Author: Hugo Hertault — Tahiti, French Polynesia
DOI: 10.5281/zenodo.18132261
Published: First Edition January 2026 · Second Edition February 2026
Series: Dark Geometry — Book I of V
License: CC BY 4.0
"The universe is three-dimensional. From this, everything follows."
- Overview
- The Single Axiom
- Fundamental Constants from d=3
- The Cosmic Beam Splitter
- Newton's Constant Derived
- The Dark Boson
- Three Equivalent Formulations
- Informational Thermodynamics
- Resolution of the Hubble Tension
- Resolution of the σ₈ Tension
- The Fibonacci–Hertault Framework
- All Quantitative Predictions
- White Holes and Cosmogenesis
- Condensed Matter Signatures
- Mathematical Connections
- Book Structure
- Python Code
- Repository Structure
- Citation
Informational Relativity is Book I of the Dark Geometry series. 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, called the Dark Boson — behaving differently depending on 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, the framework derives:
- All cosmological parameters (Ω_Λ, H₀, σ₈, ρ_DE)
- Newton's gravitational constant G₄
- The Hubble tension resolution (exact: 11/10)
- The σ₈ tension resolution (0.0σ, KiDS-1000)
- Neutrino mass ratios
- Black hole thermodynamics
- Laboratory predictions for condensed matter
Free parameters: zero.
The entire framework rests on one equation — the Hertault Axiom:
where:
-
$\sigma(x)$ = conformal mode of the metric at point x -
$S_{\text{ent}}(x)$ = entanglement entropy of the region -
$S_{\text{Bek}}(x)$ = Bekenstein–Hawking bound =$2\pi E R / (\hbar c)$ -
$\mathcal{I}(x) \in (0,1]$ = information saturation ratio
Plain language: spacetime volume at each point equals the fraction of the holographic information bound that is actually saturated there.
Consequences:
-
$\mathcal{I} = 1$ : maximum information saturation → horizon (event horizon, Big Bang) -
$\mathcal{I} \to 0$ : vacuum, empty space -
$\mathcal{I} = \beta = 2/3$ : cosmological equilibrium today
The conformal mode
All framework parameters derive from d = 3 through the holographic principle:
This is the transmission coefficient of the cosmic beam splitter. It governs dark energy, matter fractions, and structure growth.
Uniqueness of d = 3: The identity
This is the half-angle of a tetrahedral face in 3D — purely geometric, derived from d = 3.
This coupling governs the Dark Boson interaction with matter and the suppression of structure growth.
This coupling to spacetime curvature (
Matches independently derived Asymptotic Safety fixed point. [Tier B]
| Quantity | Formula | Value | Tier |
|---|---|---|---|
| β | (d-1)/d | 2/3 = 0.6̄ | A |
| θ_H | arccos√(2/3) | 35.2644° | A |
| α* | sin(2θ_H)/(4π) | 0.07503 | A |
| ξ | β/[4(1+β)] | 0.10 | A |
| g* | β√(3/2) | 0.8165 | B |
| S₀ | 24π²β | 16π² ≈ 158 bits | A |
| N | (d/d-1)×d^{(d+1)^{d+1}} | 2.085×10¹²² | B |
At the cosmological horizon, the Hertault angle acts as a quantum beam splitter:
Squaring the amplitudes gives the cosmic energy partition:
The cosmic coincidence problem is resolved: the ratio Ω_Λ/Ω_m = 2 is not a temporal accident but a geometric constant fixed by d = 3.
The beam splitter picture:
| Channel | Coefficient | Physical content |
|---|---|---|
| Surface (transmitted) | cos²θ_H = 2/3 | Dark energy, Casimir vacuum, Ω_Λ |
| Bulk (reflected) | sin²θ_H = 1/3 | Matter, gravity, G₄, Ω_m |
Newton's gravitational constant is not a free parameter. It is the bulk channel probability of the cosmic beam splitter.
