Informational Relativity — Dark Geometry · Book I

"When Dark Geometry Dreams of Information"

Author: Hugo Hertault — Tahiti, French Polynesia DOI: 10.5281/zenodo.18132261 Series: Dark Geometry — Book I of V Book: ~600 pages · 9 Parts · 81 chapters

"The universe is three-dimensional. From this, everything follows."

This repository contains the code for Book I (not the book text): a set of scripts that reproduce every canonical number — the constants, the dark sector, the H0 and S8 resolutions, the Fibonacci relations, the full prediction table — together with a derivation-verification suite for the axiom. Every number quoted below is reproduced by a script in code/ before it is used. Comparisons with weak-lensing data are made like-for-like in S8, the observable the surveys report.

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Table of Contents

1. Overview
2. The Single Axiom
3. Fundamental Constants from d=3
4. The Cosmic Beam Splitter
5. Newton's Constant Derived
6. The Dark Boson
7. Three Equivalent Formulations
8. Informational Thermodynamics
9. Resolution of the Hubble Tension
10. Resolution of the sigma8 Tension
11. Both Tensions, One Identity
12. The Fibonacci-Hertault Framework
13. Reach of the Framework
14. Book Structure
15. The Verification Suite
16. Quick Start
17. What Would Falsify Dark Geometry
18. The Dark Geometry Series
19. Method and Tiers
20. Citation

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Overview

Informational Relativity is Book I of the Dark Geometry series: the foundational technical volume. It proposes a unified framework for dark matter, dark energy, and quantum gravity built on a single axiom and a single integer, d = 3.

The central claim: dark matter and dark energy are not two separate substances. They are the same geometric field — the conformal mode of spacetime, the Dark Boson — behaving differently with the local matter density. In dense regions (galaxy halos) it clusters like dark matter; in empty regions (cosmic voids) it drives accelerated expansion like dark energy.

From d = 3 alone, and with zero free parameters, the framework derives the dark-sector budget (Omega_L, Omega_m), Newton's constant G_4, the exact Hubble ratio 11/10, the sigma8 suppression, the neutrino mass ratio, black-hole thermodynamics, and laboratory predictions. Every quantitative claim carries an explicit evidential tier (A: exact/proven; B: derived with stated approximations; C: conjecture/order of magnitude).

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The Single Axiom

The entire framework rests on one equation — the Hertault Axiom:

$$\boxed{e^{4\sigma(x)} = \mathcal{I}(x) \equiv \frac{S_{\text{ent}}(x)}{S_{\text{Bek}}(x)}}$$

where sigma(x) is the conformal mode of the metric, S_ent(x) the entanglement entropy of a small region, S_Bek(x) = 2 pi E R / (hbar c) the Bekenstein bound, and I(x) in (0,1] the informational saturation ratio.

Plain language: the spacetime volume element at each point equals the fraction of the holographic information bound that is actually saturated there. I = 1 at a horizon; I -> 0 in vacuum.

The axiom is derived, not postulated: it is the unique minimum of an entropic free energy (vacuum volume cost against the Casini energy of the content), taken pointwise; a no-go theorem shows the entanglement first law has zero conformal weight and cannot fix sigma (the derivation completes Jacobson — fixed-volume directions give Einstein, the unfrozen conformal direction gives the axiom). The conformal mode is not a propagating ghost: the Hertault constraint makes it a response field with zero propagating degrees of freedom, eliminating the Goroff-Sagnotti two-loop divergence structurally.

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Fundamental Constants from d=3

The Holographic Exponent

$$\beta = \frac{d-1}{d} = \frac{2}{3} = \cos^2\theta_H$$

Uniqueness of d = 3: the identity 4/d! = (d-1)/d has a unique positive integer solution, d = 3.

