zeroing them out causes sharp accuracy drops. Our use of the 99th percentile as a layer importance signal connects directly to this observation. Layers with more extreme outliers are presumably processing more critical information, and PALS assigns them lower sparsity. This also explains why the 99th percentile outperforms the mean as an importance metric: the mean averages out exactly the outliers that matter.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"3 Method\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Given a pretrained model with L L transformer layers and a target sparsity s target s_{\\\\text{target}} (e.g., 0.50 for 50% weight removal), PALS adjusts the per-layer sparsity based on activation statistics, then prunes each layer independently using Wanda’s scoring.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.1 Layer Importance from Activation Percentiles\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For each layer l l , we collect activations A ( l ) ∈ ℝ n × d A^{(l)}\\\\in\\\\mathbb{R}^{n\\\\times d} on n n calibration tokens across d d hidden dimensions. The layer importance score is the 99th percentile of the absolute activations:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The 99th percentile targets the outlier activations that Dettmers et al. Dettmers et al. ( 2022 ) identified as critical, while being less noisy than the maximum. To make scores comparable across layers, we standardize:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where I ¯ = 1 L ∑ l = 1 L I ( l ) \\\\bar{I}=\\\\frac{1}{L}\\\\sum_{l=1}^{L}I^{(l)} and σ I \\\\sigma_{I} is the standard deviation of the raw scores.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.2 Sparsity Allocation with Conservative Bounds\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The normalized importance I ^ ( l ) \\\\hat{I}^{(l)} determines how much each layer deviates from the global target:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"where α \\\\alpha controls adaptation strength. The clip enforces:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The ± 5 % \\\\pm 5\\\\% bounds are important. In preliminary experiments, wider bounds ( ± 20 % \\\\pm 20\\\\% ) caused sharp degradation because heavily pruned layers created information bottlenecks—the rest of the network could not compensate for the lost capacity. Conservative bounds trade off some potential gain for robustness.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.3 Weight Scoring and Pruning\"\n },\n {\n \"type\": \"para\",\n \"text\": \"After determining s ( l ) s^{(l)} , we score and prune weights using Wanda’s criterion:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For each layer l l , the bottom s ( l ) × 100 % s^{(l)}\\\\times 100\\\\% of weights by score are removed. No weight updates, no fine-tuning.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"3.4 Complete Procedure\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Algorithm 1 summarizes the pipeline. Steps 1–5 take negligible time compared to Wanda’s scoring in steps 6–9, and the activation collection is shared with Wanda—so the only overhead is the percentile computation and sparsity reallocation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"4 Experimental Setup\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Models.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We evaluate on three architectures: LLaMA-2-7B (6.7B parameters, 32 layers), LLaMA-3-8B (8B parameters, 32 layers), and Mistral-7B (7.3B parameters, 32 layers). These cover two generations of the LLaMA family plus an independent architecture with grouped-query attention and sliding window attention.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Evaluation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Perplexity is measured on the WikiText-2 Merity et al. ( 2016 ) test set using 2048-token sequences. For downstream evaluation, we report zero-shot accuracy on BoolQ Clark et al. ( 2019 ) , PIQA Bisk et al. ( 2020 ) , HellaSwag Zellers et al. ( 2019 ) , and MMLU, using the lm-evaluation-harness framework Gao et al. ( 2021 ) .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Calibration.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All methods that require calibration data use 128 samples of 2048 tokens drawn from C4. Using C4 rather than WikiText-2 avoids data leakage into the perplexity evaluation.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Baselines.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We compare against: (1) Magnitude —remove weights with smallest | w | |w| , no calibration needed; (2) Wanda Sun et al. ( 2024 ) —activation-aware scoring with uniform per-layer sparsity; (3) SparseGPT Frantar and Alistarh ( 2023 ) —Hessian-based layer-wise optimization; and (4) PALS-Gradient —our allocation formula but substituting gradient norms ‖ ∇ W ( l ) ℒ ‖ F \\\\|\\\\nabla_{W^{(l)}}\\\\mathcal{L}\\\\|_{F} for activation percentiles.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Runs and seeds.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"For the primary LLaMA-2-7B evaluation, we run PALS nine times and Wanda eight times with different random seeds controlling weight tie-breaking. This provides variance estimates and allows a two-sample test. Other configurations use single runs.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 4,\n \"text\": \"Implementation.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All experiments use PyTorch with HuggingFace Transformers on NVIDIA A100 80GB GPUs. PALS uses α = 0.05 \\\\alpha=0.05 and the 99th percentile, with ± 5 % \\\\pm 5\\\\% bounds. All pruning is one-shot with no fine-tuning or weight updates.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"5 Results\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.1 Main Comparison on LLaMA-2-7B\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 1 presents the primary result on LLaMA-2-7B at 50% unstructured sparsity.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"PALS achieves 10.96 perplexity, down from Wanda’s 12.92—a reduction of 1.96 points. The standard deviation across nine runs is 0.59 and Wanda’s across eight runs is 0.40, so the distributions barely overlap. A Welch’s t-test gives t ( 14 ) = 8.1 t(14)=8.1 , p < 0.001 p<0.001 . 1 1 1 Degrees of freedom computed via the Welch-Satterthwaite approximation: df = ( 0.59 2 / 9 + 0.40 2 / 8 ) 2 / ( 0.59 4 / 9 2 ⋅ 8 − 1 + 0.40 4 / 8 2 ⋅ 7 − 1 ) ≈ 14 \\\\text{df}=(0.59^{2}/9+0.40^{2}/8)^{2}/(0.59^{4}/9^{2}\\\\cdot 8^{-1}+0.40^{4}/8^{2}\\\\cdot 7^{-1})\\\\approx 14 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"SparseGPT lands at 13.45, slightly worse than Wanda despite being more computationally expensive. Both gradient-based methods—magnitude pruning (43.05) and PALS-Gradient (45.23)—fail badly, which we return to in Section 6.5 .\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 1 shows the comparison visually.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.2 Cross-Model Results\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 2 extends the evaluation to LLaMA-3-8B and Mistral-7B.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The pattern is clear: PALS helps LLaMA-2 substantially but does little for the other two. Mistral-7B is the most pruning-friendly model in absolute terms (6.31 at 50% sparsity, only 33% above its dense baseline of 4.74), but PALS offers no improvement over uniform allocation. LLaMA-3-8B falls between the two, with a negligible 0.03 perplexity reduction.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We think this reflects architectural differences. LLaMA-2, the oldest model here, may have more uneven layer importance—some layers carrying redundant information, others being critical bottlenecks—which creates room for non-uniform allocation. Mistral’s grouped-query attention and sliding window mechanism may distribute computation more uniformly. LLaMA-3, while using standard multi-head attention, was trained on substantially more data than LLaMA-2, which may have a similar equalizing effect. We discuss this further in Section 7.2 .\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.3 Downstream Task Performance\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Perplexity measures language modeling quality, but deployment applications care about task accuracy. Table 3 reports zero-shot accuracy on four benchmarks.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"PALS retains 93.3% of the dense model’s average accuracy, compared to 91.7% for Wanda. The 1.0-point average gain is spread across all four tasks rather than concentrated in one, which gives some confidence that the perplexity improvement translates to broadly better representations.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"That said, the downstream gaps are modest. On MMLU, PALS gets 42.2 versus Wanda’s 41.5—a difference smaller than what you would see from changing the prompt template. Downstream benchmarks at this scale may lack the sensitivity to resolve the kind of gains PALS provides.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.4 Efficiency\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 4 reports deployment metrics on an A100 GPU with batch size 1 and sequence length 512.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The 50% sparsity halves the stored model size from 13.2 GB to 6.6 GB. Throughput improves by 1.41 × \\\\times using sparse matrix kernels, while peak memory drops modestly (1.10 × \\\\times ) because activation memory is unchanged. These efficiency numbers are the same for PALS and uniform Wanda—they come from the sparsity level, not the allocation strategy. PALS gets better perplexity at the same sparsity budget.