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Dark Geometry — Behind The Horizon

Book Zero of the Dark Geometry Series

A self-contained introduction to a single-axiom, zero-parameter framework
that derives approximately 185 predictions of modern physics from one integer: $d = 3$.


DOI (concept) DOI (this version) License: CC BY 4.0 LaTeX Pages Format


Dark Geometry: Behind The Horizon — cover



📖 Read the PDF  ·  🗂 Zenodo Archive  ·  📚 Printed Edition (KDP)  ·  🧭 Series Overview

The promise

Give me one number — just one — and I will derive the mass of the electron, the mass of the Higgs boson, the fine-structure constant, the expansion rate of the universe, the temperature of a black hole, the reason dark matter has never been detected, and a research programme connecting the holographic structure of spacetime to the Riemann Hypothesis.

The number is 3. Space has three dimensions.

This is Book Zero of the Dark Geometry series — a guided tour through an unusual but internally consistent geometric framework. Written by a surgeon, not a professional theoretical physicist. Honest about what is proven, what is derived, and what remains conjecture.

Table of contents

At a glance
Title Dark Geometry: Behind The Horizon
Series position Book Zero (introductory volume to Books I–IV)
Author Hugo Hertault — Papeete, Tahiti, French Polynesia
Edition First edition, 2026
Format 6" × 9" paperback, 271 pages
Concept DOI (all versions) 10.5281/zenodo.19673186
Version DOI (this release) 10.5281/zenodo.19673187
Print availability Amazon KDP (ISBN assigned at publication)
Open access Zenodo, CC BY 4.0
Prerequisites Exponential, logarithm, $\pi \approx 3.14$. No quantum mechanics required.

The central claim

At the technical core of Dark Geometry is a single local identity between geometry and information content — the Hertault Axiom:

$$ \boxed{,;e^{d,\sigma(x)} ;=; \mathcal{I}(x) ;;\Longrightarrow;; e^{4\sigma(x)} ;=; \frac{S_{\mathrm{ent}}(x)}{S_{\mathrm{Bek}}(x)},} $$

Here

  • $\sigma(x)$ is the conformal mode of the local metric — a single scalar that describes how stretched or compressed geometry is at a point, relative to a horizon reference.
  • $\mathcal{I}(x) = S_{\mathrm{ent}}(x)/S_{\mathrm{Bek}}(x)$ is the local information saturation: the entanglement entropy actually present in a region, divided by the Bekenstein bound it could maximally contain.
  • $d$ is the dimension of space. It cannot be two (knots unstable, QFT trivial), it cannot be four or higher (no stable bound states, orbits unstable). It must be three.

The axiom says: geometry is information. Not related to. Not proportional to. Is.

From this one equation — combined with $d=3$ and the Planck mass as the only dimensional scale — the book derives, chapter by chapter:

  • the dark sector,
  • the cosmological constant and its hierarchy,
  • the masses and couplings of the Standard Model,
  • the resolution of the Hubble and $\sigma_8$ tensions,
  • the origin of the Big Bang (as the exit of a parent white hole),
  • and a physical interpretation of the Riemann zeros as resonances of the holographic fibre.

The Hertault angle and the cosmic beam splitter

A single geometric angle controls the holographic boundary of our universe:

$$ \theta_H ;=; \arctan!\frac{1}{\sqrt{2}} ;\approx; 35.264^\circ $$

It appears because $d = 3$ forces a 2-to-1 partition at the boundary: information splits into a bulk channel (gravitational, carrying $\cos^2\theta_H = 2/3$ of the flux) and a boundary channel (matter, carrying $\sin^2\theta_H = 1/3$). The so-called "cosmic coincidence" $\Omega_\Lambda/\Omega_m \approx 2$ becomes a theorem: $\cot^2\theta_H = 2$ exactly.

The EML operator

Every elementary transcendental function used in physics — $\sin$, $\cos$, $\pi$, $\arctan$, the exponential itself — can be derived from one single two-variable operator:

$$ E(x,,y) ;=; \exp(x) ;-; \ln(y) $$

The Hertault Axiom is the fixed-point equation of EML applied to spacetime:

$$ \hat{E}\bigl(d\sigma,;\mathcal{I}\bigr) ;=; 0 \qquad \Longleftrightarrow \qquad e^{d\sigma} = \mathcal{I}. $$

This reframes "spacetime" as the self-consistent configuration in which geometry computes its own information content. Every prediction of Dark Geometry is an EML evaluation of the input $d=3$.