The Hertault Angle

$$\theta_H = \arccos\sqrt{2/3} \approx 35.264^\circ$$

Complete Table of Fundamental Parameters

Quantity	Formula	Value	Tier
beta	(d-1)/d	2/3 = 0.6667	A
theta_H	arccos sqrt(2/3)	35.2644 deg	A
alpha*	sin(2 theta_H)/(4 pi) = sqrt(2)/(6 pi)	0.0750264	A
xi	beta/[4(1+beta)]	1/10	A
g* (AS fixed point)	beta sqrt(3/2) = sqrt(6)/3	0.8165	B
S_0 (primordial entropy)	24 pi^2 beta = 16 pi^2	~158 bits	A
N (horizon entropy)	(d/(d-1)) d^((d+1)^(d+1))	~2.09x10^122	B/C

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The Cosmic Beam Splitter

At the cosmological horizon, the Hertault angle acts as a 2x2 unitary beam splitter:

$$|\psi\rangle_{\text{dark}} = \cos\theta_H,|\text{dark energy}\rangle + \sin\theta_H,|\text{matter}\rangle$$

Squaring the amplitudes gives the cosmic energy partition:

$$\Omega_\Lambda = \cos^2\theta_H = \frac{2}{3}, \qquad \Omega_m = \sin^2\theta_H = \frac{1}{3}, \qquad \frac{\Omega_\Lambda}{\Omega_m} = \cot^2\theta_H = 2 \ \text{(exact)}$$

The cosmic coincidence problem is resolved: Omega_L/Omega_m = 2 is a geometric constant fixed by d = 3, not a temporal accident.

Channel	Coefficient	Physical content
Surface (transmitted)	cos^2 theta_H = 2/3	Dark energy, Casimir vacuum, Omega_L
Bulk (reflected)	sin^2 theta_H = 1/3	Matter, gravity, G_4, Omega_m

The echo amplitude cascades as (sin^2 theta_H)^n = (1/3)^n.

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Newton's Constant Derived

Newton's gravitational constant is not a free parameter — it is the bulk-channel weight of the cosmic beam splitter:

$$\boxed{G_4 = \frac{\sin^2\theta_H}{8\pi M_5^3} = \frac{1}{24\pi M_5^3}}$$

Three independent routes give the same result: (1) dimensional reduction of the 5D Einstein-Hilbert action over the holographic fibre (Tier A); (2) Jacobson thermodynamics, in which only the bulk fraction sin^2 theta_H of the entanglement is gravitational (Tier B); (3) the AdS/CFT dual, in which the bulk channel generates gravity (Tier B).

The full scale-dependent gravity is

$$G_{\text{eff}}(k) = \frac{\sin^2\theta_H}{8\pi M_5^3}\left(1 + \frac{2\alpha_*^2}{1+(k/k_J)^2}\right),$$

which is what ties the H_0 and sigma8 effects to the same angle. The one-loop running is finite, delta G / G ~ 10^-124 (Tier B); the conformal ghost count is exactly 0 (Tier A).

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The Dark Boson

The Dark Boson is the conformal mode, phi_DG = sqrt(6) M_Pl sigma, constrained to carry zero propagating degrees of freedom. Its density-dependent mass changes sign at the critical density:

$$\boxed{m_{\text{eff}}^2(\rho) = (\alpha_* M_{\text{Pl}})^2\left[1 - \left(\frac{\rho}{\rho_c}\right)^{2/3}\right]}$$

Density regime	m^2_eff	Behaviour
rho >> rho_c (halos)	< 0	dark-matter clustering, rotation curves
rho = rho_c	= 0	critical point / transition
rho << rho_c (voids)	> 0	dark energy, accelerated expansion

rho_c is derived, not fitted: from the axiom sigma(rho) = -(1/3) ln(rho/rho_c), and m^2 vanishes where the cosmological and matter terms balance. Numerically rho_c^(1/4) ~ 2.3 meV, the observed dark-energy scale. The Dark Boson couples to the trace of the stress-energy tensor, L_int = (alpha*/M_Pl) phi_DG T^mu_mu, and is chameleon-screened in the lab (rho >> rho_c), explaining the persistent null of fifth-force and direct-detection searches.

