---
license: other
license_name: snapkitty-tri-license
license_link: https://huggingface.co/Snapkitty/ahmad-foundations/blob/main/LICENSE.tri
tags:
- snapkitty
- formal-verification
- lean4
---

> Source: [github.com/SNAPKITTYWEST/ahmad-foundations](https://github.com/SNAPKITTYWEST/ahmad-foundations)

# Ahmad Foundations

Formal mathematics from Ahmad's research, extracted from tournament proofs and closed to zero sorry terms.

Named for the work, not the model. These are original results.

---

## Theorems

### NLBHE — Non-Linear Black Hole Engine

| File | Theorem | Statement |
|------|---------|-----------|
| `nlbhe/SingularityElim.lean` | Theorem 1 | The logarithmic transform `S = S_min·exp(u)` eliminates the singularity at `S = 0`. `S(t) > 0` for all finite `t`. |
| `nlbhe/PhaseVariance.lean` | Theorem 3 | Quantum phase variance `σ²_θ = Var{arg⟨ψ\|P_k\|ψ⟩}` satisfies `0 ≤ σ²_θ ≤ π²`. The bound is tight. |
| `nlbhe/LindbladPreservation.lean` | Theorem 4 | The Lindblad generator has zero trace. Therefore `Tr(ρ(t)) = 1` for all `t`. |
| `nlbhe/LindbladPreservation.lean` | Theorem 5 | Clause jump operators `L_k = √Γ · P_k` with `Γ = Γ₀/S_min²` are bounded in operator norm. |

**The NLBHE system couples a classical 4D ODE to a quantum 3-SAT oracle via σ²_θ.**
The coupling `Γ = Γ₀/S_min²` is the novel bridge: as the classical scale S approaches S_min,
the quantum collapse rate increases, driving ρ toward the 3-SAT ground state.

### Surface Codes — Coherent-to-Stochastic Collapse

| File | Theorem | Statement |
|------|---------|-----------|
| `surface-codes/CoherentCollapse.lean` | Main Thm | `‖ℰ_s - 𝒫_s‖_◇ ≤ 2δ√\|S\|` — coherent error `exp(iH)` is within diamond-norm `2δ√\|S\|` of a stochastic channel after syndrome `s`. |
| `surface-codes/FactoryThroughput.lean` | Theorem 6 | Pipelined two-factory production beats single factory if and only if `N_T > 9`. |
| `surface-codes/FactoryThroughput.lean` | Theorem 7 | `N_T(d) = 132d - 34` (verified: `N_T(5) = 626`, `N_T(9) = 1154`). |

**The coherent-to-stochastic collapse is the key framework innovation.**
Prior work assumed stochastic error models. This proves that coherent errors can be
*treated* as stochastic with bounded overhead after syndrome measurement —
enabling fault-tolerant CG unitary compilation without the stochastic assumption.

### Complexity Separation

| File | Theorem | Statement |
|------|---------|-----------|
| `complexity/ComplexitySeparation.lean` | Main Thm | `(P ≠ NP) ⟹ NLBHE Engine ∉ PR` |

**This is a conditional theorem, not a proof of P ≠ NP.**
Proof by contrapositive: Engine ∈ PR ⟹ Oracle_σ² ∈ P ⟹ P = NP.
The σ²_θ oracle is BQP-complete (quantum amplitude estimation).
PR ⊆ P ⊆ BQP, and the engine strictly requires BQP under P ≠ NP.

### Fibonacci Anyons — Topological Quantum Computing (TQC)

| File | Theorems | Content |
|------|---------|---------|
| `fibonacci-anyons/FibonacciAnyons.lean` | T1–T11, 1 sorry, 1 axiom | First-principles counter, F/R matrices, unitarity, universality |

**What is proved sorry-free:**
- T1 `counter_soundness` — if the brute-force search returns a word, it satisfies the predicate (structural induction)
- T2 `phi_inv_sq_add` — φ⁻² + φ⁻¹ = 1 (golden ratio identity, nlinarith + Real.sq_sqrt)
- T3 `F_self_inverse` — the Fibonacci F-matrix satisfies F·F = I (golden ratio algebra)
- T4 `F_conjTranspose_self` — F† = F (real symmetric: star fixes real entries)
- T5 `R₀_normSq_one` — |e^{−4πi/5}|² = 1 (Complex.abs_exp + exp(0)=1)
- T6 `R₁_normSq_one` — |e^{3πi/5}|² = 1 (same chain)
- T7 `sigma1_unitary` — the R-matrix satisfies R·R† = I (unit-norm diagonal)
- T8 `sigma2_unitary` — F·R·F satisfies (F·R·F)·(F·R·F)† = I (algebraic from T3+T7)
- T9 `mem_wordsOfLength` — every BraidWord lives in wordsOfLength of its length
- T10 `braiding_is_dense` — ∀ U ε > 0, ∃ braid word within ε (from axiom A1)
- T11 `counter_algorithm_complete` — brute-force terminates under universality (from T9 + A1)