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"5.5 Performance Across Sparsity Levels\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 2 shows WikiText-2 perplexity across sparsity levels from 30% to 70%.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"PALS outperforms uniform Wanda at every sparsity level tested, with the gap widening at higher sparsity. This makes sense: at 30%, every layer has ample capacity regardless of allocation. At 70%, capacity is scarce, and putting it where it counts matters more.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"6 Analysis\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"6.1 Learned Sparsity Patterns\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Figure 3 shows the per-layer sparsity budgets PALS assigns across our three models.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"All three models show a roughly U-shaped pattern: early layers (0–8) and late layers (21–31) get sparsity below 50%, while middle layers (9–20) get sparsity above 50%. The range stays within the ± 5 % \\\\pm 5\\\\% bounds by construction, so every layer falls between 45% and 55%.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"This aligns with what is known about transformer layer function. Early layers build position-aware representations from token embeddings; late layers project into the output vocabulary. Both are hard to replace with fewer weights. Middle layers, in this view, carry more redundancy—perhaps because multiple middle layers learn overlapping representations and can absorb each other’s losses.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"One caveat: the pattern similarity across models is partly mechanical. Because PALS normalizes importance scores and clips to ± 5 % \\\\pm 5\\\\% , the output will look roughly U-shaped for any model whose early and late layers have larger activation outliers than the middle. Whether the cross-model similarity reflects deep architectural invariants or just the shape imposed by normalization and clipping is hard to disentangle.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"6.2 Effect of Adaptation Strength α \\\\alpha\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 5 shows how α \\\\alpha affects perplexity on LLaMA-2-7B. These are single runs (except α = 0.05 \\\\alpha=0.05 , which is the 9-run mean), so differences within the run-to-run noise band ( ∼ \\\\sim 0.6 PPL) should not be over-interpreted.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"There is an interesting detail here. For α ≥ 0.07 \\\\alpha\\\\geq 0.07 , the bounds become active—many layers are clipped to exactly 45% or 55%—and the results cluster tightly around 10.1–10.2. These single-run numbers are slightly better than our chosen α = 0.05 \\\\alpha=0.05 (10.96), but the difference of ∼ \\\\sim 0.8 PPL is within the run-to-run variance (std = 0.59 over the 9-run evaluation at α = 0.05 \\\\alpha=0.05 ). A single lucky seed at α = 0.07 \\\\alpha=0.07 could easily account for the gap.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We use α = 0.05 \\\\alpha=0.05 because it represents the transition point: below it, the allocation is too timid; above it, the bounds do most of the work and the specific value of α \\\\alpha stops mattering. The data is consistent with the interpretation that a coarser binary allocation (every layer at either 45% or 55%) works roughly as well as the smooth allocation PALS provides, though confirming this would require multi-run evaluation at each α \\\\alpha value.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The α = 0.01 \\\\alpha=0.01 entry (14.23 PPL) is worse than uniform Wanda (12.92)—a very small amount of adaptation apparently hurting. We suspect this is seed noise in a single run, not a real effect. Without multi-run data at each α \\\\alpha , we cannot say whether this non-monotonicity is genuine.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"6.3 Effect of Bound Width\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 6 is the cleanest result in our ablations. Performance degrades sharply once the bounds exceed ± 10 % \\\\pm 10\\\\% , and unbounded allocation (12–88% range) is catastrophic. At ± 20 % \\\\pm 20\\\\% , the most aggressively pruned layers hit 70% sparsity, and the information loss there apparently cannot be compensated by gently pruned layers elsewhere. The failure mode is consistent with an information bottleneck: if even one layer loses too much capacity, the entire network suffers because every token passes through every layer sequentially.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The ± 5 % \\\\pm 5\\\\% setting sits in a good spot. Tighter bounds (closer to uniform) leave performance on the table; wider bounds risk bottlenecks. Whether ± 5 % \\\\pm 5\\\\% is universally best or happens to suit these three models is an open question.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"6.4 Choice of Activation Percentile\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The mean is a poor importance signal (13.78 PPL, worse than uniform Wanda), confirming that outlier activations, not average activation level, drive layer importance for pruning. Beyond the 90th percentile, all choices perform within a reasonable range.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"A careful reader will notice that the 99th percentile (10.96) is not the best in this table: both the 95th (10.67) and 99.5th (10.11) give lower perplexity. Since these are single runs and the run-to-run variance at α = 0.05 \\\\alpha=0.05 is 0.59, the differences are not statistically distinguishable. We chose the 99th percentile before running this ablation, based on its connection to the outlier phenomenon Dettmers et al. ( 2022 ) , and kept it rather than cherry-picking after the fact. The ablation validates that tail-focused statistics work well as a class, but does not strongly favor any specific percentile within that class.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"6.5 Gradient-Based Allocation Fails\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Table 8 is, in our view, the most striking result in this paper. Gradient-based importance metrics—which have a long history of success in pruning smaller networks LeCun et al. ( 1989 ) —produce allocations that are worse than random . Gradient norm gives 47.32 PPL; first-order Taylor expansion ( ∑ | g i j w i j | \\\\sum|g_{ij}w_{ij}| ) does better at 38.61 but is still far worse than random at 24.37.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"To be clear about what “random allocation” means here: each layer gets a sparsity ratio drawn uniformly from [ s target − 0.05 , s target + 0.05 ] [s_{\\\\text{target}}-0.05,\\\\;s_{\\\\text{target}}+0.05] , with no information about layer importance. It beats both gradient methods by a wide margin.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Why do gradients fail? Two possibilities, both speculative:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Convergence confound. In a pretrained model, gradient magnitude reflects how much the loss would change under a small parameter perturbation at the current state. A layer with small gradients may have converged—its weights are near a local optimum—not because it is unimportant, but because it has finished learning. Gradient-based allocation would over-prune such layers.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Discrete–continuous mismatch. Pruning removes a large, discrete fraction of weights. Gradients measure infinitesimal, continuous sensitivity. At 50% sparsity, the “perturbation” is far from infinitesimal. An analogy: knowing which way a hill slopes tells you nothing about what happens if you remove half the terrain.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Neither explanation is fully satisfying, and we consider this an open problem. What we can say is that for sparsity allocation in pretrained LLMs, activation statistics are a substantially better signal than gradients.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"6.6 Calibration Stability\"\n },\n {\n \"type\": \"para\",\n \"text\": \"PALS relies on activation statistics from 128 calibration samples. A natural concern is whether the importance scores are stable across different calibration draws.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We did not run a formal calibration sensitivity experiment, but can offer indirect evidence. The 99th percentile is computed over roughly 128 × 2048 ≈ 262 128\\\\times 2048\\\\approx 262 K tokens per layer. Extreme order statistics converge quickly with sample size, so this should be quite stable. Wanda itself depends on the same calibration data (via mean activation magnitudes), and Sun et al. Sun et al. ( 2024 ) reported that Wanda’s results are stable across calibration subsets. Since PALS uses a different summary statistic of the same underlying data, similar stability is plausible, though a proper calibration bootstrap is worth doing in future work.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Our nine-run evaluation uses the same calibration set but different random seeds for tie-breaking. The resulting std of 0.59 bounds the contribution of tie-breaking noise. Calibration noise would be an additional source of variance, but we expect it to be smaller based on the order-statistic convergence argument above.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"7 Discussion\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"7.1 Why Activations Work\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The success of activation-based importance connects to the growing literature on activation outliers Dettmers et al. ( 2022 ) . Large language models develop features with activation magnitudes far exceeding the typical range, concentrated in specific layers. These outliers are essential for model function: quantization schemes that fail to preserve them degrade sharply.