Informational Geometry (IG)

Informational Geometry: Mathematical Foundations of Dark Geometry and Particle Physics

Hugo Hertault — Book II of the Dark Geometry series

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 = M&sup4; ×_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) gives 1/136.8 at the Thomson limit (error 0.20%); with QED vacuum-polarisation running and NLO corrections this becomes 1/137.04 (4 ppm)
• 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} = 246.225 GeV at NLO (error: 22 ppm)
• The hierarchy problem is resolved: the ratio M_Pl/v_H ~ e^{4π²} 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 = d · η_QCD = 2.35 (Georgi–Jarlskog relation, d = 3 colours)

Higgs mass: m_H = v(2/π)^{3/2} = 125.07 GeV (error: 0.14%)

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 = M_Pl/(648π³) derived from the fibration gives:

• Absolute neutrino mass: m_3 = 4d⁴π³ v²/M_Pl = 49.88 meV (experimental: 49.53 ± 0.33 meV, error: +0.7%, 1.1σ)
• Mass-squared ratio: Δm²_21/Δm²_31 = 1/F_9 = 1/34 ≈ 0.02941 from the Fibonacci gap labelling theorem (current central value 0.0307; convergence toward 1/34 is the JUNO test)
• 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%)
• σ_8 = 0.766 from Δσ_8 = 2βα_*² (agrees with weak lensing measurements)
• Cosmological constant: Λ ~ e^{-b_0 × 4π²} M_Pl⁴ with b_0 from the one-loop beta function, resolving the 122-order-of-magnitude problem

Algebraic Resolution of the Hubble Tension — from the Fibre Monoid (Tier A)

The holographic fibre F = (0,1] is a multiplicative monoid with identity I = 1. At this identity: σ = 0, φ_DG = 0, and the Dark Boson coupling ξRφ² vanishes exactly. This is the unique coupling-free point on F, and it corresponds to the geometric Hubble constant:

H_0(geom) = 70.3 km/s/Mpc (from d = 3 alone)

Moving away from the identity I = 1 in either direction activates the coupling ξ = 1/10 with opposite signs, depending on cosmic epoch:

• CMB epoch (deep interior of F): the coupling reduces the sound horizon, giving H_0(Planck) = H_0(geom)/√(1+ξ) ≈ 67.0 km/s/Mpc
• Local epoch (near the fibre boundary): the same coupling amplifies local distances, giving H_0(SH0ES) = H_0(geom) × √(1+ξ) ≈ 73.7 km/s/Mpc

These are conjugate projections of H_0(geom) through the fibre structure. Their ratio is an exact algebraic identity (Tier A):

• H_0(SH0ES)/H_0(Planck) = 1 + ξ = 11/10 (exact, zero free parameters)
• H_0(geom) = √[H_0(Planck) × H_0(SH0ES)] (geometric mean, exact)

Numerical verification: √(67.36 × 73.04) = 70.14 km/s/Mpc vs H_0(geom) = 70.3 km/s/Mpc (< 0.2σ). Observed ratio: 1.084 vs predicted 1.100 (1.5%). The Hubble "tension" is the observational signature of the non-trivial topology of F = (0,1]: Planck probes the interior (CMB), SH0ES probes the boundary (local universe). The universe is never at the monoid identity — and both instruments measure the distance from that identity, from opposite sides.

Baryogenesis

The CP violation from the Hertault algebra provides the geometric CP source ε_CP ~ α_*² (equivalently the CKM phase δ_CP = 2θ_H), placing the baryon-to-photon ratio in the observed range η_B ~ 6 × 10⁻¹⁰ (order of magnitude; the absolute value depends on a non-geometric efficiency factor). 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) = arccos(−sin²θ_H) = 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, algebraic Hubble-tension ratio)
• 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)
• 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

• 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 k_2 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. Inputs: d = 3 and M_Pl. Free parameters: zero.