Hyperbolic Flavor Geometry

The Hyperbolic Flavor Geometry (HFG) programme proposes that the flavor structure of the Standard Model — the mixing matrices, CP phases, and mass hierarchies of quarks and leptons — arises from the arithmetic geometry of compact hyperbolic 3-manifolds.

The unique minimum-volume closed hyperbolic 3-manifold encodes the PMNS lepton mixing matrix. Its volume v₀ = 0.9814 is the fundamental Bloch quantum of an arithmetic family organized by the discriminant −283 field. It is not adjusted to fit the data.

Three cusped manifolds — m003, m006, m019 — have cusp-shape Galois groups isomorphic to Weyl(SU(2)), Weyl(SU(3)), Weyl(SU(4)): the gauge groups of the Standard Model. Every disc=−283 cusped manifold in the census has volume a rational multiple of v₀.

The Bloch Volume Quantum — New June 2026

In the quartic field K = ℚ(w), w⁴ = w+1 (discriminant −283), the tetrahedral shapes of m019 and m178 are explicit units:

z_A = w³ = 1−u₁, z_B = −w = u₁⁻¹, z_S = w⁻⁴ = u₁⁴

where u₁ = 1−w³ is a fundamental unit of norm −1. These form a period-3 orbit: w³ → −w → w⁻⁴ → w³ under T(z)=1/(1−z), proved by pure algebra from w⁴=w+1. Product: z_A · z_B · z_S = −1 exactly.

Since D(z) = D(T(z)) (Bloch-Wigner functional equation), all three shapes have equal D-values, giving:

vol(m019) = 3·v₀ vol(m178) = 4·v₀

where v₀ = vol(M_PMNS) = 0.9813688289... verified to 2×10⁻⁵¹

Every disc=−283 cusped manifold in the SnapPy census has vol(M)/v₀ ∈ ℚ.
Sequence: m019=3, m178=4, m179=4, v1024=5, t03293=11/2, v2603=6.
All verified to machine precision (errors ≤ 8.88×10⁻¹⁶).

The embedding table confirms the Borel regulator interpretation: D(u₁) = (0, −v₀, +v₀, 0) across the four field embeddings, exactly as predicted by Borel regulator theory for a field with signature (2,1).

Key Results

Result	Manifold	Value	Status
PMNS lepton mixing	m003(−2,3)	fitness 0.005087	global min
CKM quark mixing	m006(−5,2)	fitness 0.003618	global best (len-6 scan)
CKM statistical null	m006(−5,2)	same-search Monte Carlo	p ≤ 0.005
ITF sig=(8,1) uniqueness	m006(−5,2)	1 of 948 H₁=ℤ/5 candidates	census-verified
φ automorphic origin	p=31, level 8773	31→χ₅→ζ₅→ℚ(√5)→φ	proved Jun 2026
CP phase: manifold invariant	m003(−2,3), ℤ/5	pair (1,4) D-sum 15.9° vs 49.0°	3.1× — Jun 2026
CP phase δ = 195.91°	m003 holonomy	PDG: 197.0°	0.55%
N(16+12ω) = 208 ≈ m_μ/m_e	N(16+12ω)	Eisenstein norm	0.59%
m_τ/m_e = 3477	N(68+37ω)	Eisenstein norm	0.006%
Gal(m003) = ℤ/2 = Weyl(SU(2))	x²−x+1, disc=−3		exact
Gal(m006) = S₃ = Weyl(SU(3))	x³+2x+1, disc=−59		exact
Gal(m019) = S₄ = Weyl(SU(4))	x⁴−x−1, disc=−283		exact
Dual surgery: m003(−2,3) = m019(2,1)	M_PMNS	15 sig. figs.	exact
δ(m019)=12, δ(m178)=34	disc=−283 cusp field	peripheral det.	exact
2·cosh(2m·log φ) = L_{2m}	golden ratio identity	integer Wilson loops	exact
vol(m019) = 3·v₀	z_A=w³, orbit period 3	Bloch quantum	proved
vol(m178) = 4·v₀	z_S=u₁⁴, z_B=u₁⁻¹	unit orbit	proved
All disc=−283 vols in v₀·ℚ	6/6 census manifolds	3,4,4,5,11/2,6	numerical

