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+# SnapKitty Topological Quantum Computing
+
+**For quantum researchers. Every claim is labeled.**
+
+---
+
+## What Is Here
+
+SnapKitty contains Lean 4 formalizations of topological quantum computing (Fibonacci anyons, braid group, gate synthesis) and classical simulations of the same systems. None of this runs on physical quantum hardware.
+
+---
+
+## Established Theory (Correctly Implemented)
+
+### Fibonacci Anyon Fusion Category
+**Source:** `FibonacciAnyon.lean`  
+**Status:** ESTABLISHED THEORY — correct formalization
+
+The Fibonacci anyon model has two objects: 1 (trivial) and τ. Fusion rules:
+```
+1 ⊗ x = x ⊗ 1 = x    (trivial fuses transparently)
+τ ⊗ τ = 1 ⊕ τ         (non-abelian fusion)
+```
+
+Quantum dimension of τ: `d_τ = (1 + √5)/2 = φ` (golden ratio). **Implemented correctly** — `quantumDim FibObject.tau = (1 + Real.sqrt 5) / 2`.
+
+Hilbert space dimension for n anyons: follows Fibonacci sequence. `hilbertSpaceDim n = 2^(n-2)` for n ≥ 3. **Proved** by `fibonacci_dimension_recurrence`.
+
+**R-matrix:** Braiding eigenvalue for τ⊗τ: `e^(i·4π/5)`. Unitarity |R| = 1. **Proved** by `rmatrix_unitary`.
+
+**What is NOT proved:** Pentagon and hexagon axioms (consistency conditions for the fusion category) are stated as `axiom … True` — no content. These are the core axioms of any modular tensor category and are required for the model to be physically consistent.
+
+### Braid Group Representation
+**Source:** `BraidCompilation.lean`  
+**Status:** ESTABLISHED THEORY (structure) + SPEC (not proved)
+
+Braid group B_n presentation: generators σ₁,...,σ_{n-1} with:
+- Far commutativity: σᵢσⱼ = σⱼσᵢ for |i-j| ≥ 2
+- Yang-Baxter: σᵢσⱼσᵢ = σⱼσᵢσⱼ for |i-j| = 1
+
+Both are stated — **neither is proved** in the Lean file (both have `sorry`).
+
+R-move period 5: `(e^{i4π/5})^5 = 1`. **PROVED.**
+
+Solovay-Kitaev approximation (that any SU(2) unitary can be approximated to precision ε by a braid word of length O(poly(log(1/ε)))): **stated, not proved** (`sorry`).
+
+### Physical Realizations
+`topological_protection` and `physical_realization_12_5` (fractional quantum Hall at ν=12/5) are stated as `axiom True` — completely empty.
+
+---
+
+## Implementation (Classical Simulation)
+
+### Fibonacci Anyon Simulation
+**Source:** `carry-agent/quantum/topological.rs` (inferred location)  
+**Type:** CLASSICAL SIMULATION — explicitly disclaimed in the file  
+**Status:** Working, with tests
+
+Implements:
+- Quantum dimension φ = (1+√5)/2 for Fibonacci, √2 for Ising
+- `apply_braid`: anyon swap for B₃ generators σ₁, σ₂
+- `fuse_anyons`: fusion with correct probabilities
+  - τ ⊗ τ → 1 with probability 1/φ² (≈ 0.382)
+  - τ ⊗ τ → τ with probability 1 - 1/φ² (≈ 0.618)
+- Braid history tracking
+
+The fusion probabilities are physically correct per established Fibonacci anyon theory. The implementation is a deterministic simulation using pseudo-random draws for the outcome.
+
+### Vortex Lattice Simulation
+**Source:** `quantum-wasm/pkg/quantum_wasm_bg.wasm` (44KB compiled binary)  
+**Type:** CLASSICAL SIMULATION  
+**Status:** Working compiled WASM binary
+
+Exposes:
+- `topological_charge()`: integer (winding number sum over lattice)
+- `vortex_winding(i)`: winding number of vortex i
+- `vortex_coherence(i)`: coherence measure of vortex i
+- `Simulation.step()`: Trotter time evolution step
+
+The "topological charge" here is the classical topological charge of a vortex configuration — an integer winding number, not a quantum topological charge in the sense of anyonic braiding statistics. These are related concepts (both use the word "topological") but are distinct:
+
+| Concept | Context | Meaning |
+|---------|---------|---------|
+| Anyon topological charge | TQC theory | Labels the superselection sector; determines braiding statistics |
+| Vortex topological charge | Classical field theory | Winding number ∮(∇φ)·dl / 2π; integer invariant of a vortex configuration |
+
+The vortex simulation computes the second. The Lean formalization addresses the first.
