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 0.016482	0 free params
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

Cusp discriminant: −59

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.

Publications & Preprints

01The Sextic–Octic Decomposition of the Meyerhoff Manifold Shape Polynomial
Research in Number Theory — submitted June 2026 SSRN 6876278New ✧
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 8 2026

Third Manifold Discovered — June 10 2026. A third arithmetic hyperbolic 3-manifold m206(1,2) has been identified with resonance θ* = −60° (Eisenstein, order-6 torsion), exact symmetry λ_b = −λ_a, and near-maximal mixing angle ~74°. The cusped parent has trace field ℚ(√−3), discriminant −3. This constitutes a falsifiable geometric prediction for a BSM mixing sector.

Article IV → Article III → Explorer →

Reproduce

All results are reproducible using SnapPy and SageMath.

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