Progress Notes

The Volume Quantum v₀ — Corrected Definition

Multiple earlier papers and poster versions defined the volume quantum as v₀ = D(w³) = 0.305423177..., where D denotes the Bloch–Wigner dilogarithm evaluated at w³ under the geometric embedding. This was incorrect.

The error arose from conflating two manifolds that share a name prefix in the SnapPy database:

The clean Bloch–Wigner identity vol(m003) = 2·D(e^(iπ/3)) belongs to the figure-eight knot complement, not to the Meyerhoff manifold. The value D(w³) = 0.3054... is the Bloch–Wigner function at σ₂(w³) — the non-geometric embedding — and has no direct volume interpretation.

The integer quantization of the disc = −283 family is real and exact. The error in v₀ does not affect it — it simply clarifies that v₀ is the Meyerhoff volume itself, and m019, m178 are integer multiples of it. The dual surgery identity m003(−2,3) = m019(2,1) is unaffected.

Open: the correct Bloch group fundamental class for m019 (triangulation-dependent D-sums do not close to vol(m019)); the Chern–Simons invariants of m019 and m178; whether cv(m019) = 3·cv(Meyerhoff) mod π².

Sextic–Octic Decomposition Theorem — Proved and Submitted

The degree-8 shape polynomial of the Meyerhoff manifold is the algebraic norm of a single quadratic over the trace field K = ℚ(w), w⁴ = w+1, disc = −283.

All seven structural identities verified by exact symbolic computation in SageMath. Verification script publicly available at github.com/drmlgentry/hyperbolic-flavor-scan. Paper submitted to Research in Number Theory (June 4, 2026); preprint at SSRN 6876278.

The Period-3 Norm Orbit Theorem

The polynomial p₈ studied in the Sextic–Octic paper is one element of a period-3 orbit of octic polynomials under the Möbius transformation T(z) = (z−1)/z. This is the same T that generates the unit orbit w³ → −w → w⁻⁴ → w³ in the arithmetic of K = ℚ(w).

Three quadratics over K form a period-3 orbit under T:

Their algebraic norms are three distinct irreducible octic polynomials:

All three octrics share identical arithmetic invariants: disc = 7·11·283², Gal/ℚ = [2⁴]S₄ of order 384, and Sum D(roots) = 0 exactly. The polynomial p₈_HFG studied in the paper is therefore not the shape polynomial of the Meyerhoff manifold directly — it is the T¹-orbit image of the shape polynomial, related by the same Möbius transformation that generates the period-3 unit orbit.

All three norm identities verified by exact computation in SageMath. Being added to gentry-sextic-octic as a new theorem.

The Primes 7 and 11 in disc(p₈) — Completely Explained

The discriminant disc(p₈) = 7·11·283² is now completely explained by a single formula:

The six pairwise resultants all satisfy Norm(Res(Qᵢ,Qⱼ)) = 283⁴ — 7 and 11 do not appear there at all. The two prime sources are completely independent: 7 and 11 come from the arithmetic of the quadratic q₂ over K; 283 comes from the field discriminant disc(K) = −283.

Added to the paper as Proposition 5.2. Paper recompiled: 8 pages.

Ptolemy–Trace Field Identity for m019

The Ptolemy variety of m019 at obstruction class 1 has defining field K₁ with disc(K₁) = −283 and K₁ ≅ K — the Meyerhoff trace field itself. The Ptolemy coordinate a is algebraically exactly w³, where w⁴ = w+1.

The period-3 Möbius orbit w³ → −w → w⁻⁴ → w³ in K× and the Ptolemy coordinate for m019 are the same algebraic object. Only obstruction class 1 recovers the trace field; obstruction class 0 gives a different field with disc = −331.

Strengthened: z₂ = w⁻⁴ exactly. All three Ptolemy coordinates on the unit orbit.

