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.

On the Use of AI Assistance in HFG Research

The HFG programme 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.