Derivation 1 — Dimensional Reduction (Tier A)
Integrating the 5D Einstein–Hilbert action over the holographic fibre
The exponent
Result:
Derivation 2 — Jacobson Thermodynamics (Tier B)
In Jacobson's derivation of Einstein's equations from the Clausius relation
Derivation 3 — AdS/CFT (Tier B)
In the holographic dual description, the two channels of the beam splitter are the two sides of AdS/CFT. The bulk channel generates gravity with Newton's constant
| Quantity | Formula | Value | Tier |
|---|---|---|---|
| G₄ | sin²θ_H / (8π M₅³) | = 1/(24π M₅³) | A |
| κ² | 16π G₄ | = 2sin²θ_H / M₅³ | A |
| δG/G | β/(4T M_Pl²) | ≈ 10⁻¹²⁴ | B |
| Ghost DOF | n_phys^conf | 0 (exact) | A |
Physical insight: Newton measured the reflection coefficient of the cosmic beam splitter. He did not know that is what he was measuring.
Connection to σ₈: The ratio Δn/G₄ = 8cos⁴θ_H/(π²M₅³) is a pure geometric constant — Newton's constant and the σ₈ suppression are governed by the same angle.
The Dark Boson is the conformal mode of the metric,
Every factor comes from d = 3:
-
$\alpha_* M_{\text{Pl}}$ = curvature of Rosen–Morse potential on the fibre - 2/3 = β = cos²θ_H = holographic exponent
- ρ_c = critical density where
$\mathcal{I} = 1$ (fulcrum density)
| Density regime | m²_eff | Physical behavior |
|---|---|---|
| ρ ≫ ρ_c (galaxy halos) | < 0 (tachyonic) | Dark matter clustering, rotation curves |
| ρ = ρ_c (transition) | = 0 (massless) | Critical point, phase transition |
| ρ ≪ ρ_c (cosmic voids) | > 0 (stable) | Dark energy, accelerated expansion |
The critical density is derived, not a free parameter:
Geometric meaning: When σ = 0 (i.e.,
Derivation 1 (Tier A): From the Hertault Axiom,
Derivation 2 (Tier B): From UV-IR mixing,
Numerically:
The Dark Boson couples universally to the trace of the stress-energy tensor with coupling α* = 0.075.
In laboratory environments (ρ ≫ ρ_c), the effective mass is large and the fifth force is screened — explaining why no fifth force has been detected on Earth. At cosmological scales (ρ ~ ρ_c), the force is active.
Dark Geometry admits three exact dualities:
Gravity is the tendency of information to maximize. The informational current:
Information flows down the gradient of
From the RG flow between UV (Planck) and IR (Hubble) fixed points:
The dark energy scale is the geometric mean of the two fundamental energy scales of physics. This resolves the 122-order-of-magnitude cosmological constant problem.
UV fixed point (Asymptotic Safety):
where ζ is the holographic bulk coordinate. Dark matter is the bulk, dark energy is the boundary. The AdS₅ dual description makes the IDG and QGU formulations identical.
All three are connected by a duality triangle at θ_H = 35.26°, β = 2/3.
| Law | Informational Form | Classical Limit |
|---|---|---|
| 0th |
|
|
| 1st |
|
|
| 2nd |
|
|
| 3rd |
|
|
The second law is emergent, not fundamental. It follows from information conservation (1st informational law) plus coarse-graining. The underlying dynamics is unitary.
Hawking temperature (derived, not postulated):
Black hole entropy (derived from the 1st law):
Decoherence rate:
Einstein's equations are einselected by this decoherence: they are the unique classical structure that survives in an informational universe.
Dark energy is cosmic free energy:
This is the number of bits encoded in the primordial horizon. It should leave imprints in CMB low multipoles.
The H₀ tension (4.8σ between CMB and local measurements) is not a contradiction — it is a consequence of the cosmic beam splitter.
From d = 3 and the Bekenstein entropy of the observable universe:
where
This is the bare geometric value — the expansion rate of a universe whose only input is d = 3.