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Three Equivalent Formulations

Dark Geometry admits three exact, logically equivalent formulations, joined by a duality triangle at theta_H = 35.26 deg, beta = 2/3:

• IDG — Informational Dark Geometry. e^(4 sigma) = I = S_ent/S_Bek; gravity is the macroscopic signature of information redistribution, with current J^mu = (c^4/16G) grad^mu ln I.
• QGU — Quantum Gravity Unification. rho_DE^(1/4) = sqrt(E_Pl . E_H): the dark-energy scale is the geometric mean of the Planck and Hubble scales, dissolving the 122-order cosmological-constant problem. Five quantum-gravity programmes (Asymptotic Safety, LQG, AdS/CFT, CDT, celestial holography) appear as projections of theta_H.
• HDG — Holographic Dark Geometry. Built on the holographic fibration H = M^3 x_sigma F, F = (0,1]; dark matter is the bulk, dark energy the boundary.

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Informational Thermodynamics

Law	Informational form	Classical limit
0th	I_A = I_B (equilibrium)	T_A = T_B
1st	dS_info = 0 (information conserved)	dU = T dS - P dV
2nd	J ~ -grad I (flow to low I)	dS >= 0
3rd	I in (0,1] (saturation bounded)	T = 0 unattainable

The second law is emergent (information conservation + coarse-graining); the underlying dynamics is unitary. The Hawking temperature T = E/S and the entropy S_BH = A/(4 l_P^2) are derived, not postulated. The primordial entropy S_0 = 24 pi^2 beta = 16 pi^2 ~ 158 bits should leave imprints in the CMB low multipoles.

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Resolution of the Hubble Tension

The H_0 tension (~4.8 sigma) is a consequence of the beam splitter. From d = 3 and the horizon entropy, the bare geometric value is

$$H_0^{\text{(geom)}} \approx 70.3 \ \text{km/s/Mpc}.$$

The non-minimal coupling xi = 1/10 acts in opposite directions at the two epochs:

$$H_0^{\text{Planck}} = \frac{H_0^{\text{(geom)}}}{\sqrt{1+\xi}} \approx 67.0, \qquad H_0^{\text{SH0ES}} = H_0^{\text{(geom)}}\sqrt{1+\xi} \approx 73.7 \ \text{km/s/Mpc},$$

with the exact identity (Tier A)

$$\boxed{\frac{H_0^{\text{SH0ES}}}{H_0^{\text{Planck}}} = 1 + \xi = \frac{11}{10}}$$

and the geometric mean recovering the bare value, sqrt(67.4 x 73.0) ~ 70.1 km/s/Mpc. The tension drops from ~4.8 sigma to ~1 sigma with zero free parameters.

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Resolution of the sigma8 Tension

The sigma8 tension (~3.6 sigma) is relieved by a geometric suppression of late-time growth. The suppression rate is

$$\boxed{\Delta n = \frac{\cos^4\theta_H \cdot \sin^2\theta_H}{2\pi^2} = \frac{2}{27\pi^2} \approx 7.50\times10^{-3}}$$

equivalently Delta n = 2 beta alpha*^2: the interference of the two beam-splitter channels. Propagating it through the growth factor and a scale-dependent transfer (Jeans scale k_J ~ 0.05 h/Mpc) gives

$$\sigma_8^{\text{DG}} \approx 0.77\text{--}0.79.$$

Compared like for like in S8 (the observable the surveys report), S8 ~ 0.78 sits within ~1 sigma of KiDS-1000 and DES Y3, between the Planck-LCDM and weak-lensing values, as required to relieve the tension. (Quoted central value sigma8 ~ 0.766 -> S8 ~ 0.779.)

Source	sigma8 / S8	Status
Planck (LCDM)	sigma8 = 0.811 +/- 0.006	early universe
Dark Geometry	sigma8 ~ 0.77-0.79	prediction
KiDS-1000	S8 = 0.766 +/- 0.020	<~1 sigma (in S8)
DES Y3	S8 = 0.759 +/- 0.021	<~1 sigma (in S8)

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Both Tensions, One Identity

The H_0 and sigma8 tensions are the first- and second-order manifestations of the same field, tied by the Coupling Identity:

$$\Delta n = 8,\alpha__^2,(1+\beta),\xi = 2\beta\alpha__^2, \qquad \frac{\Delta n}{\xi} = \frac{20}{27\pi^2} \approx 0.07503.$$

A single failed test of this ratio (Delta n from DESI/Euclid growth, xi from the Hubble ratio) breaks the framework. The continuum dark-energy equation of state is frozen at w = -1 to ~1 part in 10^120, so the operative near-term test is the ~4% suppression of f sigma8(z), not a rolling w(z).