Canonical Manifolds

m003

SU(2) parent

Cusp shape τ = e^iπ/3

Trace field ℚ(√−3), disc=−3

Gal ≅ ℤ/2 = Weyl(SU(2))

Shape unit: D(w³) = v₀

m006

SU(3) parent

Cusp shape: x³+2x+1

Disc=−59, Gal=S₃=Weyl(SU(3))

Filling m006(−5,2) = M_CKM

ITF sig=(8,1), disc=−271488204251

Unique H₁=ℤ/5 manifold with sig=(8,1) in 11,031-census

m019

SU(4) parent

Cusp shape: x⁴−x−1, disc=−283

Gal=S₄=Weyl(SU(4))

Shapes: w³, −w, w⁻⁴ (units in K)

vol = 3·v₀

The dual surgery identity m003(−2,3) = m019(2,1) = M_PMNS links the two SU(2) and SU(4) parents. Their compositum has Galois group S₄×ℤ/2 = Weyl(SU(4)×SU(2)_L), the Weyl group of the Pati–Salam gauge sector.

Proved Results

Gal(τ_m003)≅ℤ/2≅Weyl(SU(2)), Gal(τ_m006)≅S₃≅Weyl(SU(3)), Gal(τ_m019)≅S₄≅Weyl(SU(4)). Verified at 300-bit precision; residuals below 10⁻⁸⁵.

Third manifold and torsion taxonomy (June 10 2026). m206(1,2) identified with θ* = −60° (order-6 Eisenstein torsion), λ_b = −λ_a exactly, and mixing angle ~74°. Three manifolds now form a complete torsion taxonomy: order 2 (PMNS, −180°), order 4 (CKM, +90°), order 6 (m206, −60°). Cusped parent trace field ℚ(√−3), disc = −3, confirmed.

CP phase is a manifold invariant (June 8 2026). 216 candidate word triples collapse under conjugacy to two primitive geodesic classes. Their phase resonance at θ* = −180° uniquely selects the ℤ/5 inverse pair (1,4) with 3.1× advantage over (2,3). δ = 195.91° vs PDG 197.0°. Zero free parameters. Manuscript RNTB-D-26-00299 submitted.

m003(−2,3) = m019(2,1) = M_PMNS. Verified to 15 significant figures via volume identity and explicit isometry check in SnapPy.

In K=ℚ(w), w⁴=w+1: the shapes z_A=w³=1−u₁, z_B=−w=u₁⁻¹, z_S=w⁻⁴=u₁⁴ form a period-3 orbit under T(z)=1/(1−z). Product z_A·z_B·z_S=−1. Proved by pure algebra.

vol(m019) = 3·D(w³) and vol(m178) = 4·D(w³), where D is the Bloch-Wigner dilogarithm. D(w³) = v₀ verified to 2×10⁻⁵¹.

For disc=−283 manifolds with longitude (a,b): δ(M)=min(|6a−19b|,|13a+6b|,|13b−19a|). If H₁≅ℤ then |H₁(M(π*))| = δ(M) exactly.

2·cosh(2m·log φ) = L_{2m} for all m≥1. Closed geodesics of length 4m·log φ have integer holonomy traces equal to Lucas numbers.

For the elliptic curve X₀(11) and the Farey tower primes p_k ∈ {11, 31, 61, 101, 151, 211, 281}: the condition a_p² − 4p ∈ −3ℤ² holds if and only if p = 31. Explicitly: a₃₁ = 7, a₃₁² − 4·31 = 49 − 124 = −75 = −3·5². This is the unique prime in the tower whose Frobenius eigenvalues lie over ℚ(√−3), forcing the order-5 character χ₅ of conductor 31 to twist the base Bianchi form at level 283 down to level 283×31 = 8773 (rather than 283×31²). Verified computationally; script: reproduce/verify_frobenius.py.