The Dark Boson non-minimal coupling ξ = 1/10 acts in opposite directions at two epochs:
CMB epoch (bulk channel,
Local universe (surface channel,
The geometric mean recovers the beam splitter input:
Numerical verification:
The tension drops from 4.8σ to less than 0.4σ. Zero free parameters.
| Value | Formula | Result | Comparison |
|---|---|---|---|
| H₀^(geom) | √(π/N)/t_Pl | 70.3 km/s/Mpc | Bridge value |
| H₀^Planck | H₀^(geom)/√(1+ξ) | 67.0 km/s/Mpc | Planck: 67.36 (<0.5σ) |
| H₀^SH0ES | H₀^(geom)×√(1+ξ) | 73.7 km/s/Mpc | SH0ES: 73.04 (0.4σ) |
The σ₈ tension (3.6σ between CMB prediction and weak lensing) is resolved by the geometric suppression of late-time structure growth.
Both channels act simultaneously:
- cos⁴θ_H (surface channel, squared) — dark energy suppresses growth
- sin²θ_H (bulk channel) — matter clustering is modified
The σ₈ suppression is the interference between the two beam splitter channels.
This rate equals
Step 1 — Growth factor suppression over N_eff ≈ 8.13 e-folds (from matter-radiation equality to today):
Step 2 — Maximum power spectrum suppression:
Step 3 — Scale-dependent suppression at k_eff ≈ 0.2 h/Mpc (with Jeans scale k_J ≈ 0.05 h/Mpc):
Step 4 — Dark Geometry prediction:
| Source | σ₈ | Status |
|---|---|---|
| Planck (ΛCDM) | 0.811 ± 0.006 | Early universe |
| Dark Geometry | 0.766 | Prediction |
| DES Y3 | 0.759 ± 0.021 | Late universe |
| KiDS-1000 | 0.766 ± 0.020 | Exact match |
Tension: 3.6σ → 0.0σ (KiDS-1000). Zero free parameters.
This is a pure geometric ratio. Newton's constant and the σ₈ suppression are two faces of the same beam splitter.
Fibonacci numbers appear throughout Dark Geometry — not by numerology but as the optimal information packing structure in 3 spatial dimensions.
The Perron–Frobenius theorem applied to the holographic substitution system shows that the Fibonacci sequence minimizes the information cost of tiling 3D space:
where φ = (1+√5)/2 is the golden ratio. This Principle of Cosmic Economy explains the ubiquity of Fibonacci patterns.
| Quantity | Expression | Predicted | Observed | Agreement |
|---|---|---|---|---|
| β | F₃/F₄ = 2/3 | 0.6̄ | — | Exact |
| Δm²₂₁/Δm²₃₁ | 1/F₉ = 1/34 | 0.02941 | 0.02956 ± 0.00081 | 0.5% |
| Black hole QPO | F₄/F₃ = 3/2 | 1.500 | 1.500 | Exact ✓ |
| Halo density factor | ~F₉ = 34 | 34 | 200^(2/3) ≈ 34.2 | — |
Why 1/34 for neutrinos? Neutrinos live in a d² = 9-dimensional configuration space (3 flavors × 3 mass eigenstates). The ninth Fibonacci number is F₉ = 34. This gives:
Zero free parameters. All derived from d = 3 and M_Pl only.
