screening-ceiling

📖 Documentation site — the portfolio narrative, the concepts, a full walkthrough, and what all of this proves (and does not).

A machine-certified impossibility result about coupling extraction, plus the concrete layouts where a plausible extractor predicts impossible physics.

Most ML-for-physics datasets are samples: here are some inputs, here are the answers, fit something. This one is different in a way worth being precise about. It carries a universal claim — a statement about every layout in a continuous four-parameter family, established by interval branch-and-bound rather than by sampling — together with existential counterexamples that refute a specific competing method.

Why this exists

When conductors are packed together, every other conductor screens the field between any two. So a pair's mutual capacitance inside an array is at most its isolated-pair value:

k = |C_full| / |C_iso|  ≤  1

Pairwise-superposition extraction — used throughout fast parasitic extraction — assumes k ≡ 1. The certified claim here is that on a particular manufacturable family it is provably never right, and quantifies by how much:

For every layout in the family box, k ≤ 0.909090909091. A pairwise extractor therefore over-predicts the worst coupling by at least 10.000002% on every member of the family — not on average, not usually, always.

And the other direction: a second-order Born correction, the obvious cheap fix when a full solve is too slow, does not merely stay inaccurate — on 27 layouts here it predicts k > 1, which is anti-screening, a physical impossibility.

30-second quickstart

No install, no dependencies, nothing from the repository that produced it:

python3 verify.py

screening-ceiling  independent re-derivation (stdlib only)

  claim              k <= 0.909090909091 for every layout in the family
  forced error       pairwise over-predicts by >= 10.0000%

  regions            256 x 25 samples = 6400 layouts
  worst sampled k    0.903974725909   (bound 0.909090909091)
  margin to bound    0.005116183182
  violations         0
  worst at           region 3.0.3.3  d0_um=55.7830 jog_mult=0.3838 pt_mult=1.0009 sep_mult=4.4492

  counterexamples    27/27 re-derived and confirmed above the ceiling

  consistent: no sampled layout exceeds the bound, and every counterexample re-derives

  scope: A complete interval theorem about the frozen MONOPOLE-CLOSURE model only. The closure-vs-BEM/PDE model gap remains additive and unresolved. This witness does not establish Maxwell, BEM, driven-S, fabrication, or measured-silicon truth.

verify.py rebuilds the electrostatics from the published geometry using the standard library alone — its own Gauss-Jordan inverse, its own potential matrix. It does not import this dataset's producer, numpy, or anything else.

Sampling cannot prove the universal claim. That is what the interval branch-and-bound in the source proof is for. What sampling can do is refute it, and that is the useful thing to hand a skeptical reader: a cheap, dependency-free way to try to catch us being wrong.

Loading

from loader import load_regions, load_counterexamples, load_theorem

theorem = load_theorem()
print(theorem["statement"])
print(theorem["honest_scope"])          # read this one

for c in load_counterexamples():
    print(c["case_id"], c["k_predicted"], c["n_pairs_violating"], "/", c["n_pairs_total"])

Optional conveniences: to_pandas("regions") and to_hf_dataset().

Contents

data/certified_regions.jsonl — 256 rows, the universal claim

The branch-and-bound partition. Each row is one region of the family box that the prover certified, having subdivided it into leaves until the interval enclosure of k fell below the bound everywhere inside.

Field	Type	Meaning
region_id	string	position in the 4×4×4×4 root partition, e.g. 3.0.3.3
bounds	object	{d0_um, pt_mult, sep_mult, jog_mult} → {lo, hi}
status	string	CERTIFIED for every row in this release
certified_leaves	int	leaves the region was subdivided into
processed_leaves	int	leaves examined, including interior splits
sup_certified_k_hi	float	largest k the enclosure admits anywhere in the region
volume_fraction_of_region	float	fraction certified (1.0 throughout)
unresolved_leaves	int	leaves left undecided (0 throughout)

Totals: 237,490 certified leaves, 474,724 processed, 0 failure regions, 0 unresolved, certified volume fraction 1.0.

The published rows are the 256-region partition, not all 237,490 leaves — the source proof records per-region certification and aggregate counts. That is a real limitation of what is published and it is stated rather than glossed: you can re-derive any region yourself, but you are not being handed every leaf.

data/counterexamples.jsonl — 27 rows, the existential refutation

Layouts where born_second_order predicts k > 1.

Field	Type	Meaning
case_id	string	e.g. born2_n6_p100_s1
model	string	born_second_order
n_conductors	int	6, 8 or 12
nominal_pitch_um	float	60, 80 or 100
seed	int	generator seed
worst_pair	[int, int]	indices of the worst-violating pair
k_predicted	float	predicted screening factor (> 1 for every row)
n_pairs_violating	int	pairs above the ceiling in this layout
n_pairs_total	int	ordered pairs in this layout
xy_um	[[float, float]]	conductor centres, micrometres
radius_um	[float]	conductor radii, micrometres
eps_r	float	relative permittivity (4.6)

2,060 violating pairs across the 27 layouts, worst k = 3.5141. Full coordinates are included deliberately: a counterexample you cannot rebuild is an anecdote, not evidence.

data/theorem.json — the claim, its scope, its provenance

The statement, the family box, the certified totals, the exact geometry definition, and the SHA-256 of the source proof witness.

Geometry

Four parallel circular conductors forming two tight pairs:

pitch      = 1.6 · d0 · pt_mult
separation = pitch · sep_mult
jog        = jog_mult · separation

centres: (0,0)  (pitch,0)  (separation,jog)  (separation+pitch,jog)
every conductor has diameter d0

Family box: d0 ∈ [25,60] µm, pt_mult ∈ [1.0,1.2], sep_mult ∈ [2.5,4.5], jog_mult ∈ [−0.4,0.4]. Self-term radius scale 1.0.

loader.family_layout(...) builds it for you.

