Implements the Next Steps of docs/SPECTRAL_CODEBOOK_ANALYSIS.md as
python/spectral_codebook.py (stdlib-only; NumPy optional fast path):
- Parses the 250 8x8 braid adjacency matrices from PIST/Matrices250.lean.
- Primary fingerprint: exact integer characteristic-polynomial
coefficients via Faddeev-LeVerrier in Fraction arithmetic (196 unique
over 238 distinct matrices, vs 179 unique lambda at 4dp; 6 cospectral
non-identical groups; 11 exact-duplicate matrix groups / 23 ids).
- Corrects the analysis doc: the 9 'mid band' lambda in (0.5,1) are
power-iteration non-convergence artifacts - exact rho = 1.0 for all 9
(peripheral spectra). spectral_radius is now the exact max root
modulus (numpy eigvals or Durand-Kerner on the exact char poly);
power-iteration lambda kept only for traceability.
- Gap-aware quantization: dedupe to 238 distinct matrices, boundaries at
gaps > 3x median gap, min-support guard (>=10 distinct per cluster),
sparse-tail outlier flagging above lambda ~= 7.66. Result: 9 clusters.
- Round trip encode(matrix) -> (codeword, index) -> decode -> equation_id
verified bijective over all 250 in tests/test_spectral_codebook.py.
- 278-row RRC/Q16_16Manifold corpus: 28 extra rows are repeated ids;
boundaries reproduce exactly, no new gaps or clusters.
- Emits data/spectral_codebook.json (schema spectral_codebook_v2) with
explicit collision classes; docs/SPECTRAL_CODEBOOK_GENERATOR.md notes
the hashMatrix base-5 injectivity gap and the ClassifyN threshold
(1.5/4.0 Q16.16) vs analysis-doc (0.5/1.0) mismatch.
Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
Key finding: both Goormaghtigh collisions have exact rho=3.0
This follows from the structure:
- Repunit digits are all 1 → collision graph is K_{m,n}
- Spectral radius of K_{m,n} = min(m,n)
- Goormaghtigh constraint m,n>=3 → rho>=3
- Both known collisions have min(m,n)=3 → rho=3
The Goormaghtigh conjecture restated spectrally:
The only integer lattice points on the eigensolid rho=3
in the (m,n) plane with m,n>=3 are (5,3) and (13,3).
Connection to Cartan gap: rho=3 is exact (integer), so the
Cartan floor is irrelevant for distinguishability. But the
Goormaghtigh collisions occupy a unique point in the codebook
that no other equation shares.
The character matrix of the 4 crossing pairs (Z₂⁴) is the fundamental
transform that preserves Sidon geometry while computing Cartan weights:
chi[i][k] = ±1 if strand i is in crossing pair k, 0 otherwise
C_cartan ∝ chi @ chi.T (Gram matrix of characters)
The Gram matrix has EXACTLY the block-diagonal structure of the Cartan:
[1 -1] → [273 256] (same structure, different scale convention)
[-1 1] → [256 273]
docs/transform_series.md: full 4-layer transform documentation
python/character_transform.py: working computation
Key: the character group Z₂⁴ preserves:
• Additive uniqueness → character orthogonality
• Power-of-2 nesting → tensor product Z₂ × Z₂ × Z₂ × Z₂
• Crossing pairs → character eigenvectors
helical_encoding.md §What This Is Not:
- New section explicitly disclaims any biological interpretation of
chiral labels (achiral_stable, chiral_scarred, etc.)
- The DNA ↔ Cartan isomorphism is at the structural level of
complementary pairing, independent of the numerical overlay
cartan_fingerprint.md:
- Same clarification added to the isomorphism section
python/cartan_dna_bridge.py:
- Constructs 8×8 Cartan crossing matrix (block diagonal: 4×2 pairs)
- Each 2×2 block [273 256; 256 273] has eigenvalues {529, 17}
- σ = 273/1792 = 39/256 (normalized diagonal weight)
- τ = 256/1792 = 1/7 (normalized adjacent weight)
- ∆ = (273-256)/1792 = 17/1792 (difference)
- The min nonzero eigenvalue 17 IS the gap numerator
docs/cartan_dna_derivation.md:
- Step-by-step spec for modifying dna_codec.py
- Replace thermodynamic weights with Cartan weights
- Expected output and verification
All derived values match the Lean reference exactly.
The DNA encoder can now witness the spectral gap chain.
CLAIMS_STATUS.md:
- §1: PROVEN (Lean theorems + exact identities)
- §2: COMPUTATIONAL WITNESS (#eval, axioms with receipts)
- §3: STRUCTURAL (proven for n=8, general pending)
- §4: UNPROVEN / EXPLORATORY (not claimed as established)
- §5: ROUTES UNDER INVESTIGATION (Noether, Rossby, E8, Ingest)
- §6: NAMING CONVENTION (theorem/axiom/conjecture/route)
noether_route.md:
- Explicitly framed as 'route under investigation'
- What's proven: Cartan arithmetic (independent of Noether)
- What's proposed: Lagrangian on S⁷ → Noether charges → same integers
- Plausibility arguments: for and against
- Prerequisites: 4-step proof chain
- Risk assessment: high difficulty but nothing reliant on it breaks
The Cartan proof stands independently. Noether is a possible
deepening, not a retroactive justification.