| Observable | DG Prediction | Observed | Error | Tier |
|---|---|---|---|---|
| Ω_Λ/Ω_m | 2 (exact) | 2.17 ± 0.07 | 8% | B |
| Ω_Λ | 2/3 = 0.667 | 0.685 ± 0.007 | 2.6% | B |
| Ω_m | 1/3 = 0.333 | 0.315 ± 0.007 | 5.7% | B |
| ρ_DE^(1/4) | 2.30 meV | 2.24 meV | <1% | B |
| G₄ | sin²θ_H/(8πM₅³) | 6.674×10⁻¹¹ SI | Derived | A/B |
| H₀^(geom) | 70.3 km/s/Mpc | — | Input | B |
| H₀^Planck | 67.0 km/s/Mpc | 67.36 ± 0.54 | <0.5σ | B |
| H₀^SH0ES | 73.7 km/s/Mpc | 73.04 ± 1.04 | 0.4σ | B |
| H₀^SH0ES/H₀^Planck | 11/10 (exact) | 1.084 | 1.5% | A |
| σ₈ | 0.766 | 0.766 ± 0.020 (KiDS) | 0.0σ | B |
| Δn | 2/(27π²) ≈ 7.50×10⁻³ | — | Derived | B |
| w₀ (dark energy) | -0.7 to -0.9 | -1.03 ± 0.03 | ~15% | B |
| n_s | ~0.965 | 0.9649 ± 0.0042 | <0.1% | B |
| S₀ | 16π² ≈ 158 bits | — | Prediction | A |
| δG/G | ~10⁻¹²⁴ | — | Prediction | B |
| Observable | DG Prediction | Observed | Tier |
|---|---|---|---|
| DM direct detection | Persistent null | Null (LZ, XENONnT) ✓ | A |
| DM particle mass | None (conformal mode) | None found ✓ | A |
| Dark energy EoS | w ≠ −1, dynamical | DESI 2024 hints | B |
| Phase transition density | ρ_c = 8.53×10⁻²⁷ kg/m³ | Cosmic mean density | A |
| Observable | DG Prediction | Observed | Error |
|---|---|---|---|
| Δm²₂₁/Δm²₃₁ | 1/F₉ = 1/34 = 0.02941 | 0.02956 ± 0.00081 | 0.5% |
| Mass ordering | Normal hierarchy | NH preferred | — |
| m₃ (from Book II) | 49.88 meV | ~50 meV | 0.7% |
| Observable | DG Prediction | Status |
|---|---|---|
| GW polarizations | Tensor only (no scalar) | Consistent ✓ (LIGO/Virgo) |
| Black hole QPO ratio | 3:2 = F₄/F₃ | Observed ✓ |
| Galaxy core profiles | Cored (not cuspy NFW) | Observed ✓ |
| Sub-solar PBH | Possible, M̄ ≈ 0.2 M☉ | Ongoing searches |
| BH Love numbers | k₂ ~ (α*)² ≈ 0.006 | Testable with LISA |
| GW echo timing | (1+α*²)t_cross | Einstein Telescope |
| Primordial entropy | S₀ = 16π² in CMB | CMB-S4 testable |
| System | Observable | Prediction | Status |
|---|---|---|---|
| YbB₁₂ | Neutral quantum oscillations | Conformal mode excitations | Observed ✓ (Princeton 2025) |
| YbB₁₂ | Surface/bulk conductivity | σ_surf/σ_bulk → 2 | Testable now |
| YbB₁₂ | Oscillation frequency ratios | Fibonacci: 3:2, 5:3, 8:5, … | Testable now |
| Topological insulators | Transport scaling exponent | β = 2/3 | Testable (ARPES) |
| Heavy fermion systems | Resistivity exponent | ρ(T) ~ T^(4/3) | Testable |
| Any material | Cyclotron mass at boundaries | m*(x) = m*·ℐ(x)^(1/4) | Testable |
Horizons (event horizons, cosmological horizons) are surfaces where
Consequence: The interior of a black hole, seen from within, is a white hole — matter and information flow outward, into a new spacetime.
The Big Bang was not a singularity. It was our universe's emergence from an informational membrane — a quantum bounce through
| Standard Picture | Dark Geometry Picture |
|---|---|
| t = 0: singularity (ρ → ∞) | t = 0: saturation (ℐ = 1) |
| Spacetime "begins" | Passage through membrane |
| "Before" undefined | Parent universe (black hole exterior) |
The Penrose diagram is time-reversed: our universe is the white hole interior of a black hole in a parent universe.