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"PALS identifies layers with large outliers (via the 99th percentile) and assigns them lower sparsity. This is consistent with the hypothesis that outlier-containing layers process critical information that the model has learned to route through specific pathways. The connection also predicts that any importance metric sensitive to the tail of the activation distribution should work—and indeed, the 90th through max percentile all outperform the mean (Table 7 ).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"7.2 Architecture Dependence\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The large improvement on LLaMA-2 (15%) versus zero on Mistral is worth understanding. Three hypotheses:\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Training data volume. LLaMA-2-7B was trained on 2T tokens; LLaMA-3-8B on roughly 15T tokens. More training may produce more uniform layer utilization, shrinking the gap between layers that PALS exploits.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Architecture. Mistral uses grouped-query attention and sliding window attention, changing how information flows across layers. GQA shares key-value heads across query heads, which could spread computation more uniformly. Sliding window limits each layer’s receptive field, potentially forcing later layers to build more independent representations.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Model maturity. Newer models may simply be better trained, with fewer “wasted” layers. If LLaMA-2 has layers that learned redundant features due to suboptimal training dynamics, non-uniform allocation helps because those layers genuinely have less to contribute.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"We cannot distinguish between these with our current experiments. The practical upshot: if you run PALS and the sparsity pattern looks nearly flat (all layers close to 50%), the model probably will not benefit, and uniform allocation is fine.\"\n },\n {\n \"type\": \"heading\",\n \"level\": 2,\n \"text\": \"7.3 Limitations\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Unstructured sparsity. The throughput gains in Table 4 require sparse matrix kernels or hardware support (e.g., NVIDIA’s 2:4 structured sparsity). Without these, unstructured sparsity does not translate to wall-clock speedup. Extending PALS to N:M or block sparsity patterns is a natural next step.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Single perplexity dataset. Our primary metric is WikiText-2 perplexity. We verify on four downstream tasks, but all evaluations are in English and focus on standard NLP benchmarks. Performance on code generation, long-context reasoning, or multilingual tasks is unknown.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"No fine-tuning. We evaluate one-shot pruning only. Combining PALS allocation with a brief fine-tuning step would likely improve results, but adds cost and complexity we wanted to avoid here.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"Model scale. All experiments are on 7–8B parameter models. Whether the gains hold at 13B, 70B, or larger is untested. Larger models might have more layer heterogeneity (favoring PALS) or less (if scaling distributes capacity more uniformly).\"\n },\n {\n \"type\": \"heading\",\n \"level\": 1,\n \"text\": \"8 Conclusion\"\n },\n {\n \"type\": \"para\",\n \"text\": \"PALS adjusts per-layer sparsity in one-shot LLM pruning using activation percentiles, reducing LLaMA-2-7B perplexity from 12.92 to 10.96 at 50% sparsity with negligible extra cost. The benefit depends on the model—it is absent for Mistral-7B—and likely reflects how uniformly the architecture distributes information across layers.\"\n },\n {\n \"type\": \"para\",\n \"text\": \"The gradient failure finding is, to us, the more interesting result. Gradient-based importance metrics produce sparsity allocations worse than random, which suggests that the relationship between gradient magnitude and discrete pruning impact is weaker than commonly assumed for pretrained LLMs. Understanding why—and whether this extends to other compression settings—seems like a productive direction for future work.\"\n }\n]","isConversational":false}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":0,"numItemsPerPage":100,"numTotalItems":1000,"offset":0,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6IjVHZDBvd0g5MTM2eDZjc1FvbE1zcktNWUZoRVFoUm5PVVVybEpjOUhaUEEifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc4NDY1NTExMSwianRpIjoiZDFiMDgwYzktYjhlZi00ODVlLWExN2UtOGI1ZWYxNDczZmUxIiwic3ViIjoiL2RhdGFzZXRzL1ByaW1lSW50ZWxsZWN0L2Rlc2h1ZmZsZS1wYXBlcnMtdjEtY29ycHVzIiwiZXhwIjoxNzg0NjU4NzExLCJpc3MiOiJodHRwczovL2h1Z2dpbmdmYWNlLmNvIn0.CY-xMwY53BJVTXUODfxLu9slPI3sBH6g6QNuR4lTfMp8MkwObvhq7Pn3Z-xxd95LFqpgmh-gXgTQidZZw-K-Bg","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":1000,"numBytesParquet":49695546}]},"discussionsStats":{"closed":0,"open":0,"total":0},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
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|---|---|---|---|---|---|---|---|---|---|
arxiv | 2607.02513 | null | 1 | LACUNA: A Testbed for Evaluating Localization Precision for LLM Unlearning | [
"Matteo Boglioni",
"Thibault Rousset",
"Siva Reddy",
"Marius Mosbach",
"Verna Dankers"
] | CC-BY-4.0 | https://arxiv.org/html/2607.02513v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "LLMs memorize sensitive training data, including personally identifiable information (PII), creating a pressing need for reliable post hoc removal methods. Unlearning has emerged as a promising solution, with state-of-the-art (SOTA) methods often following a localize-first, unlear... |
arxiv | 2607.02512 | null | 1 | Program-as-Weights: A Programming Paradigm for Fuzzy Functions | [
"Wentao Zhang",
"Liliana Hotsko",
"Woojeong Kim",
"Pengyu Nie",
"Stuart Shieber",
"Yuntian Deng"
] | CC-BY-4.0 | https://arxiv.org/html/2607.02512v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "Many everyday programming tasks resist clean rule-based implementation, such as alerting on important log lines, repairing malformed JSON, or ranking search results by intent, and are increasingly outsourced to large language model APIs at the cost of locality, reproducibility, an... |
arxiv | 2607.02507 | null | 1 | What LLM Agents Say When No One Is Watching: Social Structure and Latent Objective Emergence in Multi-Agent Debates | [
"Arman Ghaffarizadeh",
"Danyal Mohaddes",
"Aliakbar Izadkhah",
"Shahriar Noroozizadeh"
] | CC-BY-4.0 | https://arxiv.org/html/2607.02507v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "LLM agents will increasingly act in socially structured settings where role, audience, and relational context can shape what is advantageous or costly to say. We study whether such social structure, without any explicit objective in the prompt, changes what an agent expresses publ... |
arxiv | 2607.02499 | null | 1 | Beyond Adam: SOAP and Muon for Faster, Label-Efficient Training of Machine Learning Interatomic Potentials | [
"Gil Harari",
"Yoel Zimmermann",
"Ola Tangen Kulseng",
"Laura Zichi",
"Chuin Wei Tan",
"Marc L. Descoteaux",
"Boris Kozinsky"
] | CC-BY-4.0 | https://arxiv.org/html/2607.02499v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "Machine learning interatomic potentials (MLIPs) have become a hallmark of AI for scientific simulation. While efforts on new architectures and datasets have led to increasingly accurate and general models, the choice of optimizer for training has largely remained unexplored, defau... |
arxiv | 2607.02391 | null | 1 | WattGPU: Predicting Inference Power and Latency on Unseen GPUs and LLMs | [
"Mauricio Fadel Argerich",
"Jonathan Fürst",
"Marta Patiño-Martínez"
] | CC-BY-4.0 | https://arxiv.org/html/2607.02391v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0)."
},
{
"type": "para",
"text": "SuRE’26: Workshop on Sustainability and Resource-Efficiency of Artificial Intelligence, August 17, 2024, Brem... |
arxiv | 2607.02292 | null | 1 | One More Time: Revisiting Neural Quantum States from a Reinforcement Learning Perspective | [
"Juan Agustín Duque",
"Sergio García Heredia",
"Vinicius Hernandes",
"Eliška Greplová",
"Thomas Spriggs",
"Aaron Courville",
"Anna Dawid"
] | CC-BY-4.0 | https://arxiv.org/html/2607.02292v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "Neural quantum states (NQS) provide a flexible and scalable framework for approximating quantum many-body wavefunctions. Among NQS parameterizations, autoregressive models are especially attractive because they enable exact, independent sampling from the Born distribution, avoidin... |
arxiv | 2607.02288 | null | 1 | Generalization in offline RL: The structure is more important than the amount of pessimism | [
"Max Weltevrede",
"Matthijs T. J. Spaan",
"Wendelin Böhmer"
] | CC-BY-4.0 | https://arxiv.org/html/2607.02288v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "While pessimism counteracts overestimation bias in offline reinforcement learning (RL), being overly conservative has been associated with hindering certain forms of generalization. However, in this paper we demonstrate that being overly pessimistic does not inherently prevent opt... |
arxiv | 2607.02266 | null | 1 | HERMES: A Multi-Granularity Labeling Substrate for Pre-training Data Mixtures | [
"Ziyun Qiao",
"Yue Min",
"Ruining Chen",
"Yujun Li"
] | CC-BY-4.0 | https://arxiv.org/html/2607.02266v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "Most data-mixing methods assume the corpus has already been partitioned into groups, and the choice of those groups determines what a mixer can express. Existing labels (provenance, topic or format taxonomies, flat embedding clusters) commit to one semantic axis at one granularity... |