Landmark predictions

All predictions follow from $d = 3$ and the Planck mass. Free parameters: zero.

Coupling constants

Observable Formula Predicted Observed Error
$1/\alpha_{\mathrm{em}}$ $8\pi^{2}\sqrt{3} + \tfrac{2\ln 3}{3\pi}$ $136.99$ $137.036$ 0.03 %
$\alpha_s(M_Z)$ $\sin(2\theta_H)/8$ $0.11785$ $0.1179$ 0.04 %
$\sin^2\theta_W$ $d/(d^{2}+d+1)$ $0.23077$ $0.2312$ 0.19 %

Leptons and electroweak sector

Observable Formula Predicted Observed Error
Koide $Q$ $\cos^2\theta_H$ $0.\overline{6}$ $0.666661$ $9\times10^{-4}$
$m_p / m_e$ $6\pi^{5}$ $1836.118$ $1836.152$ 19 ppm
Higgs VEV $v_H$ $2\sqrt{2},M_{\mathrm{Pl}},e^{-4\pi^{2}}$ $246.225$ GeV $246.22$ GeV 22 ppm
Higgs mass $m_H$ $v_H,(2/\pi)^{3/2}$ $125.07$ GeV $125.20$ GeV $0.10$ %
$W$ boson mass tree-level $80.31$ GeV $80.38$ GeV $0.08$ %

Cosmology

Observable Formula Predicted Observed Error
$\Omega_\Lambda$ $\cos^2\theta_H$ $2/3$ $0.685$ $2.8$ %
$\Omega_m$ $\sin^2\theta_H$ $1/3$ $0.315$ $5.4$ %
$\Omega_\Lambda/\Omega_m$ $\cot^2\theta_H$ $2$ (exact) $2.17$ $7.8$ %
Hubble ratio $H_0^S/H_0^P$ $1+\xi = 11/10$ $1.100$ $1.084$ $1.5$ %
$\sigma_8$ holographic $0.766$ $0.766$ $0.0$ %
$\rho_\Lambda$ $\propto e^{-28\pi^{2}}$ $2.44\times10^{-47},\mathrm{GeV}^4$ $2.45\times10^{-47},\mathrm{GeV}^4$ $0.3$ %

Neutrinos

Observable Formula Predicted Observed Error
$m_3$ $4d^{4}\pi^{3}v_H^{2}/M_{\mathrm{Pl}}$ $49.88$ meV $49.5$ meV $0.7$ %
$\Delta m^{2}{21}/\Delta m^{2}{31}$ $1/F_9 = 1/34$ $0.02941$ $0.0307$ $4.2$ %
$\sin\theta_{13}$ $2\alpha_*$ $0.1500$ $0.1492$ $0.5$ %

Chemistry, QCD, condensed matter

Observable Formula Predicted Observed
Tetrahedral angle $\arccos(-1/3)$ $109.47^\circ$ $109.5^\circ$
Yang–Mills mass gap 8-step Mosco convergence $0.18003,\mathrm{GeV}^{2}$ $0.180,\mathrm{GeV}^{2}$ (lattice)
Proton mass $d \cdot a^{2}$ $941.5$ MeV $938.3$ MeV
Nuclear $a_V$ $m_p/60$ $15.66$ MeV $15.75$ MeV
QO freq. ratio $F_5/F_4 = 5/3$ $1.667$ $1.675$

Decisive, forward-looking predictions

  • Null signal at all direct dark-matter detectors. $\sigma_{\mathrm{SI}} \equiv 0$ for WIMP-like scattering. Consistent with LZ and XENONnT to date — predicted a priori.
  • No supersymmetric partners at any energy. SUSY forbidden at the structural level because $d=3$ is already rigid.
  • Only tensor polarisations in gravitational waves. Scalar memory prediction $\Delta\mathcal{I} \propto |\Delta h|^2$ testable at LISA / Einstein Telescope.
  • Dark-energy equation of state $w(z) = -1 + w_a,z/(1+z)$ with $w_a \approx 0.10$ — directly testable by DESI and Euclid.
  • CMB large-scale anomaly axis — hemispherical asymmetry, low quadrupole, quadrupole–octupole alignment and cold spot all aligned on a single cascade axis. CMB-S4 will see it or not.
  • Laboratory signatures of the Hertault angle — boundary-to-bulk conductivity ratio $\sigma_s/\sigma_b = \cot^2\theta_H = 2$ in layered holographic materials; Fibonacci ratios in quantum-oscillation frequencies.
  • Proton lifetime $\tau_p \approx 7\times 10^{40}$ yr, above current bounds and beyond next-generation detectors.

