"When Information Dreams of Dark Geometry — Mathematical Foundations of Dark Geometry & Particle Physics"
Author: Hugo Hertault — Tahiti, French Polynesia Series: Dark Geometry — Book II of V Companion to: Informational Relativity (Book I)
Informational Geometry develops the rigorous mathematical foundations of Dark Geometry: the holographic fibration, the Hertault algebra, spectral theory, representation theory, anomaly matching, and the complete derivation of Standard Model parameters from a single integer.
Where Book I asked what the framework predicts cosmologically, Book II shows how the entire Standard Model — gauge group, particle masses, coupling constants, mixing angles — follows mathematically from d = 3.
~170 quantitative predictions. ~80 with sub-percent accuracy. Two inputs (
d = 3and the Planck mass). Zero free parameters.
This repository contains the verification code and the master table of predictions — not the book text. The book itself (600 pages) is archived on Zenodo and available in print/ebook.
The fundamental structure is the holographic fibration:
H = M⁴ ×_σ F
where F = (0, 1] is the informational fibre, equipped with the Fisher–Rao metric of information geometry:
ds²_F = dI² / [I²(1−I)²]
The warping factor σ(x) is fixed by the Hertault Axiom e^{4σ} = I = S_ent/S_Bek. The entire Standard Model — gauge group, particle spectrum, coupling constants — emerges from the geometry, topology, and spectral theory of H.
| Category | Result | Error | Tier |
|---|---|---|---|
| Gauge group | SU(3)_C × SU(2)_L × U(1)_Y | exact | A |
| Generations | n_gen = 3 | exact | A |
| Weinberg angle | sin²θ_W = 3/13 | 0.2% | B |
| Fine-structure constant | α⁻¹ = 137.04 (with running) | 4 ppm | B+ |
| Strong coupling | α_s(M_Z) = √2/12 ≈ 0.1179 | 0.08% | B |
| Lepton masses | Koide formula, zero free parameters | < 0.006% | B |
| Electroweak scale | v_H = 2√2 M_Pl e^{−4π²−βα*²} = 246.225 GeV | 22 ppm | B |
| Higgs mass | m_H = v(2/π)^{3/2} = 125.07 GeV | 0.03% | B+ |
| W boson | m_W = 80.31 GeV | 0.08% | B |
| Z boson | m_Z = m_W√(13/10) = 91.57 GeV | 0.42% | B |
| Proton/electron ratio | m_p/m_e = 6π⁵ = 1836.12 | 19 ppm | B |
| Absolute neutrino mass | m₃ = 4d⁴π³v²/M_Pl = 49.88 meV | 0.7% (1.1σ) | B |
| Neutrino mass-squared ratio | Δm²₂₁/Δm²₃₁ = 1/34 (9th Fibonacci) | JUNO test | B |
| Hubble-tension ratio | H₀(SH0ES)/H₀(Planck) = 11/10 | exact | A |
| Tetrahedral bond angle | arccos(−1/3) = 109.47° | 0.03% | A |
| Atomic masses (118 elements) | full periodic table | ~0.39% avg | B |
A full machine-checkable list lives in docs/master_table.csv.
The fibration is fixed by four axioms — informational content e^{4σ} = I, holographic saturation I ∈ (0,1], factorisation over independent regions, and smoothness σ ∈ C^∞(M⁴). These fix β = (d−1)/d = 2/3, the Hertault angle θ_H = arccos√(2/3) ≈ 35.264°, and α_* = √2/(6π) as functions of d = 3 alone.
The Hertault algebra h_d is a semidirect product of the rotation algebra with a Heisenberg algebra at the Hertault angle. For d = 3 it satisfies the exceptional isomorphism
h_3 ≅ su(2) ⊕ u(1)
which exists only in three dimensions and is the algebraic origin of the electroweak gauge group.
SU(3)_C × SU(2)_L × U(1)_Y emerges from three independent structures of H: U(1)_Y from the fibre automorphisms Aut(F), SU(2)_L from h_3 ≅ su(2), and SU(3)_C from the Peter–Weyl decomposition on the holographic surface S² combined with the hairy-ball theorem. There is no larger group that is subsequently broken.
n_gen = 3 is not an input: the Z₃ discrete 't Hooft anomaly of h_3 on S² × F cancels only for n_gen = d = 3.