Scope, honestly

This is a theorem about the monopole-closure model, not about Maxwell. The closure is a zero-parameter analytic multiple-scattering model that matches a boundary-element reference to 0.081% in the exact two-cylinder limit, but the closure-versus-solver gap is an additive, disclosed, unresolved term. It is never absorbed into the bound.

That 0.081% comes from the boundary-element solver used to develop the closure, which is not part of this release — so unlike every other figure on this page, you cannot re-derive it from what is published here. It is quoted because it bounds how much trust the closure has earned, and a reader is entitled to know which numbers are checkable and which are taken on our word.

The scope line travels with the data, in theorem.json:

A complete interval theorem about the frozen MONOPOLE-CLOSURE model only. The closure-vs-BEM/PDE model gap remains additive and unresolved. This witness does not establish Maxwell, BEM, driven-S, fabrication, or measured-silicon truth.

Three further limits worth stating plainly:

• One family, not all layouts. Four conductors in a specific arrangement. Nothing here says anything about a different topology.
• The bound is not tight. The certified supremum is 0.90909089; the worst layout found by adversarial search is ≈0.9053. The gap is the price of a first-order interval relaxation, not a claim about physics. That 0.9053 is the second figure on this page you cannot re-derive from the release — it came from a differential-evolution search in the source prover. What you can check here is weaker but points the same way: verify.py reports a worst sampled k of 0.903974725909 at its defaults, and sampling harder climbs toward that 0.9053 without ever reaching the bound — at --seed 7, 25 / 100 / 400 samples per region give 0.902144353337, 0.903775408593, 0.904308125283, with zero violations at each.
• No measured data. Every number is computational.

Reproduction

python3 verify.py --samples 100 --seed 7   # sample harder, different seed
python3 verify.py --self-test              # prove the checker still discriminates
python3 export.py --check                  # confirm data matches a fresh export

The self-test is the part that makes a clean report worth anything. It fabricates an impossible bound, tampers with a published value, and requires the checker to reject both — then confirms an isolated pair reproduces k = 1.000000000000 exactly, since a lone pair has nothing to screen it.

Troubleshooting

verify.py reports violations — that is the interesting outcome, and we want to hear about it. Sampling cannot prove the bound but it can refute it, so a genuine violation means the theorem is wrong. Before reporting, re-run with --self-test to confirm the checker still discriminates: a checker that has stopped working can produce either verdict.

ModuleNotFoundError: numpy from verify.py — it should never import numpy. If it does, the file has been edited; the shipped version runs under env -i /usr/bin/python3 with nothing installed, and a test asserts it imports no third-party module.

loader.to_pandas or to_hf_dataset raises ImportError — those two are conveniences and do need pandas / datasets. Everything else, including verify.py, is standard library only.

The Hub viewer shows two configs and you wanted one table — regions and counterexamples have different schemas and are deliberately separate. Pick the config in the viewer's dropdown, or use load_regions() / load_counterexamples().

export.py --check says the data does not match — it re-derives the files from the committed proof witness and compares. A mismatch means either the data or the witness was edited. Counterexample regeneration also needs maxwell-lint installed, since they are produced by running its reference models.

You want every certified leaf, not the 256 regions — they are not published. The source proof records per-region certification plus aggregate counts, so the 237,490 leaves are attested but not enumerated here. That is a real limit of this release and is stated rather than glossed.

Provenance

Exported by export.py from a committed proof witness produced by an outward-rounded interval branch-and-bound prover (256 parallel roots, 243.5 s wall clock, centered/mean-value enclosure forms). The witness digest is recorded in theorem.json; export.py --check re-derives the files and compares.

Counterexamples are generated by running the open-source maxwell-lint reference models, so they are reproducible from published code alone.

Citation

@misc{screening_ceiling_2026,
  title  = {Screening Ceiling: Certified Regions and Counterexamples for
            Many-Body Coupling Extraction},
  author = {ChipletOS / Genesis contributors},
  year   = {2026},
  note   = {CC-BY-4.0}
}

The rest of the toolkit

Eight artifacts that answer one question in different places: is this model physically possible? Each is a grader — it can tell you a model is wrong; none can tell you one is right.

sparam-lint	Is an S-parameter model physically possible? Five laws + a negative control.
maxwell-lint	Does a coupling extractor predict impossible physics? Screening ceiling k ≤ 1.
abstain-bench	Does a model know when to shut up? Abstention recall, never pooled with accuracy.
sparam-conformance	11 labelled networks with verified ground truth. Grades the graders.
screening-ceiling ← you are here	A certified impossibility result + 27 counterexamples. Zero-dependency verifier.
physics-lint-action	The same checks, in your CI.
physics-lint-mcp	A physics oracle your AI agent can call.
Try it in your browser	All three checks, no install, runs client-side.

These tools grade a model. Producing one that is passive by construction — so it cannot fail these laws whatever its parameters — and accurate at speed in the many-body regime, with calibrated abstention and a fail-closed signoff certificate, is the commercial core: ChipletOS.

Licence

CC-BY-4.0. Synthetic and computational throughout; no proprietary or measured data. Attribution: ChipletOS / Genesis contributors.

Related

• maxwell-lint — run the ceiling test on your extractor
• sparam-conformance — the S-parameter analogue
• ChipletOS — the closed core: a learned many-body coupling operator that stays inside this ceiling and is accurate at speed, with calibrated abstention and a fail-closed signoff certificate

Contributing

One non-negotiable rule here: the verifier must import nothing from this project, so a skeptic can read it in one sitting and run it with nothing installed. CONTRIBUTING.md has the detail. Each sibling repository states its own, and they differ — that is deliberate, and it is why each is trustworthy on its own terms.