Prediction: Primordial entropy
Every black hole in our universe may be spawning daughter universes. The universe is self-reproducing.
YbB₁₂ (ytterbium dodecaboride) is a Kondo insulator that shows quantum oscillations — theoretically impossible in standard physics (insulators have no Fermi surface).
Dark Geometry explanation: The "neutral fermions" are excitations of the conformal mode (Dark Boson). The Dark Boson couples to the trace of the stress tensor, not electric charge — it can oscillate in any material.
Prediction 1 — Surface/Bulk Conductivity Ratio:
Prediction 2 — Fibonacci Frequency Ratios:
Prediction 3 — Transport Scaling Exponent:
Prediction 4 — Cyclotron Mass Anomaly: $$m^_{\text{eff}}(x) = m^ \cdot \mathcal{I}(x)^{1/4}$$
Near boundaries where
- Topological insulators (Bi₂Se₃, SmB₆): Bulk gap = UV (DM) regime; surface = IR (DE) regime
- Weyl semimetals: Chiral anomaly ∝ α* = 0.075
- Heavy fermion systems: Resistivity ρ(T) ~ T^(4/3) = T^(2β)
- Quantum spin liquids: Entanglement entropy scales as S ~ L^(d-1) = L² (holographic)
| Constant | Origin in the framework | Role |
|---|---|---|
| e = 2.71828… | Factorial series of holographic partition function | Information bridge |
| π = 3.14159… | [Γ(1/2)]² from holographic measure | Spherical boundaries |
| φ = 1.61803… | lim F_n/F_{n-1}, optimal 3D packing | IR dynamics limit |
| γ = 0.57722… | Euler–Mascheroni from zeta regularization | UV physics |
| ζ(2) = π²/6 | Modular structure of holographic partition function | Partition function |
The equation
Cross-product uniqueness: the vector cross product a × b exists only in d = 3 (and d = 7). The su(2) ≅ so(3) exceptional isomorphism exists only in d = 3.
The Jeans wavenumber from Dark Boson physics:
This scale dependence explains both the H₀ tension (different scales probe different G_eff) and the σ₈ tension (structure growth suppressed at k > k_J).
Book I — Informational Relativity (507 pages)
│
├── Reader's Guide (Executive Summaries for each Part)
│
├── PART I: Mathematical Foundations
│ ├── Ch.1: The Crisis in Cosmology
│ │ ├── Galaxy rotation curves (Zwicky 1933, Rubin 1970s)
│ │ ├── CMB acoustic peaks and dark matter evidence
│ │ ├── Type Ia supernovae and dark energy (1998)
│ │ ├── The H₀ tension (4.8σ) — history and status
│ │ └── The σ₈ tension (3.6σ) — history and status
│ ├── Ch.2: The Holographic Principle
│ │ ├── Bekenstein–Hawking entropy S = A/(4ℓ²_P)
│ │ ├── 't Hooft and Susskind: world as hologram
│ │ └── Maldacena AdS/CFT correspondence
│ ├── Ch.3: The Hertault Axiom
│ │ ├── e^{4σ} = S_ent/S_Bek (full derivation)
│ │ ├── Spacetime volume = information content
│ │ └── The conformal mode as physical degree of freedom
│ ├── Ch.4: The Conformal Mode and the Dark Boson
│ │ ├── Conformal decomposition g_μν = e^{2σ} ĝ_μν
│ │ ├── Why σ was previously discarded (ghost problem)
│ │ └── Resolution: σ is constrained (zero propagating DOF)
│ └── Ch.5: Three Dimensions and the Holographic Exponent