Full prediction table with ~185 entries, broken down by tier: docs/predictions-table.md.

Book structure

The book is organised in four parts, plus a technical appendix. Confidence tiers (A / B / C) are labelled throughout.

Part I — The Single Axiom
  1. Why Three Dimensions?
  2. The Hertault Axiom
  3. One Operator to Rule Them All: The EML Connection
Part II — The Holographic Structure
  1. The Cosmic Beam Splitter
  2. The Big Bang Was Not the Beginning
  3. The Holographic Fibration
  4. Spacetime as a Quantum Circuit
  5. Why $\pi$ Is Space and $2\pi$ Is Time
  6. The Dark Boson Is the Clock
  7. Newton's Constant Is a Beam Splitter Coefficient
  8. Einstein's Equations and the Ghost That Wasn't
Part III — The Dark Sector and the Constants of Nature
  1. Dark Matter and Dark Energy Are the Same Field
  2. The Fine Structure Constant: Why 1/137?
  3. The Koide Formula and Lepton Masses
  4. The Higgs Boson and the Hierarchy Problem
  5. Neutrinos, Fibonacci, and the Number 34
Part IV — Cosmology and Beyond
  1. The Cosmological Constant: Seven Faces of One Number
  2. The Hubble Tension: An Algebraic Resolution
  3. The $\sigma_8$ Tension: Structure Growth Resolved
  4. The CMB Anomalies: Four Puzzles, One Axis
  5. Black Holes, Wormholes, and Baby Universes
  6. The Standard Model Gauge Group: Derived
  7. The Hertault–Schrödinger Equation
  8. Chemistry from $d = 3$
  9. Dark Geometry in the Laboratory
  10. $E = mc^2$ in Dark Geometry
  11. The Yang–Mills Mass Gap
  12. The Riemann Hypothesis
  13. Dark Geometry and the Major Theories
  14. The Complete Picture and the Decisive Tests
Appendices
  • Summary of All Predictions
  • The Yang–Mills Mass Gap: Reference
  • The Riemann Hypothesis: Reference
  • Mathematical Companion: The Master Equations
  • Derivations from First Principles

The Dark Geometry series

# Title Role DOI (concept)
0 Behind The Horizon (this book) Accessible guided tour of the whole framework 10.5281/zenodo.19673186
I Informational Relativity Foundational volume: axiom, dark sector, cosmological constant, tensions 10.5281/zenodo.18132261
II Informational Geometry Mathematical foundations, particle physics, Koide, electroweak 10.5281/zenodo.18870211
III Quantum Geometry Topological informational gravity, Yang–Mills mass gap 10.5281/zenodo.18929646
IV The Holographic Fibration Full geometric construction, MERA, Riemann programme 10.5281/zenodo.19546658

Reading order. New readers start with Book Zero. Technical readers proceed to I → II → III → IV; each of I–IV is self-contained and can also be read independently.

Confidence tiers

Every quantitative claim in the book carries one of three labels:

Tier Meaning Example
A Rigorously proven mathematical theorem. Ryu–Takayanagi from the axiom; $d=3$ topological uniqueness.
B Derived result with strong numerical verification, error < 2 %. $1/\alpha_{\mathrm{em}}$, $m_p/m_e$, $\sigma_8$, Yang–Mills mass gap.
C Plausible conjecture, consistent with current data. $w_a \approx 0.10$ for dark energy; specific CMB axis.

The tier structure is a honesty contract with the reader. If a prediction is at Tier B, the framework stakes something concrete on it; if at Tier C, it is presented as a direction to probe rather than a result to defend.

Research programmes

The book reserves two chapters for open problems in pure mathematics.