arxiv | 2607.02247 | null | 1 | Aggregation with Exponential Weights is Optimal in Expectation | [
"Mikael Møller Høgsgaard",
"Patrick Rebeschini",
"Tobias Wegel"
] | CC-BY-4.0 | https://arxiv.org/html/2607.02247v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "The aggregation with exponential weights (AEW) estimator is not fully understood in the basic setting of model selection aggregation with squared loss. In particular, whether it is minimax-rate optimal in expectation for large enough fixed temperatures and under random design has ... |
arxiv | 2607.02203 | null | 1 | Self-explainable Operator Learning for Discovering Spatial Patterns in Functional Data | [
"Mojgan Alishiri",
"Amirhossein Arzani"
] | CC-BY-4.0 | https://arxiv.org/html/2607.02203v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "Operator learning has emerged as a powerful tool for modeling complex physical systems in functional spaces. However, their neural network–based architectures make them opaque models, obscuring the reasoning behind their predictions. In this work, we introduce a self-explainable o... |
arxiv | 2607.02199 | null | 1 | Fourier Preconditioning for Neural Feature Learning | [
"Preston Pitzer",
"Anish Pradhan",
"Harpreet S. Dhillon"
] | CC-BY-4.0 | https://arxiv.org/html/2607.02199v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "Mutual information (MI)-inspired feature learning techniques are capable of generating low-dimensional embeddings that retain nonlinear dependence structures, but direct estimations of MI suffer from noisy probability distribution estimates in the low-data regime. The H-Score obje... |
arxiv | 2607.02187 | null | 1 | Privacy-Preserving and Verifiable Approximate Distributed Coded Computing | [
"Xavier Martínez-Luaña",
"Alba Gude-Santos",
"Manuel Fernández-Veiga",
"Rebeca P. Díaz-Redondo"
] | CC-BY-4.0 | https://arxiv.org/html/2607.02187v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "Distributed machine learning enables collaborative model training without centralizing data, but it also exposes learning processes to privacy leakage and malicious manipulation. Existing defenses typically address these threats in isolation and are often tailored to specific lear... |
arxiv | 2607.02137 | null | 1 | ART for Diffusion Sampling: Continuous-Time Control and Actor-Critic Learning | [
"Yilie Huang",
"Wenpin Tang",
"Xun Yu Zhou"
] | CC-BY-4.0 | https://arxiv.org/html/2607.02137v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "We study timestep allocation for score-based diffusion sampling, where a learned reverse-time dynamics is discretized on a finite grid. Uniform and hand-crafted schedules are standard choices, but they rely on ad hoc , fixed prescriptions and can therefore be suboptimal. To addres... |
arxiv | 2607.02131 | null | 1 | AbsoluteDegradation: A Physics-Inspired Synthetic Film-Degradation Pipeline and Archival Film Restoration Benchmark | [
"Mikołaj Jastrzębski",
"Dawid Glinkowski",
"Dawid Zieliński",
"Daniel Borkowski",
"Wojciech Kozłowski",
"Kamil Adamczewski"
] | CC-BY-4.0 | https://arxiv.org/html/2607.02131v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "Restoring archival film remains a fundamentally challenging problem due to the absence of paired training data and the lack of standardized evaluation benchmarks. Pristine versions of deteriorated footage are physically unrecoverable, requiring supervised methods to rely on synthe... |
arxiv | 2607.02127 | null | 1 | Population-Scale Segmentation of Penile Tissue in DIXON MRI using Deep Learning for Quantitative Phenotyping in Male Reproductive Health | [
"Jan Ernsting",
"Gunnar Paul Kordes",
"Nils Johannaber",
"Lynn Ogoniak",
"Wolfgang Roll",
"Tim Hahn",
"Alexander Siegfried Busch",
"Benjamin Risse"
] | CC-BY-4.0 | https://arxiv.org/html/2607.02127v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "Penile measurement is clinically relevant across male reproductive and urogenital health, including conditions such as micropenis, congenital and endocrine disorders, and sexual or urinary dysfunction. However, quantitative assessment of penile size has relied mainly on external l... |
arxiv | 2607.02104 | null | 1 | Ask the Right Comparison:Bias-Aware Bayesian Active Top-$k$ Ranking with LLM Judges | [
"Jian Xu",
"Delu Zeng",
"John Paisley",
"Qibin Zhao"
] | CC-BY-4.0 | https://arxiv.org/html/2607.02104v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "Large language models (LLMs) are increasingly used as cheap, scalable judges that compare candidate outputs pairwise—to rank responses, select models, or triage papers. Yet LLM judges are both noisy and systematically biased : they favor verbose or well-formatted answers and exhib... |
arxiv | 2607.02087 | null | 1 | SUNTA: Hierarchical Video Prediction with Surprise-based Chunking | [
"Tomoshi Iiyama",
"Masahiro Suzuki",
"Yutaka Matsuo"
] | CC-BY-4.0 | https://arxiv.org/html/2607.02087v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "Hierarchical state-space models (HSSMs) offer a promising approach to long-horizon prediction by segmenting sequences into temporal chunks. However, their performance hinges on how chunk boundaries are determined. While prior HSSMs typically rely on fixed-length chunking or simila... |
arxiv | 2607.02079 | null | 1 | HaloGuard 1.0: An Open Weights Constitutional Classifier for Multilingual AI Safety | [
"Navaneeth Sangameswaran",
"Preetham S",
"Ashmiya Lenin"
] | CC-BY-4.0 | https://arxiv.org/html/2607.02079v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "Large language models (LLMs) are increasingly moving beyond chat interfaces to agentic use cases where attack surfaces are exponentially bigger and downstream failures are catastrophically expensive. A practical defence-in-depth solution requires multiple layers with a pre-generat... |
arxiv | 2607.02073 | null | 1 | Evidence-State Rewards for Long-Context Reasoning | [
"Ya Gao",
"Pekka Marttinen"
] | CC-BY-4.0 | https://arxiv.org/html/2607.02073v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "Long-context reasoning requires models to locate, revise, and synthesize evidence distributed across lengthy inputs. Existing long-context RL methods usually reward final answers or static evidence extraction, offering little feedback on how intermediate actions change the model’s... |
arxiv | 2607.02063 | null | 1 | SA-HGNN: Sample-Adaptive Hyperbolic Graph Neural Network for EEG-Based Depression Recognition | [
"Yang Li",
"Pan Hu",
"Yan Zhang",
"Wenfan Yang",
"Tao Wu",
"Lianbo Guo"
] | CC-BY-4.0 | https://arxiv.org/html/2607.02063v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "Graph Neural Networks (GNNs) have been widely used to capture spatial functional connectivity patterns to improve electroencephalography (EEG)-based depression recognition performance. However, the functional connectivity of brain networks in patients with depression exhibits an i... |
arxiv | 2607.02050 | null | 1 | A Memory Efficient Unified Algorithm for Online Learning of Linear Dynamical Systems | [
"Yuval Ran-Milo",
"Angelos Assos",
"Elad Hazan"
] | CC-BY-4.0 | https://arxiv.org/html/2607.02050v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "Motivated by the challenge of stabilizing a general unknown linear dynamical system (LDS) from observations, we study the natural prerequisite of online prediction. Our goal is to achieve sublinear regret with a memory footprint that adapts to the intrinsic complexity of the dynam... |
arxiv | 2607.02037 | null | 1 | Cross-Platform Control for Autonomous Surface Vehicles via Adaptive Reinforcement Learning | [
"Ruiheng Jiang",
"Thomas Bi",
"Raffaello D'Andrea",
"Aswin Ramachandran"
] | CC-BY-4.0 | https://arxiv.org/html/2607.02037v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "Autonomous surface vehicles vary widely in hydrodynamic and actuation characteristics, yet most controllers are designed for single-platform deployment. We present an adaptive reinforcement learning approach for trajectory tracking that enables zero-shot cross-platform deployment ... |
arxiv | 2607.02003 | null | 1 | Born Discrete, Made Smooth: Variational Formulation of Shallow Neural Networks | [
"Matej Benko",
"Pierre Bousquet",
"Iwona Chlebicka",
"Błażej Miasojedow"
] | CC-BY-4.0 | https://arxiv.org/html/2607.02003v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "Although neural networks are remarkably effective, their underlying optimization principles remain theoretically elusive, often characterized by non-convex landscapes and stochastic heuristics. In this work, we propose a paradigm shift by replacing the discrete training problem of... |
arxiv | 2607.01986 | null | 1 | Liquid Latent State Dynamics for Interpretable Turbofan Degradation Modeling | [
"Weizhi Nie",
"Weijie Wang",
"Yuting Su"
] | CC-BY-4.0 | https://arxiv.org/html/2607.01986v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "Multivariate time-series models for prognostics are often evaluated by point prediction accuracy, yet their internal states rarely expose a coherent degradation process. We study liquid neural networks as latent dynamics models for aircraft engine health monitoring on the C-MAPSS ... |
arxiv | 2607.01973 | null | 1 | Assessing VLM Reliability for Medical Image Quality Evaluation Under Corruption and Bias | [
"Sofiane Ouaari",
"Kevin Vorwalder",
"Nico Pfeifer"
] | CC-BY-4.0 | https://arxiv.org/html/2607.01973v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "1]Methods in Medical Informatics, Department of Computer Science, University of Tuebingen, Germany 2]Institute for Bioinformatics and Medical Informatics (IBMI), University of Tuebingen, Germany"
},
{
"type": "para",
