Informational Geometry (IG)

Informational Geometry: Mathematical Foundations of Dark Geometry and Particle Physics

Hugo Hertault — Book II of the Dark Geometry series

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**Overview**

This book develops the rigorous mathematical foundations of Dark Geometry, a theoretical framework in which all fundamental constants and particle masses are derived from a single geometric structure: the holographic fibration H = M⁴ ×_σ F, where F = (0,1] is the informational fibre equipped with the Fisher–Rao metric. The framework has exactly two inputs — the spatial dimension d = 3 and the Planck mass M_Pl — and zero free parameters. All coupling constants, mixing angles, particle masses, and cosmological parameters are derived, not fitted.

**The Foundational Structure**

Four axioms (informational content, holographic saturation, factorisation, and smoothness) uniquely fix the conformal factor σ(x) = (1/4) ln I(x) through the Hertault Axiom e^{4σ} = S_ent/S_Bek and determine the Hertault angle θ_H = arccos√(2/3) ≈ 35.26° as a function of d = 3 alone. The Hertault algebra h_3 ≅ su(2) ⊕ u(1), constructed from the holographic partition operators, provides the algebraic backbone from which the Standard Model gauge group and particle spectrum emerge.

**Gauge Group Derivation**

The Standard Model gauge group SU(3) × SU(2) × U(1) is derived — not postulated — from three independent geometric mechanisms: U(1)_Y from the fibre automorphism group Aut(F), SU(2)_L from the Hertault algebra h_d ≅ su(2), and SU(3)_C from the Peter–Weyl decomposition on the holographic surface S² combined with the hairy ball theorem. The number of fermion generations n_gen = 3 follows from the Z_3 discrete 't Hooft anomaly cancellation on S² × F.

**Coupling Constants (zero free parameters)**

All three gauge couplings are determined by θ_H:
- Fine structure constant: α_em = sin θ_H/(8π²) ≈ 1/137.04 (error: 0.20%, correctable to 0.03% with QED vacuum polarisation running)
- Strong coupling: α_s = sin(2θ_H)/8 = √2/12 ≈ 0.1179 (error: < 0.1%)
- Weinberg angle: sin²θ_W = d/(d² + d + 1) = 3/13 ≈ 0.2308 (error: 0.2%)
- Instanton coupling: g² = 2 at the Planck scale
- Informational coupling: α_* = sin(2θ_H)/(4π) = √2/(6π) ≈ 0.0750

**Electroweak Symmetry Breaking**

The Rosen–Morse potential on the fibre yields a unique bound state. Instanton tunnelling through this potential generates the electroweak vacuum expectation value:
- v_H = 2√2 M_Pl e^{-4π²} ≈ 246.5 GeV (error: 0.1%)
- The hierarchy problem is resolved: the ratio M_Pl/v_H ~ e^{4π²} ≈ 5.2 × 10¹⁷ is a geometric tunnelling factor, not a fine-tuning

**Particle Masses**

All three Koide parameters (Q = 2/3, r = √2, ε = 2/9) are derived from the Hertault algebra:
- Electron mass: m_e = 0.5110 MeV (error: 0.006%)
- Muon mass: m_μ = 105.653 MeV (error: 0.005%)
- Tau mass: m_τ = 1776.88 MeV (error: 0.001%)

Gauge boson masses from running α_em(0) to α_em(M_Z):
- W boson: m_W = 80.31 GeV (error: 0.08%)
- Z boson: m_Z = 91.57 GeV (error: 0.42%)

Proton-to-electron mass ratio: m_p/m_e = 6π⁵ = 1836.12 (error: 19 ppm)

Bottom-to-tau mass ratio: m_b/m_τ = 6π⁵ (same geometric origin)

Higgs mass: m_H ≈ 125.07 GeV (error: 1.4%)

Proton mass: m_p ≈ 936 MeV (error: 0.25%)

**Neutrino Mass Predictions**

The Weinberg dimension-5 operator with a see-saw scale Λ = α_s · α_*³ · M_Pl derived from the fibration gives:
- Absolute neutrino mass: m₃ = 49.88 meV (experimental: 50.34 ± 0.33 meV, error: 0.7%)
- Mass-squared ratio: Δm²₂₁/Δm²₃₁ = 1/F₉ = 1/34 ≈ 0.02941 from the Fibonacci gap labelling theorem (experimental NuFIT 6.0 + JUNO 2025: 0.02952 ± 0.00055, agreement: 0.2σ)
- Sum of neutrino masses: Σm_ν = 58.4 meV (below DESI upper limit of 72 meV)
- Predicted mass ordering: normal hierarchy

The 1/34 prediction — where 34 is the 9th Fibonacci number — is a unique, parameter-free prediction testable by JUNO with increasing precision through 2030.

**Cosmological Parameters**

- Dark energy fraction: Ω_Λ = β = 2/3 (error: 3%)
- Hubble constant: H₀ = 67.3–72.7 km/s/Mpc depending on the measurement channel, resolving the Hubble tension
- σ₈ = 0.766 from Δσ₈ = 2βα_*² (agrees with weak lensing measurements)
- Cosmological constant: Λ ~ e^{-b₀ × 4π²} M_Pl⁴ with b₀ from the one-loop beta function, resolving the 122-order-of-magnitude problem

**Baryogenesis**

The CP violation from the Hertault algebra generates ε_CP ~ α_*², giving baryon-to-photon ratio η_B ~ 6 × 10⁻¹⁰ (observed: 6.1 × 10⁻¹⁰). The strong CP problem is resolved by the informational axion, a pseudo-Goldstone boson of the fibre shift symmetry.

**Chemistry**

The framework derives the periodic table from d = 3: the shell filling rule N_n = 2n², the tetrahedral bond angle arccos(-1/3) = 109.47° (error: 0.03%), atomic masses for all 118 elements (average error: 0.39%), and the Madelung rule from a modified filling parameter n + (2/3)ℓ.

**Mathematical Rigour**

All results are classified by epistemological tier:
- Tier A: Proven theorems (axiom uniqueness, algebra structure, constraint analysis, shell rule)
- Tier B: Conjectures with sub-percent agreement and strong theoretical motivation (coupling constants, Koide masses, gauge boson masses, neutrino masses)
- Tier C: Semi-empirical fits with 1–10% agreement (quark masses, CKM parameters, Higgs mass)
- Tier D: Inputs (d = 3, M_Pl)
- Tier E: Observational coincidences awaiting derivation

The book contains approximately 170 quantitative predictions, ~80 with sub-percent accuracy.

**Testable Predictions**

Key predictions testable by current and near-future experiments:
- JUNO (2025–2030): neutrino mass-squared ratio converging toward 1/34
- DESI (2024–2028): sum of neutrino masses Σm_ν = 58.4 meV
- Gravitational wave detectors: tidal Love number k₂ for neutron stars modified by α_*²
- Cassini-class experiments: fifth force screening in the solar system
- FQHE experiments: fractional quantum Hall filling ν = 1/3 = sin²θ_H

**Relation to Book I**

This is Book II of the Dark Geometry series. Book I (Informational Relativity: A Unified Framework for Dark Geometry) develops the cosmological model. Book II provides the mathematical foundations and particle physics derivations. The two volumes are self-contained but complementary.

**Open Access**

Full PDF and LaTeX source available. DOI: 10.5281/zenodo.18870211

Inputs: d = 3 and M_Pl. Free parameters: zero.