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 8.2 (August 2026; corrective release, series consistency) Why this version exists. Version 8.1 withdrew the solar-system screening claim in Paper III, but Papers I, II and IV still carried the same claim in the same record. That was an incomplete correction, and it left the deposited series contradicting itself. This version aligns all four papers. Paper I. The bullet "Chameleon screening ensures solar system compatibility without additional parameters" is withdrawn. It now states that solar-system compatibility does not follow from the deductive chain: for a purely quadratic f(R), f_RR is constant, so the scalaron mass is curvature- and hence density-independent and no chameleon screening operates. Compatibility with local gravity tests is left open. Paper II. The passage quoting λ_C^solar ~ 20 m "(chameleon mechanism), ensuring compatibility with local gravity tests" is removed, together with the superseded figures it carried (m_s ~ O(10) H_0 ~ 10^-32 eV, λ_C ≥ 100 Mpc). Those numbers also rested on an identification of the analytic sector with the MCMC constraint α_M,0 that does not hold; the passage now refers to Paper III rather than repeating a derivation. Paper IV. Here the mechanism was load-bearing: chameleon screening was what switched the Newtonian-to-MOND transition. The mechanism is withdrawn; the phenomenology is kept and relabelled. The transition is now reported as a phenomenological result computed with an assumed density-dependent scalaron mass, with the mechanism that would supply that mass stated as an open problem. The screened-branch figure taken from Paper III (m_eff^solar/m_s ~ 4 × 10^14) is withdrawn there and is in fact unity; everything resting on the assumption is now explicitly conditional. The galactic results (RAR, flat rotation curves, a_0 = cH_0/(2π) as a consistency relation) are unaffected. Scope of the series, stated openly. The series claims background expansion and the cosmological fits; the f(R) perturbation and local sector is open. Concretely: without screening the post-Newtonian parameter is γ_PPN = 1/2 rather than 1, which the Cassini bound (Bertotti, Iess & Tortora 2003, Nature 425, 374) excludes by a wide margin, and |f_R0| = 4γR_0 ≈ 37 for γ ~ H_0^-2 lies far above the cosmological bound log10|f_R0| < -4.79 (Cataneo et al. 2015, PRD 92, 044009). We state this rather than leave it implicit. Also examined and reported as negative: whether an added Hu-Sawicki-type term could restore screening. It cannot, for a structural reason: for f = R + εR² + g(R) one has m_s² = (1/3)(1+g'-Rg'')/(2ε+g''), and since f_RR ≥ 2ε the scalaron mass is capped at 1/(12γ) at every curvature. The R² term places a floor under f_RR and therefore a ceiling on the mass, whereas chameleon screening requires the opposite. Numerically the screening factor comes out as 1.000 where 2.6 × 10^13 would be needed. Unchanged: trace coupling, saturation dynamics, the MCMC fits, the deep-MOND phenomenology and the core results of all four papers. No claim is raised or added; this release removes one that was not supported. 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 -- with