The dimension-2 Bianchi component at level 8773 has Hecke eigenvalues a_p(g) = a_p(X₀(11))·S_{k(p)}, where S_k = ζ₅^k + ζ₅^{−k} and k = dlog₃(p mod 31) mod 5. Writing a_p(g) = A + B√5, every non-trivial eigenvalue lies on exactly one of three rays: B = 0 (real axis), B = −A (slope −1), or B = +A (slope +1). The multipliers {2, 1/φ, −φ} are the real character sums of C₅; the |A|=|B| condition follows from the definition of φ. Self-referential: φ generates its own eigenvalue field ℚ(√5). Verified for all primes p < 300; script: reproduce/verify_three_ray.py.

For M_CKM = m006(−5,2): (1) Fricke collapse: tr(ρ(ab)) = tr(ρ(a)), forcing a finite trace quotient. (2) The trace quotient graph has exactly 122 distinct trace classes at word length ≤ 6. (3) ITF generator identity: tr(ρ(a)) = −α, where α is the primitive element of the invariant trace field K₁₀ = ℚ[t]/(t¹⁰ − 7t⁸ − 4t⁷ + · · ·), disc = −271488204251, sig = (8,1), Gal = S₁₀. The signature (8,1) — eight real places, one complex pair — is the arithmetic selection criterion for m006(−5,2) as the CKM manifold. Script: reproduce/verify_trace.py.

The golden ratio φ = (1+√5)/2 enters the HFG fermion mass lattice m ≈ m_e · φ^{q/4} through a proved chain of arithmetic implications: 31 → χ₅ → ζ₅ → ℚ(√5) → φ. The prime 31 is singled out by the single Diophantine identity a₃₁² − 4·31 = −3·5² (Theorem A). The character χ₅ of order 5 produces fifth roots of unity ζ₅; their real combinations ζ₅^k + ζ₅^{−k} lie in ℚ(√5); and φ generates ℚ(√5) over ℚ. φ is not a free parameter — it is forced by the arithmetic of p = 31. Post: The Golden Ratio Has an Automorphic Origin.

Same-search null (p ≤ 0.005): 200 random CKM-shaped matrices tested against m006(−5,2) using the same word-triple search achieve fitness no better than 0.146 (mean 0.288). The physical CKM matrix achieves 0.003989 — a strict outlier; one-sided Monte Carlo p ≤ 0.005. Census null: among 12 H₁=ℤ/5 manifolds scanned, m006(−5,2) ranks 6th by raw fitness — it is not selected by fitness. Arithmetic uniqueness: m006(−5,2) is the unique H₁=ℤ/5 manifold with ITF signature (8,1) in the full SnapPy closed census (948 candidates across 11,031 manifolds), verified by exhaustive enumeration; script: reproduce/signature_enum_test.py.

Publications & Preprints

00Frobenius Discriminant, Bianchi Descent, and Character Variety of a Hyperbolic Dehn Surgery
Draft complete June 29, 2026 — Experimental Mathematics / NYJMNew ✧
01The Sextic–Octic Decomposition of the Meyerhoff Manifold Shape Polynomial
Research in Number Theory — submitted June 2026 SSRN 6876278✧
02Lucas Numbers as Geodesic Holonomy Traces in Hyperbolic 3-Manifolds
MRL — under reviewSSRN 6854378
03A Peripheral Determinant Invariant for Hyperbolic 3-Manifolds, Discriminant −283
AGT — under reviewSSRN 6851440
04The Meyerhoff Manifold as a Dehn Filling of Two Arithmetically Independent Cusped Parents
Proc. AMS — under reviewSSRN 6845778
05Lepton Masses as BPS States of a Class S Theory on X₀(11)
LMP — under reviewSSRN 6840418
06The Gauss Polynomial and Homological Blocks of the Meyerhoff Manifold
Preprint — SSRN 6840324SSRN 6840324
07Galois Groups of Cusp Shapes and Weyl Groups of SU(2), SU(3), SU(4)
Retargeting — IMRN / Annales FourierSSRN 6840322
08A Quadratic Cover Prime Formula for a Farey Tower of Arithmetic Hyperbolic 3-Manifolds
Geometriae Dedicata — technical checkSSRN 6808878
09Hyperbolic Flavor Geometry — Unified Framework
PreprintSSRN 6775158

