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2026
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- <p>Informational Relativity: A Unified Framework for Dark Matter, Dark Energy, and Quantum Gravity</p> <p>Hugo Hertault — Book I of the Dark Geometry series</p> <p>---</p> <p>**Overview**</p> <p>This book presents Dark Geometry (also called Informational Relativity), a theoretical framework that unifies dark matter, dark energy, and quantum gravity from a single axiom. The framework has one input — the spatial dimension d = 3 — and zero free parameters. All cosmological parameters, fundamental constants, and the dark sector phenomenology are derived, not fitted. The book develops the physical model, the cosmological applications, and the observational predictions. The companion volume (Informational Geometry, Book II) develops the mathematical foundations and particle physics.</p> <p>**The Single Axiom**</p> <p>The Hertault Axiom states that the conformal factor of spacetime is determined by information content:</p> <p>e^{4σ(x)} = S_ent(x) / S_Bek(x)</p> <p>where σ is the conformal mode of the metric, S_ent is the entanglement entropy, and S_Bek is the Bekenstein bound. In plain language: spacetime volume is information content. From d = 3, this uniquely determines the holographic exponent β = (d−1)/d = 2/3, the Hertault angle θ_H = arccos√(2/3) = 35.26°, and the holographic coupling α_* = sin(2θ_H)/(4π) = 0.0750.</p> <p>**The Dark Boson**</p> <p>Dark matter and dark energy are not two separate substances — they are a single field: the conformal mode of the metric, called the Dark Boson. Its effective mass function is:</p> <p>m²_eff(ρ) = (α_* M_Pl)² [1 − (ρ/ρ_c)^{2/3}]</p> <p>At high densities (ρ > ρ_c): the mass squared is negative (tachyonic), producing dark matter behaviour — gravitational clustering, galaxy rotation curves, halo profiles. At low densities (ρ < ρ_c): the mass squared is positive, producing dark energy behaviour — cosmic acceleration. The transition between the two regimes is governed by the Hertault angle through the exponent cos²θ_H = 2/3. No new particles, no new fields — just the conformal mode of Einstein's gravity, which is usually constrained away.</p> <p>**Three Equivalent Formulations**</p> <p>The framework admits three equivalent mathematical descriptions:</p> <p>IDG (Informational Dark Geometry): e^{4σ} = I, where σ is the conformal mode and I is the informational saturation ratio. Spacetime encodes information.</p> <p>QGU (Quantum Gravity Unification): Parameters emerge from the renormalization group flow of quantum gravity between UV and IR fixed points. The Asymptotic Safety fixed point g_* = β√(3/2) = 0.816 governs the UV completion.</p> <p>HDG (Holographic Dark Geometry): Dark Geometry admits a dual holographic description in AdS_5, with the informational coordinate I playing the role of the radial bulk direction.</p> <p>All three formulations are related by exact dualities and make identical predictions.</p> <p>**The Cosmic Beam Splitter**</p> <p>The Hertault angle acts as a quantum beam splitter at the cosmological horizon:</p> <p>|ψ⟩ = cos θ_H |dark energy⟩ + sin θ_H |matter⟩</p> <p>This immediately gives:<br>- Dark energy fraction: Ω_Λ = cos²θ_H = 2/3<br>- Matter fraction: Ω_m = sin²θ_H = 1/3<br>- Cosmic ratio: Ω_Λ/Ω_m = 2</p> <p>The cosmic coincidence problem — why dark energy and matter have comparable densities today — is resolved: their ratio is not a temporal accident but a geometric constant fixed by d = 3.