+
+---
+
+## Braid Group As Access Control (Not TQC)
+**Source:** `carry-agent/braid.rs`  
+**Type:** CLASSICAL APPLICATION OF BRAID MATHEMATICS
+
+B₃ over three authority strands (Curry, Crystal, C3). The writhe (sum of signed crossing numbers) is used as an integrity invariant. Reidemeister-I cancellation (σᵢ · σᵢ⁻¹ = e) is used for inverse detection.
+
+This uses braid group mathematics in a software pipeline context — not topological quantum computing. The analogy is precise (braid words, writhe, Reidemeister moves are all correct), but the application domain is access control, not quantum error correction.
+
+---
+
+## GJW Wormhole Protocol (Experimental)
+**Source:** `bob-agent-hackathon-2/mqs/logic/gjw_traversability.pl`  
+**Type:** HYPOTHESIS — uses established theory as metaphor  
+**Status:** Prolog code (runnable), concept is experimental
+
+Compiles (σ₁ σ₂⁻¹)^N braid words for "ER bridge traversal" using Fibonacci anyon quantum dimensions. The Gao-Jafferis-Wall (GJW) traversable wormhole protocol (2017, 2019) is a real theoretical proposal in quantum gravity. This Prolog implementation applies the braid structure computationally.
+
+**Label:** HYPOTHESIS. Whether a Prolog-compiled braid word constitutes a "traversable wormhole" in any physical or computational sense is an open question.
+
+---
+
+## Research Status Per Component
+
+| Component | Status Label | What's established | What's experimental |
+|-----------|-------------|-------------------|---------------------|
+| Fibonacci fusion rules | ESTABLISHED THEORY | τ⊗τ = 1⊕τ, quantum dim φ | — |
+| Hilbert space dimension | IMPLEMENTATION | Fibonacci recurrence, proved | — |
+| R-matrix unitarity | IMPLEMENTATION | |R|=1, proved | — |
+| Pentagon/hexagon axioms | SPEC (empty) | Required axioms of the category | Not proved |
+| Braid group structure | IMPLEMENTATION | Far commutativity, Yang-Baxter (structure) | Not proved in Lean |
+| Solovay-Kitaev | SPEC | Known theorem in TQC | Not proved here |
+| Gate synthesis (CNOT, H, T) | SPEC | Known results | Not proved here |
+| Classical anyon simulation | IMPLEMENTATION | Correct fusion probabilities | No error correction |
+| Vortex lattice WASM | IMPLEMENTATION | Classical topological charge | Relationship to TQC = open question |
+| GJW wormhole | HYPOTHESIS | GJW theory is real physics | This implementation: unverified |
+
+---
+
+## What A Quantum Researcher Should Take From This
+
+1. The Lean 4 formalizations are structurally correct for the known theory — fusion rules, quantum dimensions, R-matrix eigenvalues are all right.
+
+2. The pentagon and hexagon axioms (which are the consistency conditions that make the Fibonacci model a valid TQFT) are empty placeholders. Without these proved, the formalization cannot be said to correctly represent a valid modular tensor category in Lean.
+
+3. The classical simulations (carry-agent topological.rs, quantum-wasm) are physically accurate for the simulation layer and explicitly disclaim physical fault tolerance.
+
+4. The braid group application in carry-agent/braid.rs is a correct classical application of braid mathematics to a non-quantum domain.
+
+5. The "quantum swarm" does not use any of this quantum computing work at runtime. See QUANTUM_SWARM.md.
+
+6. **The strongest quantum contribution** is the ANU QRNG integration + formal Born Rule specification (`BornRuleCollapse.lean`), because it bridges real physical quantum entropy to formal specification.
+
+---
+
+## Open Research Questions
+
+1. Can the pentagon and hexagon axioms for the Fibonacci category be formally proved in Lean 4 using Mathlib?
+
+2. Is the SUBLEQ attention mechanism (see ARCHITECTURE.md) related to any known quantum information model? The Born-collapse aggregation uses the φ-weighting from quantum dimension; is this connection formal or analogical?
+
+3. Does the Jordan fixed-point commutativity theorem extend to the mixed-state case (non-pure ρ*) in a way that has implications for quantum error correction?
+
+4. What is the precise relationship between the vortex lattice topological charge and the Fibonacci anyon topological charge?