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 is 0.9814. It is the most rigid geometric object of its kind. It is not adjusted to fit the data.

Three cusped manifolds — m003, m006, m019 — have cusp-shape Galois groups isomorphic to the Weyl groups of SU(2), SU(3), and SU(4): the gauge groups of the Standard Model. This identification is exact and requires no free parameters.

Key Numerical Results

Result	Manifold	Value	Accuracy
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 δ = 195.91°	m003 holonomy	PDG: 197.0°	0.55%
m_μ/m_e = 206.77	N(16+12ω) = 208	Eisenstein norm	0.59%
m_τ/m_e = 3477	N(68+37ω) = 3477	Eisenstein norm	0.006%
Gal(m003) = Z/2 = Weyl(SU(2))	cusp field x²−x+1	disc = −3	exact
Gal(m006) = S₃ = Weyl(SU(3))	cusp field x³+2x+1	disc = −59	exact
Gal(m019) = S₄ = Weyl(SU(4))	cusp field x⁴−x−1	disc = −283	exact
Dual surgery identity	m003(−2,3) = m019(2,1)	M_PMNS	15 sig. figs.
Delta invariant δ(m019) = 12	disc=−283 cusp field	\|H₁(M(π*))\| = δ(M)	exact
Lucas trace: 2cosh(2m·log φ) = L_2m	golden ratio identity	integer Wilson loops	exact

Canonical Manifolds

m003

SU(2) parent

Cusp shape τ = e^iπ/3

Trace field ℚ(√−3)

Gal ≅ ℤ/2 = Weyl(SU(2))

Filling m003(−2,3) = M_PMNS

m006

SU(3) parent

Cusp shape satisfies x³+2x+1

Disc = −59 (inert in ℤ[ω])

Gal ≅ S₃ = Weyl(SU(3))

Filling m006(−5,2) = M_CKM

m019

SU(4) parent

Cusp shape satisfies x⁴−x−1

Disc = −283 (splits in ℤ[ω])

Gal ≅ S₄ = Weyl(SU(4))

Filling m019(2,1) = M_PMNS

The dual surgery identity m003(−2,3) = m019(2,1) means the Meyerhoff manifold M_PMNS is the unique closed hyperbolic 3-manifold arising as a Dehn filling of two arithmetically independent cusped parents. Their compositum has Galois group S₄ × ℤ/2 = Weyl(SU(4) × SU(2)_L), the Weyl group of the left-handed Pati–Salam gauge sector.

Proved Results

The cusp-shape Galois groups of m003, m006, m019 are isomorphic to Weyl(SU(2)), Weyl(SU(3)), Weyl(SU(4)) respectively. All computations performed at 300-bit precision in SnapPy and Sage; residuals below 10⁻⁸⁵.

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

For a one-cusped manifold M with cusp field disc = −283 and longitude (a,b), define δ(M) = min(|6a−19b|, |13a+6b|, |13b−19a|). If H₁(M;ℤ) ≅ ℤ (torsion-free), then |H₁(M(π*);ℤ)| = δ(M) exactly, where π* is the minimising Eisenstein orbit element.

2·cosh(2m·log φ) = L_2m for all integers m ≥ 1, where φ = (1+√5)/2 and L_k is the k-th Lucas number. Closed geodesics of length 4m·log φ have integer holonomy traces equal to Lucas numbers, contributing integer Wilson loops in Chern–Simons theory.

Publications & Preprints

01 The Meyerhoff Manifold as a Volume Quantum for the Cusped Manifolds m019 and m178
Experimental Mathematics — submitted SSRN 6859979
08 The Meyerhoff Manifold as a Dehn Filling of Two Arithmetically Independent Cusped Parents
Proc. AMS — under review SSRN 6845778
02 A Peripheral Determinant Invariant for Hyperbolic 3-Manifolds with Quartic Cusp Field of Discriminant −283
AGT — under review SSRN 6851440
03 Lucas Numbers as Geodesic Holonomy Traces in Hyperbolic 3-Manifolds
MRL — under review SSRN 6854378
04 Galois Groups of Cusp Shapes and Weyl Groups of SU(2), SU(3), SU(4)
CNTP — pending submission SSRN 6840322
05 Lepton Masses as BPS States of a Class S Theory on X₀(11)
LMP — under review SSRN 6840418
06 A Quadratic Cover Prime Formula for a Farey Tower of Arithmetic Hyperbolic 3-Manifolds
Geometriae Dedicata — technical check GD submission
07 Hyperbolic Flavor Geometry — Unified Framework
Preprint SSRN 6775158

Reproduce

All results are reproducible using SnapPy and SageMath. Source code and scan data are available on GitHub.

import snappy

# Verify dual surgery identity
M1 = snappy.Manifold("m003(-2,3)")
M2 = snappy.Manifold("m019(2,1)")
print(M1.volume())              # 0.9813688289...
print(M2.volume())              # 0.9813688289...
print(M1.is_isometric_to(M2))   # True

# Verify Galois–Weyl correspondence
# m003: cusp poly x^2 - x + 1, disc = -3, Gal = Z/2
# m006: cusp poly x^3 + 2x + 1, disc = -59, Gal = S3
# m019: cusp poly x^4 - x - 1, disc = -283, Gal = S4

Note: Always use OrientableClosedCensus[1] (M_PMNS) and OrientableClosedCensus[43] (M_CKM) by index in SnapPy. Two distinct census manifolds share the name "m006".

→ github.com/drmlgentry/hyperbolic-flavor-scan
→ github.com/drmlgentry/hyperbolic-flavor-geometry
→ PyPI: latticefit