"text": "*]sofiane.ouaari@uni-tuebingen.de"
},
{... |
arxiv | 2607.01966 | null | 1 | Probabilistic Low-Voltage Peak Load Forecasting with Time Series Foundation Models Evaluated on Application-Oriented Metrics | [
"Benedikt Kaas",
"Manuel Treutlein",
"Hannes Benedikt Gerber",
"Oliver Neumann",
"Cheewan Phatthanakhuha",
"Oliver Resch",
"Ralf Mikut",
"Veit Hagenmeyer"
] | CC-BY-4.0 | https://arxiv.org/html/2607.01966v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "Low-voltage load forecasting is an important component in current and future energy systems with a high degree of electrification and decentralized generation. However, current forecasting methods require significant manual effort, often lack uncertainty estimation and proper peak... |
arxiv | 2607.01965 | null | 1 | Towards a Phonology-Informed Evaluation of Multilingual TTS | [
"Sneha Ray Barman",
"Neeraj Kumar Sharma",
"Shakuntala Mahanta"
] | CC-BY-4.0 | https://arxiv.org/html/2607.01965v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "Ray Barman Sharma Mahanta"
},
{
"type": "para",
"text": "Neural TTS systems can sound natural across languages, but naturalness does not guarantee the preservation of sound contrasts that distinguish words from their grammatical forms. Standard metrics like MOS do not ... |
arxiv | 2607.01959 | null | 1 | Autorelevance function and other feature relevance measures for univariate time series | [
"Julian Cardenas",
"Jamie Arjona",
"Pedro Delicado"
] | CC-BY-4.0 | https://arxiv.org/html/2607.01959v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "[1,2,3] \\orgdiv Departament d’Estadística i Investigació Operativa, \\orgname Universitat Politècnica de Catalunya - BarcelonaTech, \\orgaddress \\street C/ Jordi Girona 1-3, \\city Barcelona, \\postcode 08034, \\state Catalunya, \\country Spain"
},
{
"type": "para",
... |
arxiv | 2607.01958 | null | 1 | A More Accurate Algorithm Comparison through A/B Testing using Offline Evaluation Methods | [
"Koki Konishi",
"Masataka Ushiku",
"Yuta Saito"
] | CC-BY-4.0 | https://arxiv.org/html/2607.01958v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "by"
},
{
"type": "para",
"text": "A/B testing is the gold standard for selecting better algorithms in online services. While offline evaluation has attracted attention as a safer alternative due to the high experimental costs and the potential risk of degrading user ex... |
arxiv | 2607.01940 | null | 1 | Conditional Co-Ablation: Recovering Self-Repair Backups in Transformer Circuits | [
"Zhiren Gong",
"Zihao Zeng",
"Chau Yuen",
"Wei Yang Bryan Lim"
] | CC-BY-4.0 | https://arxiv.org/html/2607.01940v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "Mechanistic interpretability often relies on component-level interventions to discover how a model produces a behavior. This guides attribution, capability knockout, and model pruning downstream to operate by scoring each unit by the effect of ablation in isolation. Such first-ord... |
arxiv | 2607.01907 | null | 1 | Population-Based Multi-Objective Training of Discriminators for Semi-Supervised GANs | [
"Francisco Sedeño",
"Francisco Chicano",
"Jamal Toutouh"
] | CC-BY-4.0 | https://arxiv.org/html/2607.01907v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "Semi-supervised generative adversarial networks (SSL-GANs) can exploit large unlabeled datasets while retaining a classifier in the discriminator, but their training is often unstable. This paper proposes a population-based evolutionary training strategy in which discriminator lea... |
arxiv | 2607.01897 | null | 1 | Rank-Then-Act: Reward-Free Control from Frame-Order Progress | [
"Yuriy Maksyuta",
"George Bredis",
"Ruslan Rakhimov",
"Daniil Gavrilov"
] | CC-BY-4.0 | https://arxiv.org/html/2607.01897v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "We introduce Rank-Then-Act (RTA), a framework for learning control policies from expert video demonstrations without environment rewards. RTA trains a Vision–Language Model (VLM) offline as a progress-based ordinal scorer, using a Group Relative Policy Optimization (GRPO) objectiv... |
arxiv | 2607.01880 | null | 1 | Learning the Supports for Categorical Critic in Reinforcement Learning | [
"Jen-Yen Chang",
"Takayuki Osa",
"Tatsuya Harada"
] | CC-BY-4.0 | https://arxiv.org/html/2607.01880v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "Learning the Supports for Categorical Critic in Reinforcement Learning"
},
{
"type": "para",
"text": "Jen-Yen Chang , Takayuki Osa , Tatsuya Harada"
},
{
"type": "para",
"text": "Keywords: Classification-Based Value Learning, HL-Gauss, Upper bound of Bellma... |
arxiv | 2607.01838 | null | 1 | Adaptive Group-Based Counterfactual Explanations for Time-Series Rehabilitation Data | [
"Emmanuel C. Chukwu",
"Rianne M. Schouten",
"Monique Tabak",
"Mykola Pechenizkiy"
] | CC-BY-4.0 | https://arxiv.org/html/2607.01838v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "Counterfactual explanations (CEs) for multivariate time-series classifiers are often difficult to interpret in domains where experts reason in terms of semantic feature groups rather than individual channels. In rehabilitation movement analysis with multi-sensor inertial measureme... |
arxiv | 2607.01795 | null | 1 | Single-Channel EEG-Based Cognitive Load Assessment in Online Learning: A Hybrid Deep Learning Approach | [
"Rowan Hussein",
"Mohamed Ouf"
] | CC-BY-4.0 | https://arxiv.org/html/2607.01795v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "Monitoring cognitive load during online learning could help instructors identify content that learners find difficult, but remote settings remove the visual cues that support this judgement in a classroom. We study whether a single-channel, consumer-grade EEG device (the NeuroSky ... |
arxiv | 2607.01762 | null | 1 | Role-Aware Neural Convex Divergence Heads for Asymmetric Representation Learning | [
"He Huang",
"Lu Shen",
"Yunfeng Huang",
"Li Qi"
] | CC-BY-4.0 | https://arxiv.org/html/2607.01762v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "Many representation learning problems involve directed relations, such as lexical entailment, sentence entailment, ontology hierarchy, and citation links. Standard Euclidean, cosine, and Mahalanobis heads are symmetric, while generic neural scorers can model directionality but pro... |
arxiv | 2607.01709 | null | 1 | COMFYCLAW: Self-Evolving Skill Harnesses for Image Generation Workflows | [
"Zongxia Li",
"Dawei Liu",
"Fuxiao Liu",
"Yuhang Zhou",
"Xiyang Wu",
"Jingxi Chen",
"Jing Xie",
"Xiaomin Wu",
"Lichao Sun"
] | CC-BY-4.0 | https://arxiv.org/html/2607.01709v1 | 2026-07-02T00:00:00 | [
{
"type": "para",
"text": "Agents are increasingly used to construct workflows and help humans complete recurring tasks more efficiently. As these workflows become repeated and domain-specific, agent memory and reusable skills become increasingly important: agents should be able to recall workflow patterns,... |
biorxiv | 10.64898/2026.07.01.735929 | 10.64898/2026.07.01.735929 | 2 | The Premotor Language Area Encodes a Full Acoustic-to-semantic Speech Hierarchy | [
"Guo, S.",
"Huth, A."
] | CC-BY-4.0 | https://www.biorxiv.org/content/10.64898/2026.07.01.735929v2 | 2026-07-02T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "Introduction"
},
{
"type": "para",
"text": "Spoken language processing is one of the most complex motor–cognitive behaviors humans perform, requiring the seamless integration of sensory analysis, motor planning, and linguistic computation. Achieving ... |
biorxiv | 10.64898/2026.07.01.735911 | 10.64898/2026.07.01.735911 | 1 | ELAVL1 and ELAVL4 are required for Musashi-dependent translational activation | [
"Bronson, K.",
"Reddick, M. M.",
"MacNicol, K. B.",
"Bolen, C. R.",
"Hardy, L. L.",
"Lagasse, A. N.",
"Odle, A. K.",
"Childs, G. V.",
"MacNicol, M. C.",
"MacNicol, A. M."
] | CC-BY-4.0 | https://www.biorxiv.org/content/10.64898/2026.07.01.735911v1 | 2026-07-02T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "Introduction"
},
{
"type": "para",
"text": "The Musashi (MSI) family of evolutionarily conserved, sequence-specific mRNA binding proteins (MSI1 and MSI2) has been shown to regulate the translation of target transcripts involved in the promotion of st... |
biorxiv | 10.64898/2026.07.01.735890 | 10.64898/2026.07.01.735890 | 1 | Particle size determines mucociliary transport mechanisms in normal and cystic fibrosis airways | [
"Scott, M.",
"Bierstedt, K. C.",
"Du, W.",
"Riley, M. J.",
"Fischer, A. J.",
"Xie, Y."
] | CC-BY-4.0 | https://www.biorxiv.org/content/10.64898/2026.07.01.735890v1 | 2026-07-02T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "Introduction"
},
{
"type": "para",
"text": "With each breath, a wide spectrum of microparticles (1-1000 μm in diameter) is inhaled, including dust (10-100 µm), particulate matter (PM 10 ), and bacteria (1-10 µm). Due to their size, inhaled microparti... |
biorxiv | 10.64898/2026.07.01.735889 | 10.64898/2026.07.01.735889 | 1 | Learning Shapes the Energy Cost of Neural Tasks | [
"Xue, K.",
"Rezayat, F.",
"Qi, T.",
"Shen, L.",
"Zhao, B.",
"Huang, X.",
"Marvin, J. S.",
"Ye, L."
] | CC-BY-4.0 | https://www.biorxiv.org/content/10.64898/2026.07.01.735889v1 | 2026-07-02T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "Main Text"
},
{
"type": "para",
"text": "In contrast to the gigawatt scale energy consumption of AI, the biological brain operates at incredibly low power. For example, it is estimated that the human brain operates on a modest, nearly constant 20 W o... |
biorxiv | 10.64898/2026.07.01.735859 | 10.64898/2026.07.01.735859 | 1 | Zinc Differentially Modulates Tau Aggregation, Fibril Morphology, and Prion-like Seeding in a Construct-Dependent Manner | [
"Poirier, E. L.",
"Stainton, A.",
"Simon, O.",
"Mittal, S. S.",
"Varona Ortiz, A. B.",
"Kim, S. A.",
"Rauch, J. N."
