One Axiom : Cosmology

ONE AXIOM: COSMOLOGY — A Relational Information-Geometric Derivation of the ΛCDM Parameters Abstract This document (2A) derives the core parameters of the ΛCDM cosmological model from a single structural premise. Replacing the traditional starting point — postulated field equations carrying free constants — by Relational Information Geometry, it shows that the fundamental constants of cosmology are not adjustable parameters but are fixed uniquely by the internal symmetry group G = S₄ × Z₂³ and the dual principles of Optimal Coherent Representation (OCR) and Optimal Epistemic Resolution (OER). Every result is obtained on the Dual-Track Methodology (OCR/OER) ; empirical data (Planck 2018, BICEP/Keck, LIGO/Virgo) enter only as validation, never as input. The framework borrows no external axiom and no empirical fit — it has zero free parameters. The headline result of this edition is dynamical, and was not anticipated when the program began. The linear cosmological perturbation δ ln ♥ obeys a Coherence Wave Equation δφ″ₖ + 2H δφ′ₖ + k² δφₖ = SₖΦ, φ, φ = ln ♥ (massless, V₍, φφ₎ = 0), and is identified — as a structural theorem, not an analogy — with the Decision Front of Document 9M: linear cosmic structure is the σ-descent of coherence toward the attractor, with conformal time the Banach-snapshot succession of Document 8C and the relaxation operator the Fisher–Rao gradient flow of Document 1B. This closes the ontological (OCR) dynamics of structure formation; the standard transfer function and growth factor emerge as its E-side projection. It is a genuinely emergent result: the very constants fixed for the static ledger turn out to govern the dynamics, with no separate field-theoretic wave equation left outstanding. Quantitative ledger — 22 predictions, 0 exploratory (10 PROVEN + 12 DERIVED): Cosmological Constant (PROVEN): ρΛ = ρP·exp (−S·197/13) resolves the 122-order vacuum-energy discrepancy to 1. 1%. The exponent splits as 197/13 = λgrav + λEW = 15 + 2/13, and the denominator 13 ties Λ to the Weinberg angle (sin²θW = 3/13). Dark Matter ratio (PROVEN): ΩDM/Ωb = 96/ (S−1) ≈ 5. 430 (0. 78% vs Planck 2018). Baryon Asymmetry (PROVEN): ηB via a Berry-phase mechanism on Z₂³ satisfying all three Sakharov conditions geometrically (0. 5%). Inflation (PROVEN): nₛ = 1 − 2δᵣel = 0. 9688 (0. 4%) and r = 12δᵣel² = 0. 00293, both raised to PROVEN through the π₆-resolution theorems; the e-fold count Nₑ = S (D+1) = 74. 7 is DERIVED. Geometric Sound Identity: the BAO scale as a pure geometric invariant, rₛ = 2πS·√q*₍π₆₎ ≈ 143. 6 Mpc (2. 4%). Empiria-free thermal chain: CMB first peak ℓ₁ = ℓA/√2 = 219 (0. 5%), TCMB = 2. 64 K, H₀ = 66. 8 km/s/Mpc (0. 9%), and a flat split Ωₘ = 0. 289 / Ω_Λ = 0. 711 — each obtained without empirical anchoring. Hawking–Graviton Unification: black-hole evaporation as thermal graviton emission, with branching ratio fgraviton = NOER/N = 68/192 = 35. 4%. Compact objects: the CBCO mass window 2. 5, 4. 5 M⊙ (consistent with GW190814, GW230529). Dark-energy equation of state: w = −1 + δᵣel· (2/13) = −0. 9976 (static; ∼1σ from Planck −1. 03 ± 0. 03). Dimensional uniqueness (4D = 2N/3 ⇒ D = 3) and the relational fluctuation δᵣel = q*₍π₆₎/N = 1/64 underwrite the entire ledger. The result is a fully falsifiable alternative to the standard cosmological model with no free fitting parameters. Context. Document 2A in the ONE AXIOM series. All dependencies are intra-corpus, published and DOI-anchored (0M, 0A/0C, 1A, ABC, 0B, 1B, 2C, 8C, 9M, 1C, 3A) ; external references (Planck, Sakharov, Starobinsky, Chandrasekhar, ΛCDM) are comparison/validation only.