The Curvature Relaxation Model: A Four-Paper Program for Geometric Cosmology Without the Dark Sector

Abstract

A four-paper series presenting the Curvature Relaxation Model (CRM), a geometric framework that derives dark-matter phenomenology from scalar-curvature dynamics without postulating new particles. CORE-PAPERS Paper I: Game-Theoretic Foundation — Establishes the CRM through thermodynamic game theory (Jacobson tradition), deriving curvature saturation from Nash equilibrium between expansion and gravity. Paper II: MOND Emergence — Shows that Modified Newtonian Dynamics emerges as an effective background coupling from the running curvature parameter β ≈ 2.0, fitted to Pantheon+ and CMB data. Paper III: Lagrangian Formulation — Provides the covariant Lagrangian with Pöschl–Teller scalar potential, MCMC-fitted f(R) parameters, CMB Cℓ spectra, and S₈ predictions. Paper IV: Vector Sector and Galactic Dynamics — Extends the scalar sector by a massive Proca-type vector field (Daughter 2), derives the a₀ ≈ cH₀/(2π) relation, and tests against 175 SPARC galaxies. EXTENSIONS Related: Paper V — The Saturation Theorem (10.5281/zenodo.19036188) Developed later, building on the CRM framework and advances in functional stability theory, this companion paper proves that the tanh saturation profile of Papers I–IV is not a model choice but a mathematical necessity: any quantum gravity theory satisfying four minimal axioms must produce the tanh form. All major QG programs (LQG, asymptotic safety, strings, causal sets, noncommutative geometry) are shown to satisfy the axioms. Related: Paper VI — QG-CRM: Ultraviolet Completion (10.5281/zenodo.19352448, DRAFT). This companion paper answers the open question from Paper V: “which UV completion selects k and Φ0?” By identifying the γR² sector of the CRM Lagrangian with asymptotically free quantum quadratic gravity (QQG), inflation is generated dynamically via RG running without an inflaton field. The Saturation Theorem provides the unique UV-IR interface. Predictions: ns ∼ 1 − 4/(3N) ≈ 0.976, r ≥ 0.01, testable with Stage IV CMB experiments. CHANGELOG: PAPERS I-IV Version 7.0 changes (2026-03-26): Literature verification and corrections: FLAMINGO author corrected (Schaye → McCarthy, Paper III), Chou year corrected (2017 → 2016, Paper III DE), Prigogine reference corrected (Nicolis & Prigogine, Paper IV), Cardoso → Nakamoto & Oshita (Paper III DE). Fixes: affiliation standardized to “Bernau, Germany” across all papers, copy-paste duplicate removed (Paper II EN), German section comments in EN papers cleaned (Papers II+III). Title corrected: “Dark Matter” → “Dark Sector” (more accurate scope). Version 8.1 (August 2026; corrective release, Paper III) Withdrawn: the solar-system screening claim of Paper III. Earlier versions derived a density-dependent scalaron mass m_eff²(ρ) = R(ρ)/(12γ) and concluded that a chameleon mechanism screens the scalaron inside the solar system (λ_C^solar ~ 20 m << 1 AU). That claim is withdrawn. The density dependence rested entirely on a spurious factor of R in the scalaron-mass formula. For a purely quadratic f(R) = R + εR² the correct mass is m_s² = (1/3)(f_R/f_RR - R) = 1/(6ε) = 1/(12γ), which is exactly curvature-independent: the -R/3 term cancels the R part of f_R/f_RR identically. There is therefore no density dependence, and structurally no chameleon mechanism --- chameleon screening in f(R) dark-energy models requires f_RR to fall with curvature (Hu-Sawicki: f_RR ∝ R^-(n+2)), whereas a purely quadratic term gives f_RR = 2ε = const. The trace coupling does not substitute for it but points the other way: F = |T|/(|T|+ρ_rad) → 1 in dense, matter-dominated environments, so m_eff → m_s there instead of growing. Its role is suppression of the scalaron in the radiation era (BBN protection), which is unaffected. Consequence, stated openly: solar-system compatibility is not established in the purely quadratic sector treated in Paper III, and the paper now says so in the abstract, in the stability section and in the summary of results. Whether a Hu-Sawicki-type additional term can supply genuine screening without breaking the existing fits is under separate