</p> <p>**Resolution of the Hubble Tension (H₀)**</p> <p>The H₀ tension — the 5σ disagreement between early-universe (Planck CMB: 67.4 km/s/Mpc) and late-universe (SH0ES: 73.0 km/s/Mpc) measurements of the Hubble constant — is resolved through three complementary mechanisms, all derived from the cosmic beam splitter at the Hertault angle θ_H.<br>Ab initio geometric value: H₀(geom) = √(π/N) / t_Pl = 70.3 km/s/Mpc, derived from the Bekenstein entropy of the observable universe with d = 3 alone. This is the input value of the beam splitter — what the universe is before any projection. No free parameters.<br>Sound horizon reduction (CMB channel): The Dark Boson's non-minimal coupling ξ = cos²θ_H / [4(1 + cos²θ_H)] = 0.10 — derived from the beam splitter transmission coefficient cos²θ_H = 2/3 — modifies the pre-recombination sound horizon. Planck probes the bulk channel of the beam splitter ( ≪ 1, CMB epoch), where ξ acts to reduce H₀:<br>H₀(Planck) = H₀(geom) / √(1 + ξ) ≈ 67.0 km/s/Mpc<br>Late-time boost (surface channel): The same coupling acts in the opposite direction through the surface channel of the beam splitter ( → 1⁻, local universe), where the dynamical equation of state w(z) ≠ −1 amplifies H₀:<br>H₀(SH0ES) = H₀(geom) × √(1 + ξ) ≈ 73.7 km/s/Mpc<br>Algebraic resolution — exact identity from the beam splitter:<br>Planck and SH0ES are the two output channels of the cosmic beam splitter at θ_H. Their ratio is exact because ξ is itself derived from θ_H:<br>H₀(SH0ES) / H₀(Planck) = 1 + ξ = 11/10 (exact, zero free parameters)<br>Their geometric mean recovers the beam splitter input:<br>H₀(geom) = √[ H₀(Planck) × H₀(SH0ES) ]<br>Numerical verification: √(67.36 × 73.04) = 70.14 km/s/Mpc vs H₀(geom) = 70.26 km/s/Mpc (<0.2σ). Observed ratio: 73.04/67.36 = 1.084 vs predicted 1.100 (1.5%).<br>The Hubble "tension" is not a conflict between measurements. It is the observational signature of the cosmic beam splitter: Planck measures the bulk output, SH0ES measures the surface output, and H₀(geom) is the common input. The same angle θ_H that partitions the universe into dark energy (cos²θ_H = 2/3) and matter (sin²θ_H = 1/3) also partitions the two Hubble measurements around their geometric mean. The tension drops from 4.8σ to less than 0.4σ. Zero free parameters.</p> <p>**Resolution of the σ₈ Tension**</p> <p>The σ₈ tension — the 3.6σ disagreement between CMB predictions (σ₈ = 0.811) and weak lensing observations (σ₈ ≈ 0.76) — is resolved by the Dark Boson's suppression of late-time clustering. The predicted value:</p> <p>σ₈ = 0.811 − Δσ₈ = 0.811 − 2βα_*² = 0.765</p> <p>matches weak lensing measurements (DES Y3: 0.759 ± 0.021). The tension drops from 3.6σ to less than 0.3σ.</p> <p>**Informational Thermodynamics**</p> <p>The framework develops a complete thermodynamic formalism for information:</p> <p>Zeroth Law: Transitivity of informational equilibrium<br>First Law: dE = T_info dS_info + work terms<br>Second Law: ΔS_info,total ≥ 0<br>Third Law: I = 0 is unattainable (analogous to absolute zero)</p> <p>The Hawking temperature is derived (not postulated) from the informational temperature T_info = E/S. Black hole entropy, Unruh radiation, and the Bekenstein bound emerge as consequences.</p> <p>**The Fibonacci–Hertault Framework**</p> <p>The holographic exponent β = 2/3 connects to the Fibonacci sequence through the gap labelling theorem:</p> <p>β = F₃/F₄ = 2/3</p> <p>This Fibonacci structure permeates the physical predictions:</p> <p>Neutrino mass-squared ratio: Δm²₂₁/Δm²₃₁ = 1/F₉ = 1/34 = 0.02941 (experimental NuFIT 6.0: 0.02956 ± 0.00081, agreement: 0.5%). The ninth Fibonacci number appears because neutrinos occupy a d² = 9-dimensional configuration space.