] | CC-BY-4.0 | https://www.biorxiv.org/content/10.64898/2026.07.01.735859v1 | 2026-07-02T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "INTRODUCTION"
},
{
"type": "para",
"text": "Microtubule-associated protein tau (tau) is an intrinsically disordered protein that is highly enriched in neuronal axons 1 . In diseases such as Alzheimer’s Disease (AD), tau aggregates into neurofibrillar... |
arxiv | 2607.08756 | null | 1 | MulTTiPop: A Multitrack Transcription Dataset for Pop Music | [
"Nathan Pruyne",
"Benjamin Stoler",
"William Chen",
"Chien-yu Huang",
"Shinji Watanabe",
"Chris Donahue"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08756v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "In recent years, automatic music transcription (AMT) systems for converting audio to note-level symbolic representations of music have evolved from transcribing solo piano music [ 11 ] to targeting perfor... |
arxiv | 2607.08746 | null | 1 | Dimensionality Reduction Meets Network Science: Sensemaking on UMAP's kNN Graph | [
"Duen Horng Chau",
"Donghao Ren",
"Fred Hohman",
"Dominik Moritz"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08746v1 | 2026-07-09T00:00:00 | [
{
"type": "para",
"text": "0 \\vgtccategory Research \\vgtcinsertpkg \\teaser Standard graph algorithms applied to UMAP’s internal k k NN graph reveal structure lost in 2D scatter plot layouts. A. PageRank on the Fashion MNIST k k NN graph identifies representative data points. The highest-scoring points ex... |
arxiv | 2607.08741 | 10.1145/3811284 | 1 | ARDY: Autoregressive Diffusion with Hybrid Representation for Interactive Human Motion Generation | [
"Kaifeng Zhao",
"Mathis Petrovich",
"Haotian Zhang",
"Tingwu Wang",
"Siyu Tang",
"Davis Rempe"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08741v1 | 2026-07-09T00:00:00 | [
{
"type": "para",
"text": "by"
},
{
"type": "heading",
"level": 1,
"text": "1. Introduction"
},
{
"type": "para",
"text": "Learning to generate realistic 3D human motions has become a promising direction with applications ranging from character animation and simulation to humanoi... |
arxiv | 2607.08733 | null | 1 | Super Weights in LLMs and the Failure of Selective Training | [
"Shreyas Subramanian",
"Adewale Akinfaderin",
"Akarsha Sehwag"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08733v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Parameter-efficient fine-tuning (PEFT) methods like LoRA (Hu et al., 2022 ) achieve performance comparable to full fine-tuning while updating only 0.1–1% of parameters. Aghajanyan et al. ( 2021 ) showed t... |
arxiv | 2607.08725 | null | 1 | Pose-to-Biomechanics: Bridging 3D Human Pose Estimation and Biomechanical Attribute Prediction | [
"Ayda Eghbalian",
"Kevin Desai"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08725v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "3D human pose estimation models have become increasingly effective at recovering geometric skeletons from images and videos, yet a gap remains between kinematic pose and the biomechanical quantities requi... |
arxiv | 2607.08724 | null | 1 | Latent Memory Palace: Reasoning for Control as Autoregressive Variational Inference | [
"Chuning Zhu",
"Eva Xu",
"Jose Barreiros",
"Krishnan Srinivasan",
"Paarth Shah",
"Abhishek Gupta"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08724v1 | 2026-07-09T00:00:00 | [
{
"type": "para",
"text": "https://weirdlabuw.github.io/lmp/"
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Human decision-making ranges from the reflexive (e.g. walking) to the deliberate (e.g. playing chess), with variability in the ... |
arxiv | 2607.08703 | null | 1 | MPFlow: Learning Budgeted Max-Flow Optimization on the Lightning Network with Deep Graph Reinforcement Learning | [
"Harrison Rush",
"Vincent Davis",
"Simone Antonelli",
"Vikash Singh",
"Jesse Shrader",
"Emanuele Rossi"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08703v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Payment-channel networks such as the Bitcoin Lightning Network (LN) enable fast, low-cost payments by moving transactions off-chain. Their performance, however, hinges on where liquidity is placed: poor p... |
arxiv | 2607.08690 | null | 1 | A Practical Investigation of Training-free Relaxed Speculative Decoding | [
"Guoxuan Xia",
"Luka Ribar",
"Paul Balanca"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08690v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Large language models (LLMs) dominate modern applications of machine learning ( sajadieh2026aiindex ) . Language modelling as a task is itself in turn dominated by autoregressive ( AR ) models ( transform... |
arxiv | 2607.08659 | null | 1 | EdgeRefine: Privacy-Utility Balance for Graphs via Jaccard Sampling under Edge Differential Privacy | [
"Wenxiu Ding",
"Muzhi Liu",
"Zheng Yan",
"Mingjun Wang",
"Yifan Zhao",
"Qiao Liu"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08659v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1. Introduction"
},
{
"type": "para",
"text": "Graph Neural Networks (GNNs) [ 7 ] are powerful methods for learning from graph-structured data [ 39 ] . They have shown great success in many areas, such as social network analysis [ 42 ] , recommendati... |
arxiv | 2607.08643 | null | 1 | BiSCo-LLM: Lookup-Free Binary Spherical Coding for Extreme Low-Bit Large Language Model Compression | [
"Yuantian Shao",
"Peisong Wang",
"Zhilei Liu",
"Chuangyi Li",
"Yuanteng Chen",
"Pengcheng Xie",
"Yiwu Yao",
"Zhihui Wei",
"Jian Cheng"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08643v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Large language models (LLMs) have become foundation components for language understanding, reasoning, generation, and general-purpose AI services [ 5 , 57 , 1 ] . Despite their effectiveness, the continuo... |
arxiv | 2607.08641 | null | 1 | Steering Neural Network Training through Interpretable Constraints Based on Partial Dependence | [
"Yann Claes",
"Pierre Geurts",
"Vân Anh Huynh-Thu"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08641v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Over the past decades, progress in machine learning (ML) methods has enabled practitioners in a variety of fields to tackle increasingly complex problems, in which such models serve as surrogates for firs... |
arxiv | 2607.08605 | null | 1 | When Structured Sparse Autoencoders Learn Consistent Concepts Across Modalities | [
"Weiduo Liao",
"Yunqiao Yang",
"Ying Wei"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08605v1 | 2026-07-09T00:00:00 | [
{
"type": "para",
"text": "Ying Wei https://wei-ying.net/"
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Recent large vision-language models (VLMs) have achieved strong performance across diverse vision-language tasks, including image ... |
arxiv | 2607.08595 | null | 1 | Federated Deep Learning for Privacy-Preserving Cardiovascular Disease Risk Prediction | [
"Hyunho Mo",
"Djura Smits",
"Mahlet A. Birhanu",
"Maarten J. G. Leening",
"Daniel Bos",
"Pim van der Harst",
"Esther E. Bron"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08595v1 | 2026-07-09T00:00:00 | [
{
"type": "para",
"text": "1 Department of Radiology & Nuclear Medicine, Erasmus MC University Medical Center Rotterdam, Dr. Molewaterplein 40, Rotterdam, 3015 GD, The Netherlands 2 Netherlands eScience Center, Matrix THREE, Science Park 402, Amsterdam, 1098 XH, The Netherlands 3 Department of Epidemiology,... |
arxiv | 2607.08579 | null | 1 | ImputeViz: A Visual Analytics Dashboard for Diagnosing Missing Data and Comparing Imputation Methods | [
"Aitik Dandapat",
"Lalith Punepalle Raveendrareddy",
"Mithilesh Kumar Singh",
"Klaus Mueller"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08579v1 | 2026-07-09T00:00:00 | [
{
"type": "para",
"text": "1359 \\vgtccategory Research \\vgtcpapertype application/design study \\authorfooter Aitik Dandapat is with Stony Brook University. E-mail: adandapat@cs.stonybrook.edu. Lalith Punepalle Raveendrareddy is with Stony Brook University. E-mail: lpunepallera@cs.stonybrook.edu. Mithiles... |
arxiv | 2607.08545 | null | 1 | Structural Bottlenecks on Frequency Representation in End-to-End Audio Models | [
"Nicole Cosme-Clifford"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08545v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Modern neural audio systems achieve state-of-the-art performance across compression and generation, with scaling consistently improving empirical results Défossez et al. ( 2022 ); Kumar et al. ( 2023 ); E... |
arxiv | 2607.08497 | null | 1 | Cognitive-structured Multimodal Agent for Multimodal Understanding, Generation, and Editing | [
"Feng Wang",
"Canmiao Fu",
"Zhipeng Huang",
"Chen Li",
"Jing Lyu",
"Ge Li"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08497v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Unified multimodal models have recently shown that a single architecture can jointly perform vision-language understanding and image generation and editing. Recent approaches [ 6 , 35 , 39 , 22 , 7 , 12 ]... |
arxiv | 2607.08449 | null | 1 | Predicting Viticulture Potential through an Ensemble of U-Net and a Geospatial Foundation Model | [
"Jorge Ignacio Perez",
"Hwaai Kang Kee",
"Lucas Rassbach"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08449v1 | 2026-07-09T00:00:00 | [
{
"type": "para",
"text": "Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0)."