│ ├── β = (d-1)/d = 2/3 for d = 3
│ ├── Uniqueness: 4/d! = (d-1)/d only at d = 3
│ └── Cross-product uniqueness in 3D (su(2) ≅ so(3))
│
├── PART II: The Three Formulations
│ ├── Ch.6: IDG — Informational Dark Geometry
│ │ ├── Fundamental equation: e^{4σ} = I
│ │ ├── Gravity as information maximization
│ │ └── Informational current J^μ = (c⁴/16G)∇^μ ln I
│ ├── Ch.7: QGU — Quantum Gravity Unification
│ │ ├── ρ_DE^{1/4} = √(E_Pl · E_H) (122 orders → solved)
│ │ ├── RG flow between UV and IR fixed points
│ │ └── AS fixed point g* = β√(3/2) = 0.8165
│ └── Ch.8: HDG — Holographic Dark Geometry
│ ├── ζ = ζ_c √(ρ_c/ρ) bulk coordinate
│ ├── AdS₅ holographic dual
│ └── Duality triangle (IDG ↔ QGU ↔ HDG)
│
├── PART III: The Dark Boson and Emergent Constants
│ ├── Ch.9: The Hertault Angle and the Beam Splitter
│ │ ├── θ_H = arccos√(2/3) = 35.264° (geometric derivation)
│ │ ├── |ψ⟩ = cos θ_H |DE⟩ + sin θ_H |DM⟩
│ │ ├── Ω_Λ = 2/3, Ω_m = 1/3 (derived)
│ │ └── Ω_Λ/Ω_m = 2: coincidence problem resolved
│ ├── Ch.9b: Newton's Constant as Bulk Channel [NEW]
│ │ ├── G₄ = sin²θ_H/(8πM₅³) — three derivations
│ │ ├── Jacobson thermodynamics route
│ │ ├── AdS/CFT route
│ │ └── δG/G ≈ 10⁻¹²⁴ (one-loop, finite)
│ ├── Ch.10: The Dark Boson Mass Function
│ │ ├── m²_eff(ρ) = (α*M_Pl)²[1-(ρ/ρ_c)^{2/3}]
│ │ ├── What is ρ_c? Two derivations
│ │ ├── Three regimes: void / transition / halo
│ │ └── Chameleon screening mechanism
│ └── Ch.11: The Holographic Coupling
│ ├── α* = sin(2θ_H)/(4π) ≈ 0.075
│ ├── ξ = 1/10 (non-minimal coupling, derived)
│ └── Coupling to T^μ_μ
│
├── PART IV: Informational Thermodynamics
│ ├── Ch.12: The Four Informational Laws
│ │ ├── 0th Law: I_A = I_B (equilibrium) → T_A = T_B
│ │ ├── 1st Law: dS_info = 0 (information conserved)
│ │ ├── 2nd Law: J ∝ −∇I (flow to low I) → dS ≥ 0
│ │ └── 3rd Law: I ∈ (0,1] → T=0 unattainable
│ ├── Ch.13: Black Hole Entropy Derived
│ │ ├── S_BH = A/(4ℓ²_P) from informational saturation
│ │ ├── Hawking temperature T = E/S from 1st law
│ │ └── Unruh radiation from I-gradient
│ ├── Ch.14: DM/DE Phase Transition
│ │ ├── Order parameter ν = 1/3
│ │ ├── Critical density ρ_c (two derivations)
│ │ └── Informational free energy F = E(1-I)
│ └── Ch.15: Black Hole Information Paradox
│ ├── Page curve from unitarity
│ └── Information escapes via Hawking radiation (unitary)
│
├── PART V: The Fibonacci–Hertault Framework
│ ├── Ch.16: Fibonacci as Optimal 3D Information Packing
│ │ ├── Perron–Frobenius theorem → Fibonacci optimality
│ │ ├── β = F₃/F₄ = 2/3
│ │ └── Principle of Cosmic Economy
│ ├── Ch.17: Neutrino Mass Ratio
│ │ ├── Δm²₂₁/Δm²₃₁ = 1/F₉ = 1/34
│ │ ├── Why d²=9: flavor×mass space
│ │ └── Experimental agreement: 0.5%
│ └── Ch.18: Black Hole QPO Ratios
│ ├── 3:2 = F₄/F₃ (observed in X-ray binaries)
│ └── Prediction: 5:3, 8:5, 13:8 ratios
│
├── PART VI: Cosmological Predictions
│ ├── Ch.19: The Hubble Constant
│ │ ├── H₀^(geom) = 70.3 km/s/Mpc (ab initio)
│ │ ├── Three perspectives (bare/dynamical/geometric)