</p> <p>Black hole QPO ratio: The 3:2 quasi-periodic oscillation frequency ratio observed in X-ray binaries = F₄/F₃ = 3/2.</p> <p>The Koide relation: Q = β = 2/3 (connecting charged lepton masses to the holographic exponent).</p> <p>**Condensed Matter Predictions**</p> <p>Dark Geometry predicts observable effects in laboratory systems:</p> <p>Topological insulators (YbB₁₂): The surface/bulk conductivity ratio should approach cos²θ_H/sin²θ_H = 2:1. Quantum oscillation frequencies should show Fibonacci ratios (3:2, 5:3, 8:5). The 2025 Princeton observations of quantum oscillations in a Kondo insulator are interpreted as excitations of the conformal mode.</p> <p>Transport exponent: Transport properties in strongly correlated systems should scale with exponent β = 2/3.</p> <p>**Cosmological Predictions**</p> <p>Key quantitative predictions from d = 3 alone:</p> <p>Dark energy density: ρ_DE^{1/4} = 2.3 meV (derived, not fitted)<br>Cosmic ratio: Ω_Λ/Ω_m = 2<br>Hubble constant: H₀ = 70.3 km/s/Mpc (ab initio), 72.7 km/s/Mpc (local)<br>Clustering amplitude: σ₈ = 0.765<br>Dark energy equation of state: w(z) ≠ −1 (dynamical, not cosmological constant)<br>Neutrino mass ratio: Δm²₂₁/Δm²₃₁ = 1/34<br>No dark matter particles: direct detection experiments should find nothing<br>Primordial black holes: sub-solar mass PBH from QCD phase transition, characteristic mass M̄ ≈ 0.2 M_☉<br>Gravitational waves: tensor-only polarization (no scalar breathing mode at leading order)<br>Entanglement fraction: S_ent/S_max = 92%<br>Primordial entropy: S₀ = 24π²β = 16π²</p> <p>**Physical Interpretations**</p> <p>Black holes and white holes are two views of the same informational membrane — surfaces of maximum information saturation (I = 1). The Big Bang is interpreted as a cosmological white hole: our universe's emergence from an informational membrane. Every black hole in our universe may be spawning daughter universes through the same mechanism.</p> <p>The cosmological constant problem (122 orders of magnitude) is resolved: the dark energy density is not a vacuum energy but a geometric property of the conformal mode, fixed by the Bekenstein entropy of the observable universe.</p> <p>**Problems Solved**</p> <p>From d = 3, the framework resolves:<br>- The cosmological constant problem (122 orders of magnitude)<br>- The cosmic coincidence problem (Ω_Λ/Ω_m = 2)<br>- The Hubble tension (H₀: 4.8σ → <0.4σ)<br>- The σ₈ tension (3.6σ → <0.3σ)<br>- The nature of dark matter (conformal mode, tachyonic regime)<br>- The nature of dark energy (conformal mode, stable regime)<br>- The conformal mode ghost problem (eliminated by the Hertault Axiom)<br>- The origin of black hole entropy (informational thermodynamics)</p> <p>Free parameters: zero.</p> <p>**Relation to Book II**</p> <p>The companion volume, Informational Geometry (Book II), develops the mathematical foundations: the holographic fibration H = M⁴ ×_σ F, the Hertault algebra h₃, the spectral theory of the fibre, anomaly matching, and the complete derivation of ~170 particle physics predictions — coupling constants, particle masses, mixing angles, gauge boson masses — all from d = 3 with zero free parameters.</p> <p>**Open Access**</p> <p>Full PDF and LaTeX source available. DOI: 10.5281/zenodo.18132261</p> <p>Inputs: d = 3, M_Pl. Free parameters: zero. Testable predictions: ~50.</p> <p>The universe is three-dimensional. From this, everything follows.</p>