Koide's empirical relation becomes a theorem:
Q = (m_e + m_μ + m_τ) / (√m_e + √m_μ + √m_τ)² = β = 2/3 ⟺ d = 3
The complete lepton mass formula has zero free parameters given the electroweak scale v:
√m_k = √(3 α*³ v) · [ 1 + √2 · cos( 2π k/3 + 2/9 ) ], k = 0 (e), 1 (μ), 2 (τ)
with phase excess ε = 2/9 = β(1−β) from beam-splitter interference and amplitude √2 = cot θ_H.
| Lepton | Predicted | Experimental | Error |
|---|---|---|---|
| m_e | 0.5110 MeV | 0.5110 MeV | 0.006% |
| m_μ | 105.653 MeV | 105.658 MeV | 0.005% |
| m_τ | 1776.88 MeV | 1776.86 MeV | 0.001% |
Every coupling has the form α = f(θ_H) / n_top, with f(θ_H) a bulk/boundary projection and n_top a topological normalisation (a sphere volume):
α_em = sin θ_H / (8π²) → 1/136.8 (Thomson); 1/137.04 after QED running (4 ppm)
α_s = sin(2θ_H) / 8 = √2/12 ≈ 0.1179 (0.08%)
sin²θ_W = d / (d²+d+1) = 3/13 ≈ 0.2308 (0.2%)
α_* = sin(2θ_H) / (4π) = √2/(6π) ≈ 0.0750
Instanton tunnelling through the Rosen–Morse ground state on the fibre generates the electroweak scale with no fine-tuning:
v_H = 2√2 · M_Pl · e^{−4π² − β α*²} = 246.225 GeV (observed 246.220; 22 ppm)
The hierarchy M_Pl/v_H ~ e^{4π²} is a geometric tunnelling factor, not a tuning. The Higgs quartic follows from fibre curvature, λ_H = (d−1)³/(2π³) = 4/π³, giving m_H = v(2/π)^{3/2} = 125.07 GeV (0.03%).
The Weinberg dimension-5 operator with a see-saw scale derived from the fibration,
Λ = α_s · α*³ · M_Pl = M_Pl / (648 π³),
m₃ = 4 d⁴ π³ v² / M_Pl = 49.88 meV (observed 49.53 ± 0.33 meV; 0.7%, 1.1σ)
and the mass-squared ratio Δm²₂₁/Δm²₃₁ = 1/F₉ = 1/34 from the Fibonacci gap-labelling theorem (where 34 is the 9th Fibonacci number) — a parameter-free prediction that JUNO is testing through 2030.
The fibre F = (0,1] is a multiplicative monoid with identity I = 1, where the Dark Boson coupling vanishes exactly. This unique coupling-free point gives H₀(geom) = 70.3 km/s/Mpc from d = 3 alone. Moving off the identity activates ξ = 1/10 with opposite signs by epoch:
H₀(Planck) = H₀(geom)/√(1+ξ) ≈ 67.0 (CMB, deep interior of F)
H₀(SH0ES) = H₀(geom)·√(1+ξ) ≈ 73.7 (local, near the boundary)
H₀(SH0ES)/H₀(Planck) = 1 + ξ = 11/10 (exact, zero free parameters)
H₀(geom) = √[H₀(Planck)·H₀(SH0ES)] (geometric mean, exact)
The "tension" is the observational signature of the non-trivial topology of F: the two instruments probe the fibre from opposite sides of the monoid identity.
Every canonical number in the book is reproduced by a script before it is quoted. The scripts are that discipline made public — each one ends in ALL PASS.
| Script | What it checks |
|---|---|
verify_identities.py |
Master identities from d = 3: θ_H, the budget ratio d−1, ξ = 1/10, Δn = 2βα*² = 2/(27π²) (all algebraic forms coincide exactly), u_max = π/α*; plus honest INFO lines (Hubble ratio at 0.9σ, S₈ at 0.6–0.9σ). |
verify_masses.py |
The mass machinery: μ_LO = (α*/2) M_Pl e^{−π/α*} ≈ 299 MeV (NLO book value 313.7); v_H(NLO) = 246.225 GeV, 23 ppm from observation (the LO formula gives 247.2, 0.38%); m_p/m_e = 6π⁵ at 18.8 ppm of CODATA. |
verify_spectrum.py |
Spectral theory of the monoid fibre (12 checks): s(s−1) = β; the canonical Rosen–Morse potential as a perfect square with no L² bound state; the SUSY partner carrying ψ_S at exactly −1/4; the unit tail rate 2κ = 1; the Robin datum κ* = 1/2 to 12 digits; the shooting test at E = −1/4; and the log-chart Pöschl–Teller picture. |
pip install -r requirements.txt
cd code
for f in *.py; do python "$f"; done # each script must end in ALL PASSDependencies: numpy, scipy (standard scientific Python).