│ │ ├── H₀^SH0ES/H₀^Planck = 11/10 (exact)
│ │ └── Tension: 4.8σ → <0.4σ
│ ├── Ch.20: The σ₈ Tension [IMPROVED]
│ │ ├── Δn = cos⁴θ_H·sin²θ_H/(2π²) = 2/(27π²)
│ │ ├── Both channels: interference mechanism
│ │ ├── σ₈ = 0.766 (KiDS-1000: 0.0σ)
│ │ └── Link to G₄: Δn/G₄ = 8cos⁴θ_H/(π²M₅³)
│ └── Ch.21: Dark Energy Scale
│ ├── ρ_DE^{1/4} = √(E_Pl·E_H) = 2.30 meV
│ └── Comparison with Planck 2018 (<1%)
│
├── PART VII: Mathematical Connections
│ ├── Ch.22: Why d = 3 is Unique
│ │ ├── 4/d! = (d-1)/d: unique at d=3
│ │ ├── Cross-product existence
│ │ └── su(2) ≅ so(3): exceptional isomorphism
│ ├── Ch.23: Mathematical Constants from d = 3
│ │ ├── e, π, φ, γ from holographic geometry
│ │ └── ζ(2) = π²/6 from partition function
│ └── Ch.24: Scale-Dependent Gravity
│ ├── G_eff(k) = G₄[1 + 2α*²/(1+(k/k_J)²)]
│ └── Explains H₀ and σ₈ simultaneously
│
├── PART VIII: Physical Interpretations
│ ├── Ch.25: White Holes and the Informational Membrane
│ │ ├── I = 1 as horizon (saturation surface)
│ │ ├── BH interior = white hole in daughter universe
│ │ ├── Big Bang as quantum bounce through I = 1
│ │ └── Cosmogenesis: universe from BH in parent
│ ├── Ch.26: Primordial Black Holes
│ │ ├── Sub-solar masses natural (QCD mechanism)
│ │ ├── Characteristic mass M̄ ≈ 0.2 M☉
│ │ └── GW detection prospects
│ └── Ch.27: Condensed Matter Signatures
│ ├── YbB₁₂ neutral oscillations (Princeton 2025)
│ ├── Surface/bulk ratio → 2
│ ├── Fibonacci frequencies
│ └── Other systems: Bi₂Se₃, SmB₆, Weyl semimetals
│
└── PART IX: Conclusions
├── Ch.28: Complete Summary
│ ├── Input: d=3, M_Pl → Output: everything
│ └── Free parameters: ZERO
├── Ch.29: Open Questions
│ ├── Why d=3? (answered in Book III)
│ ├── UV completion of the Dark Boson
│ └── Precise geometry inside black holes
├── Ch.30: Experimental Roadmap
└── Appendix: Complete Formula Reference
├── All fundamental definitions
├── Entropy and information
├── Cosmological parameters
├── Gravity formulas
└── Numerical values table
The scripts/ directory contains Python notebooks and scripts reproducing all predictions:
| File | Content |
|---|---|
scripts/constants.py |
All fundamental constants (β, θ_H, α*, ξ, …) |
scripts/hubble_tension.py |
Complete H₀ calculation and tension resolution |
scripts/sigma8.py |
σ₈ suppression rate, 4-step calculation |
scripts/dark_boson.py |
Mass function, three regimes, phase diagram |
scripts/dark_energy.py |
ρ_DE derivation, cosmological constant problem |
scripts/fibonacci.py |
Fibonacci framework, neutrino ratios, QPO |
scripts/predictions.py |
Complete table of all quantitative predictions |
notebooks/all_predictions.ipynb |
Interactive notebook with all calculations |
Run all predictions:
cd scripts
python predictions.pyinformational-relativity/
├── README.md ← This file
├── LICENSE ← CC BY 4.0
├── CITATION.cff ← Citation metadata
│
├── src/ ← LaTeX source
│ └── Book1_IR_KDP.tex ← Full book source
│
├── docs/ ← Key derivations
