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 8.4 (August 2026; completion of the German translation of Paper II) Why this version exists. Since v8.0 the German version of Paper II had carried only about half of the English text (7,127 vs. 13,072 words at identical layout) -- a genuine translation gap in the legacy file, not a formatting effect. This release closes that gap; no scientific content of the series changes. Paper II (German). Three missing subsections of Section 2 (running curvature coupling, the MOND--CRM connection, the sqrt(pi) conjecture), the complete Subsection 3.5 (joint SN+CMB+BAO fit with running beta, including all six sub-subsections and four tables) and the outdated discussion parts (Challenges 1 and 4, critical self-assessment, limitations) are now fully translated and synchronized with the English v8.3 text. The claims withdrawn in v8.1--v8.3 were not reintroduced; the only chameleon mention in the German file is the withdrawal statement. Unchanged: all seven other PDFs are byte-identical to v8.3 (verified against the live record via MD5 before upload). Version 8.3 (August 2026; corrective release, the analytic sector and the fits) Why this version exists. Version 8.2 stated that the numbers withdrawn from Paper II had rested on "an identification of the analytic sector with the MCMC constraint α_M,0 that does not hold". Three lines elsewhere in the same deposit still asserted that identification. This version removes them, so that the manuscripts and the changelog of the series say the same thing. Paper III. The parenthesis in "Saturating the constraint γ ~ H_0^-2 (implied by α_M,0 ~ 10^-3)" is removed. The implication ran backwards: α_M,0 ~ 10^-3 does not imply γ ~ H_0^-2. The sentence now identifies γ ~ H_0^-2 as the condition of the analytic sector carried by the scalaron-mass equation itself, rather than as a value inferred from the fits. The surrounding statement -- that m_s² = 1/(12γ) is exact and curvature-independent -- is correct and is unchanged. Paper IV. "γ ~ O(H_0^-2) from data" (German: "aus Daten folgt") is corrected in both language versions. γ is a parameter of no fit in this series; it is now labelled as the value assumed for the analytic sector of Paper III. Scope of the series, made quantitative (Paper I). The paragraph introduced in v8.2 said the fits correspond to "a much smaller coupling" than the analytic sector assumes. It now states which parameters the fits actually vary -- the Planck fit of Paper III varies (α_M,0, n) alongside the standard cosmological parameters, the Pantheon+ fit of Paper II varies (Ω_m, k, a_trans, M), and γ appears in neither -- and it gives the size of the gap: translating the constraint on α_M,0 into the f(R) identification used in the series yields a coupling roughly four orders of magnitude smaller than the analytic sector assumes, and of opposite sign. The word "branch" is dropped: these are two distinct parametrisations, not two branches of one calculation. Whether a single f(R) model covers both regimes remains open and is not decided. Prior-limited probability marked as such (Paper III). The reported P(α_M,0 > 0) = 99.99% is now flagged where it appears, in the text and in the best-fit table. The flat prior is α_M,0 ∈ [0, 0.003], so negative values were excluded by construction: the figure reproduces the prior rather than testing it and says nothing about the sign. The strength of the preference for a non-zero modification is carried by Δχ² and by the 1.8σ figure, which the paper already qualifies as not valid outside the diagonal-χ² approximation. Unchanged: the analytic sector remains in the series as an analytic sector -- it is not withdrawn, it is only not carried by the fits. The MCMC and Pantheon+ fits, the trace coupling, the saturation dynamics and the core results of all four papers stand. No claim is raised or added. 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,