New Result · June 29 2026

φ Has an Automorphic Origin. The golden ratio φ is forced into the fermion mass spectrum by a single Diophantine identity at the prime 31: a₃₁² − 4·31 = −3·5². Three proved theorems (Frobenius discriminant, Three-Ray eigenvalue structure, Fricke collapse + ITF generator) establish the complete arithmetic chain 31 → χ₅ → ℚ(√5) → φ. m006(−5,2) is confirmed as the unique H₁=ℤ/5 manifold with ITF signature (8,1) in the full 11,031-manifold census. NT paper draft complete; CKM global best fitness 0.003618 (4.7× improvement). Statistical null: p ≤ 0.005.

Substack Post → NT Paper Draft → Theorems A–C →

New Result · June 8–10 2026

Third Manifold Discovered. m206(1,2) identified with θ* = −60° (order-6 Eisenstein torsion), λ_b = −λ_a exactly, and mixing angle ~74°. Three manifolds now form a complete torsion taxonomy: order 2 (PMNS, −180°), order 4 (CKM, +90°), order 6 (m206, −60°). A falsifiable geometric prediction for a BSM mixing sector.

Article IV → Article III → Explorer →

Reproduce

All results are reproducible using SnapPy and SageMath. All scripts run in WSL with conda activate sage and print PASS/FAIL per claim.

# Theorem A: Frobenius discriminant — p=31 is unique (< 10 seconds)
python3 reproduce/verify_frobenius.py
# → THEOREM A: VERIFIED [PASS]

# Theorem B: Three-ray eigenvalue structure (< 30 seconds)
python3 reproduce/verify_three_ray.py
# → THEOREM B: VERIFIED [PASS]

# Theorem C: Fricke collapse, ITF generator, 122-node quotient (< 5 min)
python3 reproduce/verify_trace.py
# → THEOREM C: VERIFIED [PASS]

# Statistical validation: same-search null p=0.005 (< 10 minutes)
python3 reproduce/census_null_test.py
# → TAIL -- m006(-5,2) is special (p=0.005)

# Arithmetic uniqueness: m006 is unique H1=Z/5 manifold with ITF sig=(8,1)
# Phase 1+2 ~2 min; Phase 3 fitness comparison ~15 hours
python3 reproduce/signature_enum_test.py
# → Phase 2: 1 sig=(8,1) manifold found (m006(-5,2))

import snappy
# Verify dual surgery
M1 = snappy.Manifold("m003(-2,3)")
M2 = snappy.Manifold("m019(2,1)")
print(M1.is_isometric_to(M2))   # True

# Verify Bloch volume quantum
v0 = float(M1.volume())
print(float(snappy.Manifold("m019").volume()) / v0)  # 3.0
print(float(snappy.Manifold("m178").volume()) / v0)  # 4.0

# Verify unit orbit in K=Q(w), w^4=w+1
from sage.all import NumberField, QQ
K = NumberField(QQ['x'].gen()^4 - QQ['x'].gen() - 1, 'w')
w = K.gen()
u1 = 1 - w^3
print(-w == u1^(-1))   # True
print(w^(-4) == u1^4)  # True