},
{
"type": "para",
"text": "CLEF 2026 Working Notes, 21 – 24 September 2026, Jena, Germany"
},
{
"type": "para",
"text": "[ o... |
arxiv | 2607.08400 | null | 1 | TRACE: A Two-Channel Robust Attribution Watermark via Complementary Embeddings for LLM-Agent Trajectories | [
"Zheng Gao",
"Xiaoyu Li",
"Xiaoyan Feng",
"Jiaojiao Jiang",
"Yang Song",
"Yulei Sui",
"Zhenchang Xing",
"Liming Zhu"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08400v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Large language model agents no longer merely produce text: they invoke search APIs, file tickets, send messages, book services, execute code, and respond to security incidents ( yao2022react ; schick2023t... |
arxiv | 2607.08384 | null | 1 | Revisiting One-Zero and Two-Zero Neutrino Mass Textures in Light of Recent Oscillation and Cosmological Data | [
"Haruto Kitagawa",
"Coh Miyao",
"Satsuki Nishimura",
"Hajime Otsuka"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08384v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "The Standard Model (SM) of particle physics, completed by the discovery of the Higgs boson in 2012 ATLAS:2012yve ; CMS:2012qbp , is a remarkably successful theory that describes most current experimental ... |
arxiv | 2607.08370 | null | 1 | Tubular Neighbourhoods of Pfaffian Sets and Applications to Neural Networks | [
"Paul Lezeau",
"Martin Lotz"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08370v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1. Introduction"
},
{
"type": "para",
"text": "Pfaffian functions are functions that satisfy triangular systems of first-order partial differential equations with polynomial coefficients. The sets they define generalise algebraic and semi-algebraic s... |
arxiv | 2607.08349 | null | 1 | Certified Interventional Fidelity: Anytime-Valid, Adaptive Evaluation of Causal Claims in Mechanistic Interpretability | [
"Amir Asiaee"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08349v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Deep neural networks can match or exceed human performance, but understanding how they compute remains challenging. Mechanistic interpretability aims to provide explanations that are faithful simplificati... |
arxiv | 2607.08347 | null | 1 | Prediction-Powered Active Testing | [
"Kianoosh Ashouritaklimi",
"Valentin Kilian",
"Daolang Huang",
"Tom Rainforth",
"François Caron"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08347v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Active learning ( settles2009active ; al_model_data_survey ; selective_sampling_networks ; queries_concept_learn ) reduces the cost of training, but in many applications the cost of evaluation is just as ... |
arxiv | 2607.08335 | null | 1 | Bayesian Experimental Design via Score Matching | [
"Angus Phillips",
"Gavin Kerrigan",
"Tom Rainforth"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08335v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "The optimal design of experiments is critical for conducting informative and cost-effective scientific experiments in any field. Bayesian Experimental Design (BED) (Lindley, 1956 ; Chaloner and Verdinelli... |
arxiv | 2607.08303 | null | 1 | Learning $\mathsf{AC}^0$ under Locally Sampleable Graphical Models | [
"Weiming Feng",
"Xiongxin Yang",
"Yixiao Yu",
"Yiyao Zhang"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08303v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Learning small-depth Boolean circuits is a classical and fundamental problem in learning theory. In a seminal result, Linial, Mansour and Nisan [ LMN93 ] showed that every polynomial-size 𝖠𝖢 0 \\mathsf{... |
arxiv | 2607.08256 | null | 1 | Best-of-$N$ TTS Evaluation is Confounded by ASR Family Alignment | [
"Taehyung Yu",
"Seongjae Kang"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08256v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Recent flow-matching zero-shot TTS systems—F5-TTS (Chen et al. , 2025 ) , E2 TTS (Eskimez et al. , 2024 ) , CosyVoice 2 (Du et al. , 2024 ) , MaskGCT (Wang et al. , 2025 ) , Seed-TTS (Anastassiou et al. ,... |
arxiv | 2607.08241 | null | 1 | Closing the Null Space: Guidance-Aware Quantization for Classifier-Free Diffusion | [
"Abdullah Al Shafi",
"Sumaiya Rahim Suma"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08241v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Diffusion models [ 1 ] have become the dominant framework for high-quality conditional image generation, with classifier-free guidance (CFG) [ 2 ] providing the primary control mechanism at inference time... |
arxiv | 2607.08234 | null | 1 | RhyMix: A Lightweight Adaptive Multi-Rhythm Network for Long-Term Time Series Forecasting | [
"Sumit Satishrao Shevtekar",
"Chandresh Kumar Maurya"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08234v1 | 2026-07-09T00:00:00 | [
{
"type": "para",
"text": "Lightweight dual-path forecasting architecture (40K parameters) integrating explicit multi-period cyclic priors"
},
{
"type": "para",
"text": "Multi-Scale Temporal Convolution with Channel Attention (MSTCN-CA) for contextual temporal representation learning"
},
{
... |
arxiv | 2607.08202 | null | 1 | PIT-SUN: A Deployable Empirical Marginal Transform Framework with Expectation-Consistent Recovery for Regression in Recommender Systems | [
"Mingyu Zhao",
"Zhaohan Li",
"Zhenxiong Miao",
"Xu Zhang",
"Dewei Leng",
"Yanan Niu",
"Kun Gai"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08202v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "Introduction"
},
{
"type": "para",
"text": "Many recommender tasks—including dwell time, GMV, and LTV forecasting—predict continuous business values for ranking, allocation, and multi-objective decisions. Here, dwell time denotes user video watching ... |
arxiv | 2607.08193 | null | 1 | Open-ended Multi-agent Autocurricula via Visual Inspection of Policies with Multi-modal LLMs | [
"Lorenzo Pantè",
"Andrea Fanti",
"Roberto Capobianco"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08193v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Many real-world Reinforcement Learning (RL) tasks are difficult or impossible to solve by simply starting from a random policy and directly trying to optimize its downstream performance (Wang et al. , 201... |
arxiv | 2607.08168 | null | 1 | MuScriptor: An Open Model for Multi-Instrument Music Transcription | [
"Simon Rouard",
"Michael Krause",
"Axel Roebel",
"Carl-Johann Simon-Gabriel",
"Alexandre Défossez"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08168v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "The task of Automatic Music Transcription (AMT) consists of converting an audio recording of a piece of music into some kind of symbolic representation, typically MIDI. While significant progress has been... |
arxiv | 2607.08124 | null | 1 | TTHE: Test-Time Harness Evolution | [
"Jun Nie",
"Yonggang Zhang",
"Jun Song",
"Qianshu Cai",
"Dahai Yu",
"Yike Guo",
"Xinmei Tian",
"Bo Han"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08124v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Modern LLM agents are increasingly built as executable systems rather than single-call predictors: they retrieve context, call tools, execute code, inspect intermediate states, and recover from failures. ... |
arxiv | 2607.08122 | null | 1 | Workload-Preserving Differentially Private Synthetic Data for Causal Inference via Maximum-Entropy Calibration | [
"Amir Asiaee",
"Kaveh Aryan"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08122v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Differential privacy is increasingly used for public release of sensitive tabular datasets in domains such as health, education, and social science [ dwork2014foundations ] . A common deployment pattern i... |
arxiv | 2607.08109 | null | 1 | Contrastive Order Learning: A General Framework for Ordinal Regression | [
"Chaewon Lee",
"BeomJun Shim",
"Kwang Pyo Choi",
"Chang-Su Kim"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08109v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Ordinal regression is a task to estimate the discrete or continuous rank of an object instance. For example, facial age estimation aims to predict a person’s age given their facial photograph, while image... |
arxiv | 2607.08107 | null | 1 | BACH: A Bayesian Admixture of Contrastive Heads for Multi-Interest Two-Tower Retrieval | [
"Quoc Phong Nguyen",
"Paul Albert",
"Long Vuong",
"Vuong Le",
"Julien Monteil"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08107v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Two-tower retrievers have become the workhorse of large-scale candidate generation in recommender systems and dense information retrieval [ 6 , 26 , 14 ] . The standard formulation encodes a user (or quer... |
arxiv | 2607.08103 | null | 1 | Stochastic Order Learning: An Approach to Rank Estimation Using Noisy Data | [
"Chaewon Lee",
"Seon-Ho Lee",
"Chang-Su Kim"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08103v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Rank estimation — a task to predict the rank or ‘ordered class’ of an object — is a fundamental problem in machine learning, with applications including facial age estimation (Ricanek and Tesafaye, 2006 ;... |
arxiv | 2607.08093 | null | 1 | CausalDS: Benchmarking Causal Reasoning in Data-Science Agents | [
"Andrej Leban",
"Yuekai Sun"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08093v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Modern LLMs are increasingly powerful in agentic settings and are routinely used in data-science workflows (Jing et al. , 2025 ; Chan et al. , 2025 ; Gu et al. , 2024 ; Majumder et al. , 2025 ) . Their ac... |
arxiv | 2607.08084 | null | 1 | ConRad: Efficient Conformal Prediction for Radiomics | [
"Matt Y. Cheung",
"Ashok Veeraraghavan",
"Guha Balakrishnan"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08084v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Radiomic measurements derived from segmentation masks are increasingly used as downstream quantities in medical imaging pipelines. These measurements, including shape, intensity, and texture features, are... |
arxiv | 2607.08063 | null | 1 | Holographic Neural PCFG for Unsupervised Parsing | [
"Ryosuke Yamaki",
"Daichi Mochihashi",
"Nobutaka Shimada",
"Tadahiro Taniguchi"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08063v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Unsupervised constituency parsing is the task of inducing the syntactic tree structure of natural language from raw text alone without any annotations. This task is not merely a benchmark for parsing accu... |
arxiv | 2607.08059 | null | 1 | When Thinking Hurts: Epistemic Signals in the Reasoning Chains of Visual Language Models | [
