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:
m003 (cusped) = figure-eight knot complement, disc(K) = −3
m003(−2,3) (closed) = Meyerhoff manifold, disc(K) = −283
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.
WRONG: v₀ = D(w³) = 0.305423177...
WRONG: vol(Meyerhoff) = 3·D(w³)
WRONG: −D(w³) = vol(M)
CORRECT: v₀ = vol(Meyerhoff) = vol(m003(−2,3)) = 0.981368828892232...
CORRECT: vol(m019) = 3·v₀ (exact, verified)
CORRECT: vol(m178) = 4·v₀ (exact, verified)
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.
What survives unchanged
- vol(m019) = 3·v₀, vol(m178) = 4·v₀ (now with correct v₀)
- Dual surgery identity m003(−2,3) = m019(2,1)
- Sextic–octic decomposition theorem (unaffected)
- PMNS fitness 0.005087, CKM fitness 0.016482
- CP phase δ = 195.91° (0.55% from PDG)
- All Galois–gauge correspondences
- Eisenstein norm lepton mass ratios
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.
volume quantum
Bloch group
m019
m178
Meyerhoff
correction
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.
p₈(y) = Normₖ₋ℑ(q₂(y)) [exact, zero free parameters]
p₆ = Q₂·Q₃·Q₄ over splitting field L
Normₖ₋ℑ(p₆) = p₈³
disc(p₈) = 7·11·283² [Gal = [2⁴]S₄, order 384]
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.
sextic-octic
norm decomposition
Galois theory
SageMath verified
submitted
The Primes 7 and 11 in disc(p₈)
The discriminant disc(p₈) = 7·11·283² contains two primes beyond the trace field
discriminant 283. Both 7 and 11 split in K with decomposition type 1+3 (Frobenius
cycle type a 3-cycle in S₄). The class number of K is 1, so no ideal-class
obstruction is present.
Both 7 and 11 are norms from K. Both appear in the PMNS covering tower prime set
{2,3,5,7,11,13,29}. The question is whether they arise as prime divisors of
pairwise resultants Res(Qᵢ, Qⱼ) of the four conjugate quadratics — which would
give a complete geometric decomposition of disc(p₈).
discriminant
Galois
open
covering tower
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.
methodology
AI assistance
verification