Yang–Mills mass gap (Tier A/B)

A bound on the spectral gap of four-dimensional Yang–Mills theory:

$$ \Delta^{2}_{\mathbb{R}^{4}} ;\geq; 0.18003,\mathrm{GeV}^{2} ;>; 0 $$

obtained in 8 steps via Mosco convergence from the compactified geometry $S^{2}\times\mathcal{F}$ to $\mathbb{R}^{4}$. All constants fixed by $d=3$; lattice agreement at the $0.015,%$ level. One convergence lemma awaits independent verification.

The Riemann Hypothesis (Tier A/C — research programme, not a proof)

A rigorous algebraic identity (Tier A) is established between the spectral zeta function of the holographic fibre and the Riemann zeta function. The book proposes a physical interpretation of the non-trivial zeros as resonances of cosmic expansion, and outlines a strategy based on log-concavity of the holographic flow. One analytical lemma is explicitly flagged as unverified. No proof is claimed.

What the book does not do

To pre-empt misunderstandings, here is an honest list of limitations:

  • Book Zero is not a proof of the framework. It is an accessible tour. The derivations live in Books I–IV.
  • A proof of the Riemann Hypothesis is not claimed. The book offers a research direction with clearly marked open steps.
  • Several Tier-B results disagree with experiment at the 1–5 % level. Loop-level refinements would be required to match precision data.
  • Peer-reviewed validation is in progress. The work has not yet been independently verified by the broader physics community.
  • The author is not a professional theoretical physicist. He is a surgeon who has spent several years on this framework. He asks to be read on the merit of the arguments, not on credentials.

Repository layout

dark-geometry/
├── README.md                          ← you are here
├── LICENSE                            ← CC BY 4.0
├── CITATION.cff                       ← academic citation metadata
├── CHANGELOG.md                       ← version history
├── .gitignore                         ← LaTeX auxiliary files
│
├── book/
│   ├── interior/
│   │   └── DarkGeometry_BookZero.tex  ← main LaTeX source (271 pages)
│   └── cover/
│       └── Book0_FullCover_KDP.tex    ← KDP wraparound cover (12.86″ × 9.25″)
│
├── pdf/
│   ├── DarkGeometry_BookZero_interior.pdf   ← interior only (upload to KDP)
│   ├── DarkGeometry_BookZero_FULL.pdf       ← reading copy: front cover + interior + back cover
│   └── Book0_FullCover_KDP.pdf              ← wraparound cover (upload to KDP)
│
├── zenodo/
│   └── description.html               ← rich HTML description for the Zenodo record
│
├── docs/
│   ├── predictions-table.md           ← full ~185-prediction table
│   ├── glossary.md                    ← fifteen key concepts in plain language
│   ├── reading-guide.md               ← how to read the book by background
│   └── faq.md                         ← frequently-asked questions
│
├── assets/
│   ├── cover-front.png                ← front cover at 200 dpi
│   ├── cover-back.png                 ← back cover at 200 dpi
│   └── cover-fullwrap.png             ← full KDP wraparound at 120 dpi
│
└── .github/
    ├── workflows/
    │   └── build.yml                  ← auto-compiles PDF on every push
    └── ISSUE_TEMPLATE/
        ├── bug-report.md              ← typos, LaTeX issues
        ├── scientific-question.md     ← physics/math questions
        └── prediction-feedback.md     ← report experimental match/mismatch

Getting the book

📥 Download the PDF

🗂 Zenodo (open access, long-term archive)

Citable, DOI-backed, CC BY 4.0:
https://doi.org/10.5281/zenodo.19673186 (concept DOI — always latest version)

📚 Print edition (Amazon KDP)

The printed paperback is being published through Amazon KDP. A link will be added here once the ISBN is assigned.

Build from source

If you want to rebuild the PDF yourself, you need a full TeX Live (or MiKTeX) installation including pdflatex, latexmk, and the packages listed below.

Required packages

All are part of any modern TeX Live: geometry, inputenc, fontenc, mathpazo, microtype, amsmath, amssymb, amsthm, mathtools, mathrsfs, bm, setspace, xcolor, booktabs, array, longtable, multirow, tabularx, adjustbox, enumitem, graphicx, float, caption, epigraph, needspace, tikz, pgfplots, fancyhdr, truncate, tocloft, hyperref, titlesec, pdfpages.