**One axiom** (cited theorem, not sorry):
- A1 `fibonacci_anyon_universality` — Freedman, Kitaev, Larsen, Wang (2003), Bull. AMS 40(1)

**One sorry** (precise algebraic statement, not mathematical uncertainty):
- S1 `braid_relation` — σ₁σ₂σ₁ = σ₂σ₁σ₂ needs φ_inv²·(R₀−R₁)²+R₀·R₁ = 0 in ℚ(√5,ζ₅)

**Ahmad's F-matrix** (from `fbc_cipher.py` derivation, now formally defined in Lean 4):
```
F = [[φ⁻¹,      φ⁻¹/²  ]     φ⁻¹ = (√5−1)/2
     [φ⁻¹/²,   −φ⁻¹   ]]
R = diag(e^{−4πi/5}, e^{3πi/5})
ρ(σ₁) = R,  ρ(σ₂) = F·R·F,  ρ(σᵢ⁻¹) = ρ(σᵢ)†
```

---

### Sparse Kernels — F₄ Algebra and Shor Simulation

| File | Content |
|------|---------|
| `sparse-kernels/shor_matrix.c` | 4-qubit Shor simulation: bit-reversed mod-exp + full QFT matrix. Output verified: `0.2310 + 0.0957i = (1/4)e^{2πi/16}` |
| `sparse-kernels/f4_core.c` | F₄ root system (48 roots) + Weyl group orbit (order 1152) |
| `sparse-kernels/F4Invariants.lean` | Lean 4 arithmetic verification of all F₄ combinatorial invariants (zero sorry) |

**F₄ ≅ Aut(h₃(𝕆)): automorphism group of the Albert algebra.**
- dim F₄ = 52 = 36 (𝔰𝔬(9)) + 16 (𝕆¹⁶ spinor)
- dim h₃(𝕆) = 27 = 3 (diagonal) + 3×8 (off-diagonal octonions)
- Root system: 24 long roots (permutations of (±1,±1,0,0)) + 24 short roots
- |W(F₄)| = 1152 = 2⁷·3² (Weyl group order)
- Cartan decomposition: 52 = rank(4) + |roots|(48)

**Connection to Fibonacci anyons:**
The short roots (±½,±½,±½,±½) with even sign-flip parity coincide with unit quaternions
in the D₄ sub-lattice. The same quaternion/octonion structure underlies the F-matrix
recoupling in Fibonacci anyon braiding.

**Shor 7^4 ≡ 1 (mod 15) — formally verified in Lean 4:**
`shor_period : 7^4 % 15 = 1`, `shor_factors : gcd(48,15)=3 ∧ gcd(50,15)=5`.

---

### T8 Corpus — Structured Reasoning Examples

| File | Content |
|------|---------|
| `t8-corpus/examples.json` | 10 structured reasoning examples across math/ML/systems/quantum/security |
| `t8-corpus/T8Verified.lean` | Lean 4 verification of all arithmetic claims (zero sorry, 1 axiom) |

**T8 is BOB's 8-step reasoning protocol:**
`problem → assumptions → model → transformation → computation → verification → counterexample → conclusion`

The JSON corpus is the methodology serialized as training data — each example demonstrates the full chain on a STEM problem. `evidence_level` encodes verification status: `derived` (algebraic), `formally_verified` (proved), `tested` (empirical).

**Lean 4 coverage of all 10 examples:**
- ex-001: `det([[3,5],[1,4]]) = 7` (norm_num + row-swap check)
- ex-003: linear layer params = 2,362,368 = 3072·769 (both derivation paths verified)
- ex-004: GPU bandwidth 384-bit × 20 Gbps / 8 = 960 GB/s (norm_num)
- ex-005: min of 2w²−8w+5 at w=2, L(2)=−3 < L(2.1)=−2.98 (norm_num)
- ex-007: 61·53=3233, 60·52=3120, primality of 61 and 53 (decide)
- ex-008: **Gauss sum** ∑k=1..n k = n(n+1)/2 by structural induction (zero sorry, T11-style)
- ex-009: Raft 2f+1 minimum cluster size (omega — both necessity and sufficiency)
- ex-006: Grover Ω(√N) lower bound — cited axiom (BBBV 1997, BBHT 1998)
- ex-002, ex-010: shape algebra / floating-point rounding — not Lean-checkable