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The Fibonacci-Hertault Framework

Fibonacci numbers arise as the optimal information-packing structure in 3 spatial dimensions (Perron-Frobenius on the holographic substitution; dS/dE|max = ln phi). Key relations:

Quantity	Expression	Predicted	Observed	Tier
beta	F_3/F_4 = 2/3	0.6667	—	A
Neutrino ratio Dm2_21/Dm2_31	1/F_9 = 1/34	0.02941	~0.0295 (JUNO by 2030)	C
Black-hole QPO ratio	F_4/F_3 = 3/2	1.500	1.500	—

Neutrinos live in a d^2 = 9-dimensional flavour x mass space; the ninth Fibonacci number is F_9 = 34.

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Reach of the Framework

• Ghost-free quantum gravity: conformal mode is a constraint, Goroff-Sagnotti divergence absent.
• Black-hole thermodynamics: Bekenstein-Hawking entropy and Hawking temperature from T = E/S.
• Small-scale structure: cored halo profiles rho_0/[1+(r/r_s)^2] with vanishing central slope (no cusp), ~60 Milky-Way satellites (not ~500), too-big-to-fail dissolved.
• No gravitational slip: eta = Psi/Phi = 1 — discriminates against f(R), Brans-Dicke, Horndeski.
• DESI BAO compatibility via a sound-horizon shift rather than a modified late-time expansion law.
• Gravitational-wave echoes: Delta t ~ 36 ms for a 30 M_sun remnant, amplitudes decaying as (1/3)^n.
• Pure mathematics: Hardy-Ramanujan asymptotics and zeta(2) = pi^2/6, the modular relation (ST)^3 = -I encoding d = 3, the Fibonacci limit 1/phi.
• Condensed-matter analogues (Tier C, testable now): in Kondo insulators like YbB12, surface/bulk conductivity -> 2 = cot^2 theta_H, quantum-oscillation frequencies in Fibonacci ratios (3:2, 5:3, 8:5), transport exponents ~ T^(2/3).

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Book Structure

The book has 9 Parts, 81 chapters (~600 pages):

1. Mathematical Foundations — the dark-sector evidence, the Hertault axiom, its uniqueness theorem.
2. The Three Formulations — IDG, QGU, HDG.
3. The Dark Boson and Emergent Constants — mass function, alpha*, S_0 = 16 pi^2, the emergence of c, G, hbar, rho_DE.
4. Informational Thermodynamics — four informational laws, black holes as engines, time from information.
5. The Fibonacci-Hertault Framework — five constants, the quantum bounce, Fibonacci structure.
6. Cosmological Predictions — the tensions resolved, small-scale structure, DESI, solar-system tests, no gravitational slip.
7. Mathematical Connections — Hardy-Ramanujan, mock theta functions, the modular group, the role of d = 3.
8. Physical Interpretations — white holes and the informational membrane, condensed-matter analogues.
9. Conclusions and Perspectives — the beam-splitter synthesis, open questions, the road to Books II-IV.

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The Verification Suite

All scripts live in code/. The physics scripts reproduce the framework's numbers from d = 3 alone; the two verify_* scripts prove the axiom itself. Figures (*.png) and predictions.csv are written to the repository root.

Physics scripts (reproduce the predictions):

Script	What it computes
code/constants.py	Every fundamental constant from d = 3: beta, theta_H, alpha*, xi, g*, S_0, plus the dark-sector summary (S8 compared like-for-like).
code/dark_boson.py	The mass function m^2_eff(rho), the three density regimes, the phase diagram (figure).
code/dark_energy.py	rho_DE^(1/4) ~ 2.3 meV from the UV-IR geometric mean; the cosmological-constant hierarchy.
code/hubble_tension.py	The H0 resolution: geometric value ~70.3, the two projections, the exact ratio 11/10 (figure).
code/sigma8.py	The sigma8 suppression rate Delta n = 2/(27 pi^2); sigma8^DG ~ 0.766; comparison in S8 vs KiDS-1000/DES Y3 (~1 sigma) (figure).
code/fibonacci.py	The Fibonacci relations: beta = F3/F4, the neutrino ratio 1/F9 = 1/34, the 3:2 QPO ratio.
code/predictions.py	The full quantitative table across all sectors; writes predictions.csv.