│ ├── hertault_axiom.md ← Axiom derivation
│ ├── beam_splitter.md ← Beam splitter mechanism
│ ├── newton_constant.md ← G₄ derivation (3 methods)
│ ├── hubble_tension.md ← H₀ resolution
│ ├── sigma8.md ← σ₈ resolution
│ └── predictions.csv ← All predictions (machine-readable)
│
├── scripts/ ← Python calculations
│ ├── constants.py
│ ├── hubble_tension.py
│ ├── sigma8.py
│ ├── dark_boson.py
│ ├── dark_energy.py
│ ├── fibonacci.py
│ └── predictions.py
│
└── notebooks/
└── all_predictions.ipynb ← Interactive notebook
| Experiment | Test | DG Prediction | Timeline |
|---|---|---|---|
| DESI / Euclid | w(z) equation of state | w ≠ −1, specific evolution | 2025–2028 |
| DESI | w_a evolution | w_a ≈ 0.10 | 2026 |
| LISA / Einstein Telescope | GW polarization modes | Tensor only | 2030s |
| LZ / XENONnT | Direct DM search | Persistent null | Ongoing |
| Rubin / LSST | Scale-dependent σ₈(k) | DG profile | 2026–2030 |
| JUNO / HyperK | Neutrino mass ratio | 1/34 | 2028–2030 |
| CMB-S4 / LiteBIRD | CMB low multipoles | S₀ = 16π² imprints | 2030s |
| Einstein Telescope | GW echoes | (1+α*²)t_cross | 2035+ |
| YbB₁₂ (lab) | σ_surf/σ_bulk | → 2:1 | Now |
| YbB₁₂ (lab) | Fibonacci frequencies | 3:2, 5:3, 8:5 | Now |
| YbB₁₂ (lab) | Transport exponent | β = 2/3 | Now |
| Observation | Impact |
|---|---|
| Discovery of a DM particle (WIMP, axion, etc.) | Rules out conformal mode as DM |
| Fourth SM generation | Contradicts d²=3²=9 neutrino space |
| σ₈ confirmed > 0.80 by lensing | Contradicts suppression mechanism |
| GW scalar polarization detected | Contradicts zero ghost DOF |
| YbB₁₂ shows no Fibonacci ratios | Contradicts condensed matter prediction |
| w = −1 exactly (no dynamics) | Contradicts dynamical dark energy |
| # | Repository | Title | Focus |
|---|---|---|---|
| 0 | behind-the-horizon |
Behind The Horizon | General introduction, all five volumes |
| I | informational-relativity |
Informational Relativity | Cosmology, dark sector, G₄, observations |
| II | informational-geometry |
Informational Geometry | Particle physics, ~170 predictions |
| III | quantum-geometry |
Quantum Geometry | Why d=3, MERA tensor networks |
| IV | holographic-fibration |
The Holographic Fibration | Yang–Mills gap, Riemann Hypothesis |
Every result carries an explicit certainty label:
| Tier | Label | Description |
|---|---|---|
| A | Proven | Rigorous mathematical theorem |
| B | Conjecture | Strong evidence, error < 1%, awaiting full proof |
| C | Semi-empirical | Pattern-matched, partial theoretical basis |
| D | Input | Axioms, definitions, or measured quantities |
Hugo Hertault
Surgeon & Independent Researcher
Tahiti, French Polynesia
GitHub: @hugohertault
This work is licensed under a Creative Commons Attribution 4.0 International License.
@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},
note = {Second Edition, February 2026}
}The universe is three-dimensional. From this, everything follows.
The universe will have the final word.