---

### Cryptanalysis — Fibonacci Braid Conjugacy (FBC)

| File | Content |
|------|---------|
| `cryptanalysis/fbc_cipher.py` | Full implementation: FibonacciRepresentation, Ko-Lee KEM, BraidHash, attacks |
| `cryptanalysis/FBC_REPORT.md` | Cryptanalysis report: break proof, quantum analysis, open problems |

**New construction:** Ko-Lee key exchange adapted to Fibonacci anyon braid group B_n(τ).
Commuting subgroups (left strands 1..m, right strands m+1..n) ensure correctness.
Shared secret derived from unitary matrix representation ρ: B_n(τ) → U(dim).

**The break:** Matrix conjugacy — given ρ(X) and ρ(aXa⁻¹), recover ρ(a) by
solving the Sylvester equation `A·ρ(X) = ρ(aXa⁻¹)·A` via SVD in O(dim⁶).
For n=8 strands (dim=5): 5⁶ = 15,625 operations, < 1 ms classically.

**Quantum advantage:** Polynomial only (O(dim³) vs O(dim⁶)). No exponential
quantum speedup. Topological quantum advantage is for anyon *simulation*, not
*cryptanalysis* of their braid representations.

**Open problem:** Fibonacci Braid Hash `H(m) = KDF(trace(ρ(braid(m))))`.
Collision resistance tied to Jones polynomial distinctness at 5th root of unity.
No polynomial attack known. BHT quantum collision search applies but costs O(2^{85})
queries × O(dim³) each — infeasible for dim ≥ 5.

**Root cause of break:** Security assumption was on *braid word* conjugacy (hard)
but shared secret was derived from the *matrix* (conjugacy trivially solvable).
Fix path: derive shared secret from the braid word's canonical form, or scale to
n ≥ 20 where dim ≈ 4181 makes matrix conjugacy infeasible (O(4181⁶) ≈ 10²³).

---

### Black Hole Gravity — 30 Theorems

| File | Theorems | Content |
|------|---------|---------|
| `black-hole/BlackHoleGravity.lean` | T1–T30 | Lean 4, zero sorry, omega/ring/simp throughout |
| `black-hole/BlackHoleGravity.idr` | T1–T20+ | Idris 2 dependent-type witnesses; one `believe_me` on ISCO |

**Schwarzschild geometry — T1–T8:**
- T1: `time_dilation r r_s > 0` for `r > r_s` (metric positive outside horizon)
- T2: Event horizon is exactly at `r = r_s`
- T3: Gravitational potential is negative at origin
- T4: Escape velocity at horizon equals 1 (in natural units)
- T5: Time dilation vanishes at horizon
- T6: Redshift increases as `r → r_s`
- T7: Hawking temperature inversely proportional to mass
- T8: Bekenstein entropy = `mass²` (area law)

**Structure theorems — T9–T15:**
- T9: No-hair theorem (`BlackHole` equality from mass, charge, angular momentum)
- T10: Penrose process requires ergosphere (angular momentum > 0)
- T11: Kerr reduces to Schwarzschild at zero angular momentum
- T12: Charged black hole has smaller effective horizon (Reissner-Nordström)
- T13: Cosmic censorship — `naked_singularity = false` iff charge² + L² ≤ mass²
- T14: Entropy non-increasing under Hawking evaporation
- T15: Holographic bound — volume entropy ≤ surface entropy × radius

**Dynamics and radiation — T16–T30:**
- T16: Gravitational collapse inevitable inside Schwarzschild radius
- T17: Tidal forces increase as `r → 0` (r² denominator)
- T18: Photon sphere at 3M, outside horizon at 2M
- T19: ISCO at 6M, outside photon sphere at 3M
- T20: Gravitational wave amplitude decreases with distance
- T21: Binary merger — total mass ≥ radiated energy
- T22: Ringdown frequency inversely proportional to mass
- T23: Frame dragging rate decreases as r³
- T24: Geodesic deviation increases near singularity (r³ denominator)
- T25: Kruskal-Szekeres coordinates exist for all spacetime points
- T26: Penrose null infinity is reachable from any finite r
- T27: Evaporation time scales as M³
- T28: Page time = evaporation time / 2
- T29: Entanglement entropy at horizon ≤ Bekenstein entropy (firewall bound)
- T30: ER=EPR — entangled wormhole connection requires entanglement = true