Book II — Informational Geometry
Part I — Geometric Foundations
1. The Hertault Axiom and the Informational Manifold
2. The Holographic Fibration H = M⁴ ×_σ F
3. Spectral Theory of the Monoid Fibre
4. Necessity of the Fundamental Mathematical Constants
Part II — Algebraic Structure
5. The Hertault Algebra h_d
6. Representation Theory of h_3
Part III — Topological Constraints
7. Informational Cohomology
8. 't Hooft Anomalies and the Three Generations
Part IV — Gauge Structure
9. Emergence of the Standard Model Gauge Group
Part V — Physical Applications
10. Applications to Particle Physics
11. The Mass Operator on the Holographic Fibration
12. The Instanton Prefactor
13. The Universal Mass Scale
14. Quark–Lepton Mass Relations
15. The Mass of the Dark Boson
16. The Absolute Neutrino Mass Scale
17. Geometric Grand Unification
18. Nuclear Physics from the Sub-Planckian Anchor
19. Applications to Chemistry
20. Condensed Matter, Astrophysics, and Testable Predictions
21. Compact Astrophysical Objects: Neutron Stars, Magnetars, Hypermassive Remnants
Part VI — Informational Dynamics
22. Informational Thermodynamics
23. Quantum Entanglement in the Holographic Fibration
24. Emergent Quantum Mechanics
25. Supplementary Structures and Detailed Calculations
Part VII — Compilation
26. The Periodic Table of Informational Geometry
27. Master Table: Constants from d = 3
- JUNO (2025–2030): neutrino mass-squared ratio converging toward 1/34
- DESI (2024–2028): sum of neutrino masses Σm_ν = 58.4 meV (below the current 72 meV bound)
- 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
Every quantitative claim is classified by evidential tier:
| Tier | Meaning | Examples |
|---|---|---|
| A | Proven / algebraically exact | Q_Koide = β, n_gen = 3, N_n = 2n², h_3 ≅ su(2), Hubble ratio 11/10 |
| B | Derived, sub-percent, with stated approximations | α, α_s, sin²θ_W, m_H, m_W, v_H, m₃ |
| C | Semi-empirical / order of magnitude (1–10%) | quark masses, CKM, η_B |
| D | Inputs | d = 3, M_Pl |
| E | Observational coincidences awaiting derivation | — |
Observational comparisons are made like for like (weak-lensing comparisons in S₈, the observable the surveys report). Canonical numbers are reproduced by a script before being quoted.
Book II — Informational Geometry is archived on Zenodo: doi:10.5281/zenodo.18870211 (the DOI resolves to the latest version; print/ebook editions are listed there). ISBN 979-8250814348.
| Volume | Book (Zenodo) |
|---|---|
| Book 0 — Dark Geometry: Behind the Horizon | 10.5281/zenodo.19673186 |
| Book I — Informational Relativity | 10.5281/zenodo.18132261 |
| Book II — Informational Geometry | 10.5281/zenodo.18870211 |
| Book III — Quantum Geometry | 10.5281/zenodo.18929646 |
| Book IV — The Holographic Fibration | 10.5281/zenodo.19546658 |
This book began as a calculation on a napkin. In 2024, sitting in a café in Tahiti, I wrote β = (d−1)/d and set d = 3. The Hertault angle followed: θ_H = 35.26°. Then, on the same napkin: α = sin θ_H/(8π²) = 1/136.8 — close enough to the fine-structure constant to make my hands shake. I am not a professional physicist. I am a surgeon from Tahiti with a passion for fundamental questions.
— Hugo Hertault, Tahiti, 2026
See CITATION.cff. If you use this code, cite the book and this repository:
@book{hertault2026geometry,
author = {Hertault, Hugo},
title = {Informational Geometry: Mathematical Foundations of
Dark Geometry and Particle Physics},
series = {Dark Geometry},
volume = {II},
year = {2026},
publisher = {Self-published (KDP)},
address = {Tahiti, French Polynesia},
doi = {10.5281/zenodo.18870211}
}Code: MIT (see LICENSE). The books are separate works, licensed CC BY 4.0.
The Standard Model is not a theory. It is a theorem — a consequence of three-dimensional space viewed through the holographic principle. The universe will have the final word.