"Mayank Singal"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08059v1 | 2026-07-09T00:00:00 | [
{
"type": "para",
"text": "marginparsep has been altered. topmargin has been altered. marginparpush has been altered. The page layout violates the ICML style. Please do not change the page layout, or include packages like geometry, savetrees, or fullpage, which change it for you. We’re not able to reliably ... |
arxiv | 2607.08054 | null | 1 | Who Analyses the Analyser? Self-Validating LLM Hazard Analysis with Constitutional Meta-STPA | [
"Samuel Tetteh",
"Udip Shrestha",
"Joshua R. Waite",
"Cody Fleming"
] | CC0-1.0 | https://arxiv.org/html/2607.08054v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Large language models (LLMs) have crossed a threshold. In the span of a few years they have moved from producing fluent text to shaping consequential decisions: prompted to reason step by step [ 21 ] , th... |
arxiv | 2607.08041 | null | 1 | An exact information theory of generalization phase transitions in Bayesian diffusion models | [
"Henry Hunt",
"Mason Kamb",
"Surya Ganguli"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08041v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction and related work"
},
{
"type": "para",
"text": "Remarkably, generative AI can now learn complex distributions over high dimensional spaces with tractable amounts of training data. For example, diffusion models [ 1 , 2 , 3 ] robustly an... |
arxiv | 2607.08032 | null | 1 | What to Keep, What to Forget: A Rate--Distortion View of Memory Compaction in LLMs and Agents | [
"Ashwin Gerard Colaco",
"Nada Lahjouji"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08032v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1. Introduction"
},
{
"type": "para",
"text": "Run a transformer over a hundred thousand tokens and the cached keys and values of everything it has already read take up more of the accelerator than the model’s own weights (Kwon et al. , 2023 ; Liu et... |
arxiv | 2607.08029 | null | 1 | Rethinking Small VLM Quantization: From Component-Wise Analysis to Hardware-Aware Edge Deployment | [
"Hyeju Shin",
"Chorwon Kim",
"Ryangsoo Kim",
"Hark Yoo",
"Jaein Kim"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08029v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Recently, large multimodal models (LMMs) have opened a new horizon in artificial intelligence (AI) research owing to their innovative reasoning capabilities, which combine visual understanding and languag... |
arxiv | 2607.08012 | null | 1 | Provably Optimal Learning Algorithms for Assistance Games | [
"Nivasini Ananthakrishnan",
"Mark Bedaywi",
"Michael I. Jordan",
"Stuart Russell",
"Nika Haghtalab"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08012v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Consider a repeated interaction between two cooperative agents who share a common objective but have asymmetric access to information. One agent observes a changing latent state—such as a preference, type... |
arxiv | 2607.08007 | null | 1 | Unit-Independent Low-Rate Wrist GSR Processing for Stress Detection Using Phasic nSCR Features | [
"Zequan Liang",
"Sally Hang",
"Geneva M. Jost",
"Ning Miao",
"Wei Shao",
"Mahdi Pirayesh Shirazi Nejad",
"Hossein Sayadi",
"Ehsan Kourkchi",
"Setareh Rafatirad",
"Camelia E. Hostinar",
"Houman Homayoun"
] | CC-BY-4.0 | https://arxiv.org/html/2607.08007v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Galvanic skin response (GSR), also known as electrodermal activity (EDA), is a widely used physiological signal that reflects sweat gland activation through changes in skin conductance [ 1 ] . Because GSR... |
arxiv | 2607.07996 | null | 1 | SpO$_2$ Predictor-Guided Stage-Wise Time-Frequency Reconstruction of Low-Quality Dual-Wavelength PPG for Oxygen Saturation Estimation | [
"Zequan Liang",
"Elahe Hosseini",
"Ning Miao",
"Mahdi Pirayesh Shirazi Nejad",
"Wei Shao",
"Ehsan Kourkchi",
"Setareh Rafatirad",
"Houman Homayoun"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07996v1 | 2026-07-09T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "I Introduction"
},
{
"type": "para",
"text": "Oxygen saturation (SpO 2 ) is an important physiological indicator for monitoring respiratory and cardiovascular status [ 3 ] . Conventional pulse oximetry estimates SpO 2 from dual-wavelength photoplethy... |
arxiv | 2607.07978 | null | 1 | A Quantum Reservoir Architecture for Chaotic Forecasting and a Test of Whether Its High Dimension Helps | [
"Tushar Pandey"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07978v1 | 2026-07-08T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Reservoir computing replaces the gradient-trained recurrent network with a fixed nonlinear dynamical system whose state is read out by a trainable linear map. The two design choices that survive in such a... |
arxiv | 2607.07957 | null | 1 | Evaluating the Effect of Frame Rate in Sequence-Based Classification of Autism-Related Self-Stimulatory Hand Idiosyncrasies | [
"Raunak Mondal",
"Peter Washington"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07957v1 | 2026-07-08T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Autism spectrum disorder (ASD) is a neurodevelopmental condition characterized by differences in social communication and the presence of restricted, repetitive behaviors [ 2 ] . Current estimates place w... |
arxiv | 2607.07953 | null | 1 | Linear Attention Architectures: Mechanisms, Trade-offs, and Cross-Layer Routing | [
"Tommaso Cerruti",
"Tim Rieder",
"George Rowlands",
"Lingfeng Jin",
"Imanol Schlag"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07953v1 | 2026-07-08T00:00:00 | [
{
"type": "para",
"text": "tommasocerruti/linear-attention-architectures"
},
{
"type": "para",
"text": "Keywords Linear attention ⋅ \\cdot Recurrent associative memory ⋅ \\cdot DeltaNet ⋅ \\cdot Cross-layer routing"
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
}... |
arxiv | 2607.07935 | null | 1 | path_boost: A Python Package for Interpretable Graph-Level Prediction using Path-Based Gradient Boosting | [
"Claudio Meggio",
"Johan Pensar",
"Riccardo De Bin"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07935v1 | 2026-07-08T00:00:00 | [
{
"type": "para",
"text": "[1,] \\fnm Claudio \\sur Meggio"
},
{
"type": "para",
"text": "1] \\orgdiv Department of Mathematics, \\orgname University of Oslo, \\orgaddress \\city Oslo, \\postcode 0371, \\state Oslo, \\country Norway"
},
{
"type": "heading",
"level": 1,
"text": "1... |
arxiv | 2607.07847 | null | 1 | When Does Continual Learning Require Learning | [
"Anne Harrington",
"Nayan Saxena",
"Michael Murphy",
"Anastasia Borovykh",
"Zeyu Yun",
"Sridhar Kamath",
"Ara Eindra Kyi",
"Trevor Darrell",
"Jitendra Malik",
"Yutong Bai"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07847v1 | 2026-07-08T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Traditionally, continual learning has been defined as mitigating catastrophic forgetting [ 40 ] : how do we learn task B B without forgetting task A A ? One approach to this problem is to augment or compo... |
arxiv | 2607.07707 | null | 1 | Co-LMLM: Continuous-Query Limited Memory Language Models | [
"Yair Feldman",
"Linxi Zhao",
"Nathan Godey",
"Dongyoung Go",
"Yilun Hua",
"Kilian Q. Weinberger",
"Jennifer J. Sun",
"Yoav Artzi"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07707v1 | 2026-07-08T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Recently, there has been increasing interest in large language models (LLMs) that are trained to externalize knowledge (Ghosal et al., 2025 ; Zhao et al., 2026 ; Pouransari et al., 2026 ) . A particularly... |
arxiv | 2607.07690 | null | 1 | Agon: Competitive Cross-Model RL with Implicit Rival Grading of Reasoning | [
"Vladislav Beliaev"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07690v1 | 2026-07-08T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Reinforcement learning from verifiable rewards has become the standard tool for sharpening the reasoning of large language models (LLMs) on tasks such as mathematics and code (Guo et al., 2025 ; Shao et a... |
arxiv | 2607.07674 | null | 1 | Max Out GRPO Signal: Adaptive Trace Prefix Control for Hard Reasoning Problems | [
"Vladislav Beliaev"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07674v1 | 2026-07-08T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Reinforcement learning from verifiable rewards is now the standard tool for improving the reasoning of large language models (LLMs) on tasks such as mathematics and code (Guo et al., 2025 ; Lambert et al.... |
arxiv | 2607.07673 | null | 1 | MedPMC: A Systematic Framework for Scaling High-Fidelity Medical Multimodal Data for Foundation Models | [
"Hyunjae Kim",
"Dain Kim",
"Pan Xiao",
"Serina S. Applebaum",
"Younjoon Chung",
"Xuguang Ai",
"Yu Yin",
"Roy Jiang",
"Yuexi Du",
"Yawen Wei",
"Yiming Kong",
"Tuo Guo",
"Zhiyuan Cao",
"Mengmeng Du",
"Yuelei Fu",
"Yan Hu",
"Rui Shi",
"Gui Yang",
"Kevin W. Jin",
"Yuntian Liu",
"... | CC-BY-4.0 | https://arxiv.org/html/2607.07673v1 | 2026-07-08T00:00:00 | [
{
"type": "para",
"text": "\\ul"
},
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Medicine is multimodal in nature [ 2 , 68 ] . Triage, diagnosis, prognosis, and treatment planning routinely require clinicians to integrate images, text, l... |
arxiv | 2607.07670 | null | 1 | Does Bielik Know What It Doesn't Know? Activation Dispersion Separates Entity Familiarity from Factual Reliability Across Model Scale | [
"Grzegorz Brzezinka"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07670v1 | 2026-07-08T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "When an LLM is asked about something it does not know, does its internal state look different from when it is on familiar ground? A physically motivated intuition says yes: knowledge retrieval should beha... |
arxiv | 2607.07665 | null | 1 | Guidance Breaks the Fitted Operator: A Terminal-Fitted Repair for Classifier-Free Guidance | [
"Shiheng Zhang"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07665v1 | 2026-07-08T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Diffusion and flow-matching samplers integrate a probability-flow ODE whose velocity field stiffens as the noise scale σ → σ min \\sigma\\to\\sigma_{\\min} : on a coordinate normal to the data manifold th... |
arxiv | 2607.07557 | null | 1 | PALS: Percentile-Aware Layerwise Sparsity for LLM Pruning | [
"Yazdan Jamshidi",
"Alexey Shvets"
] | CC-BY-4.0 | https://arxiv.org/html/2607.07557v1 | 2026-07-08T00:00:00 | [
{
"type": "heading",
"level": 1,
"text": "1 Introduction"
},
{
"type": "para",
"text": "Large language models routinely reach billions of parameters. A 7B-parameter model takes roughly 13 GB in half precision and processes billions of multiply-accumulate operations per token, which puts real... |
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