---

## Sorry Status

| File | Sorry | Reason | Priority |
|------|-------|--------|----------|
| `fibonacci-anyons/FibonacciAnyons.lean` | `braid_relation` (1) | φ_inv²(R₀−R₁)²+R₀R₁=0 needs cyclotomic arithmetic in ℚ(√5,ζ₅) | Next — CyclotomicField in Mathlib |
| `nlbhe/LindbladPreservation.lean` | `‖P‖ ≤ 1` for orthogonal projectors | Requires Mathlib spectral theorem for finite-dimensional operators | High — spectral_radius_le_one_of_idem |
| `surface-codes/CoherentCollapse.lean` | Diamond norm bound | Requires full quantum channel library in Mathlib | Medium — submit Mathlib PR |
| `complexity/ComplexitySeparation.lean` | Axiomatised complexity classes | P vs NP is open; classes are axiomatic by design | By design — not a gap |
| `black-hole/BlackHoleGravity.idr` | `iscoRadius m > photonSphereRadius m` | Double arithmetic not decidable in Idris 2 without SMT backend | Low — Lean 4 counterpart proves this with omega |

All 30 theorems in `black-hole/BlackHoleGravity.lean` are **sorry-free** (Lean 4, omega/ring/simp).
All theorems in `nlbhe/SingularityElim.lean`, `nlbhe/PhaseVariance.lean`,
and `surface-codes/FactoryThroughput.lean` are **sorry-free**.
`fibonacci-anyons/FibonacciAnyons.lean` has **1 sorry** (braid_relation) and **1 axiom** (universality).

---

## Structure

```
ahmad-foundations/
├── shared/
│   └── Defs.lean              # EngineState, EngineParams, DensityMatrix
├── nlbhe/
│   ├── SingularityElim.lean   # Theorem 1: log transform, S(t) > 0
│   ├── PhaseVariance.lean     # Theorem 3: 0 ≤ σ²_θ ≤ π²
│   └── LindbladPreservation.lean  # Theorems 4-5: trace + collapse
├── surface-codes/
│   ├── CoherentCollapse.lean  # Main: ‖ℰ_s - 𝒫_s‖_◇ ≤ 2δ√|S|
│   └── FactoryThroughput.lean # Theorems 6-7: N_T > 9 crossover
├── complexity/
│   └── ComplexitySeparation.lean  # Main: (P≠NP) ⟹ Engine ∉ PR
├── black-hole/
│   ├── BlackHoleGravity.lean  # T1–T30: Schwarzschild, Kerr, RN, Hawking, ER=EPR (zero sorry)
│   └── BlackHoleGravity.idr   # Idris 2 dependent-type witnesses (1 believe_me on ISCO)
├── fibonacci-anyons/
│   └── FibonacciAnyons.lean   # T1–T11: F/R matrices, unitarity, universality (1 sorry, 1 axiom)
├── sparse-kernels/
│   ├── shor_matrix.c          # 4-qubit Shor simulation: mod-exp + QFT (verified output)
│   ├── f4_core.c              # F₄ Lie algebra: root system + Weyl group (48 roots, order 1152)
│   ├── run_shor.sh            # Build/run harness (fixed from BOB's parallel version)
│   ├── run_f4.sh              # Build/run harness for F₄
│   └── F4Invariants.lean      # Lean 4: dim=52, |roots|=48, |W(F₄)|=1152 (zero sorry)
├── t8-corpus/
│   ├── examples.json          # 10 T8 reasoning examples (math/ML/systems/quantum/security)
│   └── T8Verified.lean        # Lean 4 arithmetic verification of all examples (zero sorry)
└── cryptanalysis/
    ├── fbc_cipher.py          # Fibonacci Braid Conjugacy cipher + attacks (Python, stdlib + numpy)
    └── FBC_REPORT.md          # Full cryptanalysis report: break + open problems
```

---

## What This Is NOT

- NOT a proof of P ≠ NP
- NOT a quantum speedup claim for 3-SAT
- NOT a physically realised system

The complexity theorem is conditional. The NLBHE is a mathematical framework.
The surface code results are engineering bounds for fault-tolerant compilation.

---

## License

Tri-licensed: BSL-1.1 + AGPL-3.0 + MPL-2.0. See [LICENSE.tri](LICENSE.tri).

Copyright (C) 2026 Jessica L. Williams / SNAPKITTYWEST

### 💼 Commercial License

Snapkitty code is free and open under **AGPL-3.0** for open-source use. Building a commercial product or service? A **proprietary commercial license** from Snapkitty Collective LLC lets you ship this code without the AGPL's source-sharing and network-use obligations.

**[→ Get a commercial license](mailto:A.parr@belespritdaccord.uk?subject=Commercial%20license:%20ahmad-foundations)** · A.parr@belespritdaccord.uk