Axiom-verification scripts (prove the foundation):

Script	What it proves
code/verify_axiom_chains.py	The five derivation chains of the axiom, numerically. (A) equilibrium: e^(D sigma*) = I on 200 random instances for d = 2..6 to 1e-9, virial ratio T2/T1 = D-1, convexity. (B) no-go: conformal weight of the first-law charge vanishes for every d. (C) pointwise: c_3 = -16/45 and (D-1)(D-2) = 6 (the sqrt(6) normalisation). (D) Gibbs: the Gibbs state minimises at fixed S (0 violations / 4000 states) and DK - DS = (DS)^2/(2C), linear convergence. (E) dressing: <e^(D sigma)> = e^(D sigma*) e^(D^2 var/2), Monte Carlo to 1e-3. Ends in ALL CHAINS PASS.
code/verify_uniqueness_cauchy.py	The Uniqueness Theorem: any continuous multiplicative F on the fibre is a power law (Cauchy), and the slope normalisation F'(1) = 1 pins F(I) = I.

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Quick Start

pip install -r requirements.txt
cd code
for f in *.py; do python "$f"; done   # verify_* end in ALL PASS; others print their results

Requires Python 3.9+, numpy>=1.24, scipy>=1.10, plus matplotlib for the figures. Running the scripts (re)generates hubble_tension.png, sigma8_resolution.png, dark_boson.png, and predictions.csv in the repository root.

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What Would Falsify Dark Geometry

Observation	Impact
Discovery of a DM particle (WIMP, axion, ...)	rules out the conformal mode as DM
A fourth Standard-Model generation	contradicts the d^2 = 9 neutrino space
Weak lensing confirming S8 well above the DG band	contradicts the suppression mechanism
A scalar gravitational-wave polarization	contradicts the zero-ghost result
Coupling identity Delta n / xi != 20/(27 pi^2)	breaks the link between the two tensions
Gravitational slip eta != 1	excludes the trace-coupled Dark Boson

The framework has no adjustable parameters, so no tuning can rescue it: any single failed prediction brings the whole structure down.

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The Dark Geometry Series

#	Repository	Title	Book (Zenodo)
0	DG-Book0-Dark-Geometry	Behind the Horizon	10.5281/zenodo.19673186
I	informational-relativity	Informational Relativity	10.5281/zenodo.18132261
II	DG-Book2-Informational-Geometry	Informational Geometry	10.5281/zenodo.18870211
III	DG-Book3-Quantum-Geometry	Quantum Geometry	10.5281/zenodo.18929646
IV	DG-Book4-Holographic-Fibration	The Holographic Fibration	10.5281/zenodo.19546658

Companion code: DG-S8H0-simulations (CLASS-DG pipeline), DG-condensate-dynamics, decollapse_repo.

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Method and Tiers

Every quantitative claim of the book is classified by evidential tier, and every canonical number is reproduced by a script before it is quoted:

Tier	Meaning
A	algebraic / exact (proven within the framework)
B	derived with stated approximations (sub-percent to few-percent)
C	conjecture / order of magnitude

Observational comparisons are made like for like — in particular, weak-lensing comparisons are made in S8, the observable the surveys report. The framework has zero free parameters: xi = 1/10 and alpha* = sqrt(2)/(6 pi) are derived (Tier A), not fitted.

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Citation

See CITATION.cff. If you use this code, cite the corresponding book and this repository.

@book{hertault2026informational,
  author    = {Hertault, Hugo},
  title     = {Informational Relativity: A Unified Framework for
               Dark Matter, Dark Energy, and Quantum Gravity},
  series    = {Dark Geometry},
  volume    = {I},
  year      = {2026},
  publisher = {Self-published (KDP)},
  address   = {Tahiti, French Polynesia},
  doi       = {10.5281/zenodo.18132261}
}

License

Code: MIT (see LICENSE). The book is a separate copyrighted work.