Bloch group story completed (June 4 2026): The Neumann–Zagier regulator NZ(z) = Im(Li₂(z)) + Re(log z)·Im(log(1−z)) is T-invariant — NZ(1/(1−z)) = NZ(z) for all z. Since w³, −w, w⁻⁴ are T-orbit-related (exact), all three yield the same value NZ(σ₂(w³)) = v₀. With 3 tetrahedra of shapes (w³, w³, w⁻⁴):

The volume quantum identity vol(m019) = 3·v₀ is now explained by the T-invariance of NZ combined with the T-orbit structure of the tetrahedral shapes. Paper: 10 pages.

Volume Quantum Theorem — Full disc=−283 Family

Census scan of all 25 orientable hyperbolic 3-manifolds (16 cusped + 9 closed) with invariant trace field disc = −283 confirms the volume quantum structure universally:

For m019 and m178, all tetrahedral shapes lie in the period-3 orbit {w³, −w, w⁻⁴} and each contributes exactly v₀ via NZ T-invariance. For other manifolds, fractional contributions sum to half-integer multiples. The denominator is always at most 2.

m006 (the CKM manifold) has vol(m006) = 2·v₀ — a new exact relation between the two flavor manifolds.

Bloch Class of t03293 and Half-Integral Volume Quantum

All 8 tetrahedral shapes of t03293 identified in K = ℚ(w), w⁴ = w+1:

Since B(K)⊗ℚ has rank 1 (Borel, r₂=1) and NZ is injective, each shape contributes [zᵢ] = qᵢ·[w³] where qᵢ = NZ(zᵢ)/v₀. Summing all 8 contributions:

Census scan of all 16 cusped disc=−283 manifolds confirms the half-integral volume quantum:

Open question: why is the denominator at most 2? Likely reflects 2-torsion in H₁(M,ℤ) or the extended Bloch group.

Bloch Decomposition: [t03293] = 11/2 · [w³] in B(K)⊗ℚ

All 8 tetrahedral shapes of t03293 identified in K = ℚ(w). Using B(K)⊗ℚ ≅ ℚ (Borel, r₂=1) and injectivity of NZ:

For ALL 16 cusped disc=−283 manifolds: [M] = (n/2)·[w³], n ∈ {6,8,10,11,12,16,18,20}. The denominator-2 phenomenon (why n/2 and not arbitrary rational) is the central open question — likely connected to ℤ/2 torsion in H₁(M) or the extended Bloch group.

Key exact Bloch identities established: [z₃] = [w²] via double application of [1/z]=−[z] and [1−z]=−[z]. Five-term relations R1–R4 verified symbolically.

Records Not Kept Contemporaneously for This Period

This log has no entries between June 5, 2026 and August 24, 2026, despite substantial work happening in that window — including a new proved theorem (the four-field Galois product / 576-element result and its subsequent Artin-conductor and cusp-order extensions), an exact resolution of an open character-variety question, and changes to paper submission status. That work is documented elsewhere (the papers themselves, the GitHub commit history, and internal notes), but it was not logged here as it happened.

Per the standing policy for this log (see CLAUDE.md), that gap is being flagged explicitly rather than backfilled with reconstructed entries — a partial, memory-based reconstruction would make this register less trustworthy, not more, since it would imply a completeness it doesn't actually have. Entries from August 25, 2026 onward are being kept contemporaneously going forward.

On the Use of AI Assistance in HFG Research

The HFG program is developed with the assistance of AI language models (Claude, GPT-4, DeepSeek). These tools are valuable for computation scaffolding, LaTeX drafting, and exploring conjectures. However, AI-generated mathematical claims require the same verification as any other source.

The v₀ error documented in Entry 1 propagated partly because a plausible AI-suggested identity was accepted without independent numerical verification. The correction was itself found through systematic computation — the right response to any such claim.

Policy going forward: every numerical identity stated in an HFG paper must be verified by direct computation before submission. AI assistance is used for scaffolding and exploration; the computation is the authority.