Interior

cd book/interior
latexmk -pdf -interaction=nonstopmode DarkGeometry_BookZero.tex

Output: DarkGeometry_BookZero.pdf, 271 pages, 6″ × 9″.

Wraparound cover

cd book/cover
pdflatex Book0_FullCover_KDP.tex

Output: Book0_FullCover_KDP.pdf, 1 page at 12.8603″ × 9.25″ (matching the 271-page spine of 0.6103″).

Continuous integration

A GitHub Actions workflow in .github/workflows/build.yml rebuilds both PDFs on every push and uploads them as artifacts.

How to cite

Plain text

Hertault, H. (2026). Dark Geometry: Behind The Horizon. Book Zero of the Dark Geometry Series. Zenodo. https://doi.org/10.5281/zenodo.19673186

BibTeX

@book{Hertault_BookZero_2026,
  author       = {Hertault, Hugo},
  title        = {{Dark Geometry: Behind The Horizon}},
  subtitle     = {{Book Zero of the Dark Geometry Series}},
  year         = {2026},
  publisher    = {Zenodo},
  address      = {Papeete, Tahiti, French Polynesia},
  edition      = {First},
  doi          = {10.5281/zenodo.19673186},
  url          = {https://doi.org/10.5281/zenodo.19673186},
  note         = {Concept DOI; always resolves to the latest version.}
}

Other formats

A CITATION.cff file is included at the root of the repository: GitHub will automatically render a "Cite this repository" button, and tools such as Zotero, the Citation File Format converter, and Zenodo itself can ingest it directly.

Keywords and topics

dark-geometry · hertault-axiom · hertault-angle · holographic-principle · holographic-fibration · conformal-mode · information-saturation · bekenstein-bound · ryu-takayanagi-formula · tensor-networks · mera · bond-dimension · eml-operator · fine-structure-constant · proton-electron-mass-ratio · koide-formula · higgs-mass · electroweak-vacuum · neutrino-masses · fibonacci-hierarchy · dark-matter · dark-energy · dark-boson · cosmological-constant · vacuum-energy · lambda-cdm · hubble-tension · sigma8-tension · cmb-anomalies · white-hole-cosmology · horizon-problem · baby-universes · quantum-gravity · emergent-spacetime · standard-model-derivation · unified-theory · zero-free-parameters · yang-mills-mass-gap · clay-millennium-problems · riemann-hypothesis · spectral-zeta-function · prime-number-distribution · gravitational-waves · scalar-memory · celestial-holography · bms-symmetries · condensed-matter-holography · fibonacci-quantum-oscillations · three-spatial-dimensions · topology-of-space · foundations-of-physics

License

All content in this repository (text of the book, LaTeX source, documentation, cover design) is released under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

You are free to:

  • Share — copy and redistribute the material in any medium or format
  • Adapt — remix, transform, and build upon the material for any purpose, even commercially

Under the following terms:

  • Attribution — you must give appropriate credit, provide a link to the license, and indicate if changes were made.

See LICENSE for the full legal text.

Author

Hugo Hertault is a surgeon based in Papeete, Tahiti, French Polynesia. Dark Geometry is the result of several years of work done outside clinical hours, driven by a question that the night sky of French Polynesia makes difficult to ignore: what world are we living in?

He approaches this work without academic affiliation in theoretical physics, and with the explicit understanding that the framework must survive scrutiny from the professional community to become more than a personal effort. This is a work of ideas offered openly to that community.

Acknowledgements and disclaimer

This book is a work of theoretical speculation, presented with the rigour the author could muster and labelled honestly by confidence tier. It is not a peer-reviewed publication. It has benefited from every physics and mathematics textbook, paper, and open lecture its author has been able to consult; errors and naïvetés remain entirely his own.

If you find a mistake — a typo, a miscalculated number, a missing citation, a derivation that does not go through — please open an issue. That is what this repository is for.

Contact and feedback


"The universe will decide which predictions survive."

— Dark Geometry, Book Zero, Preface


Dark Geometry — Behind The Horizon — Book Zero — First Edition, 2026
© 2026 Hugo Hertault — Licensed under CC BY 4.0

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