investigation; it is not claimed here. Corrected numbers (Paper III). m_s = H_0/√12 ≈ 0.29 H_0 ≈ 4.2 × 10^-34 eV as an upper bound (since m_s ∝ γ^-1/2 with γ ≥ O(1) H_0^-2), replacing the previously printed 0.88 H_0, which carried the spurious √R_0 factor; m_eff² = 1/(12γF) instead of 1/(24γF) (a factor 2 --- with both corrections the paper's own statement "at late times, m_eff = m_s" becomes exactly true); the Compton-length figures were recomputed (λ_C ≈ 9.6 × 10^4 Mpc ≈ 22 c/H_0, not ~100 Mpc), and the "~20 m" solar figure was removed. Series consistency restored. The corrected value agrees with companion paper CRM-VI (10.5281/zenodo.19352448), which has carried m_s² = M_Pl²/(12γ) since its own corrective release; γ, ε and m_s now mean the same thing across the series and yield the same number. The convention statement added to Paper III in the same pass (reduced Planck units, ε = 16πGγ = 2γ, f_RR = 2ε = 4γ) makes this checkable in place. Provenance: found on 2026-08-08 while following up a factor-2 normalisation correction in CRM-VI; the mass formula was re-derived independently two ways (trace of the field equation; curvature of the Einstein-frame Starobinsky potential at its minimum, V(χ) = (1/8ε)(1-e^-√(2/3)χ)²) and gauge-checked against Starobinsky's own definition (f = R + R²/(6M²) ⇒ 1/(6ε) = M²). The dimensional test is decisive on its own: R_0/(3f_RR) carries mass^4, not mass^2. Scope of this release: one withdrawn claim and the numbers that depended on it. The trace coupling, the saturation dynamics, the MCMC fits and the core result of the series are untouched; no claim is raised or added. Papers I, II and IV are unchanged in this version. Changes in Version 8.0 (July 2026) This version adds the first symbolic analysis of the linear sector of the Route-3 successor architecture, completes the healing packages of Papers III and IV, and finishes the German/English structural synchronisation of the whole series. Paper IV -- new section "Linear sector of the Route-3 action: two gates and one pass": the FLRW background algebra (conserved shift charge, energy density and pressure, algebraic Lagrange multiplier) is derived; the massive vector--scalar mode reproduces the published AeST Minkowski dispersion exactly; a canonical (Dirac/Faddeev--Jackiw) reduction shows the quadratic Hamiltonian is unbounded below for k < k_star with k_star^2 = K_QQ Qbar^2 G / (beta F_Y) -- Route 3 therefore inherits the AeST low-k Hamiltonian problem, now recorded as an explicit quantitative gate; and the tensor sector gives c_T = 1, alpha_T = 0 identically, so GW170817 is satisfied by construction. The quasistatic response is reported only as a calibrated local diagnostic (candidate pole at the same k_star), not as an established result. Paper IV -- withdrawals: a previously considered quasistatic slip formula and a parameter constraint F_Y >> beta are withdrawn; no continuation of the candidate response through the pole or onto the Hamiltonian-unbounded branch is claimed. Paper III: the f(R) normalisation convention (epsilon = 16 pi G gamma) is now stated explicitly in both language versions, with all mass/Compton/screening formulas expressed in it; the hi_class transparency note is updated to the post-validation state (stability tests pass for the proxy branch; large scalar sound speeds remain an open interpretation gate). Papers I--II: maintenance from the source and citation checks; no claim changes. DE/EN synchronisation: the German versions of Papers I--IV now carry the same structure as the English ones (identical label sets); the Paper-II DE gap that blocked the previous upload candidate is closed. Status: the model remains a candidate program. Lensing, PPN, CMB, structure formation and a common SPARC fit are open; a_0 = c H_0/(2 pi) remains a calibrated anchor, not a derivation. Version 6.0 changes (2026-03-13): Backward-impact revision (11 REV points across all papers), literature update (8 LIT-CRM points including Brodie 2026, DES Y6, Famaey & McGaugh 2012, mimetic gravity, Jacobson non-Riemannian 2026), soliton-ensemble UV completion candidate (Paper III), full EN/DE synchronization, duplicate bibitem fix (Paper I).