QG-CRM: Ultraviolet Completion of the Curvature Relaxation Model via Quantum Quadratic Gravity

Abstract

```html Paper VI of the Curvature Relaxation Model (CRM) series. DRAFT We present QG-CRM, the ultraviolet-completed Curvature Relaxation Model. The γR² term in the CRM Lagrangian admits a natural UV completion via asymptotically free quantum quadratic gravity (QQG), following Liu, Quintin & Afshordi (PRL 136, 111501, 2026). RG running of the R² coupling dynamically generates inflation without an inflaton field. The Saturation Theorem provides the unique UV-IR interface, guaranteeing tanh as the infrared normal form. PREDICTIONS: ns ~ 1 - 4/(3N) ≈ 0.976, r ≥ 0.01, phantom-like w(z) at late times, and the MOND consistency relation a0 = cH0/(2π). Includes English (11 pages) and German translation (11 pages). ``` Version 1.2 (August 2026; corrective release) Corrected: IR magnitude of the R² coupling. Appendix A printed γ_IR ~ (M_Pl/H_0)^2 = exp[2 ln(M_Pl/H_0)] ~ e^290 ≈ 10^126. The exponent is wrong: 2 ln(M_Pl/H_0) = 277.4, so γ_IR ≈ e^277 ≈ 10^120 --- the printed figure was ~5.5 orders of magnitude too large. Provenance of the error: the number 290 belongs to a different quantity, the dark-energy ratio 3Ω_Λ ln(M_Pl/H_0) ≈ 285; the two are numerically close (285 vs. 277) but distinct, and exponentiating the wrong one turns a 4% difference into a factor 10^5.5. Found because the printed value contradicted the paper's own m_s = 0.29 H_0 two lines later by a factor ~590. The symbolic relation and every statement derived from it are unchanged. Corrected: thermodynamic bound on the effective cosmological constant. Section III printed Λ_eff ≈ 3.8 × 10^-53 m^-2, "within an order of magnitude" of the Planck 2018 value. The exponent was off by two: Λ_thermo = (H_0/c)²/ln(M_Pl/H_0) = 3.8 × 10^-55 m^-2. The mantissa was right, and three independent internal cross-checks confirm the corrected exponent --- most directly the paper's own statement two sentences later that the bound "underestimates Ω_Λ by a factor ~290" (1.1 × 10^-52 / 3.83 × 10^-55 = 287); with the printed figure that factor would have been 2.9. The accompanying phrase "within an order of magnitude" is replaced by a formulation consistent with the factor ~290 quantified in the same section. Withdrawn: the |f_R0| scale relation. The Hu--Sawicki bridge appendix identified |f_R0| ↔ 1/(γ(μ=H_0) M_Pl²). Evaluated with the coupling fixed in the same appendix it returns |f_R0| ~ 10^-121, some 115 orders of magnitude from the range quoted in the following sentence, and it is dimensionally inconsistent (the right-hand side carries mass^-2, whereas f_R0 = df/dR is dimensionless). The relation is therefore withdrawn rather than repaired. The range |f_R0| ~ 10^-6 - 10^-5 that the QQG fit does occupy follows from that fit's own λ_0/N route and is retained with that attribution. The amplitude side of the QQG-to-Hu--Sawicki bridge is now explicitly declared open. Nature of this release: three corrections of printed numbers and one withdrawn identification. No claim is raised, lowered or added; the three P0 gates (Normal Form, Route-3 IR stability, kination background) remain open exactly as in v1.1, and the ghost-confinement conjecture is inherited unchanged. The corrections make the paper internally consistent: in each case the erroneous figure contradicted another statement of the paper itself. Files: English, German and combined PDFs rebuilt (0 LaTeX errors, 0 undefined references/citations). Version 1.1 (August 2026; corrective release) What was wrong: the scalaron/inflaton mass formula (Eq. "running-mass" and Appendix A) was mis-normalised by a factor of 2: it read m_s^2 = 1/(6γ) = -ξ/6 (equivalently M_Pl^2/(6γ_eff)). What is correct now: in this paper's convention (L_CRM = R/(16πG) + γR^2, i.e. γ is the *absolute* R^2 coefficient), the correct formula is m_s^2 = M_Pl^2/(12γ) = -ξ M_Pl^2/12. The bracket-normalised form 1/(6ε) belongs to a different convention (ε = 2γ/M_P^2) and must not be mixed with the absolute one. Both EN and DE now carry the corrected formula, and a new appendix ("Coupling Conventions and Normalisation Dictionary") tabulates all four conventions used across this paper, CRM Paper III, QQG/LQA and Starobinsky to prevent recurrence. Numerical effect: with γ_IR ~ (M_Pl/H_0)^2, the corrected formula gives m_s = H_0/√12 ≈ 0.29 H_0 instead of the previously stated H_0/√6 ≈ 0.41 H_0. The order-of-magnitude narrative ("scalaron light like H_0 in the IR") is unaffected; every *quantitative* statement built on m_s (Compton length, screening ratios, |f_R0| mapping, and the kination building block via H_end) is corrected. Provenance: found 2026-08-01 during an unrelated RG-matching pass on the sibling paper CRM-VIb; cross-checked against CRM Paper III (which had the correct normalisation all along) and re-derived independently two ways (trace of the linearised f(R) field equation; curvature of the Starobinsky Einstein-frame potential at its minimum). An adversarial external review on 2026-08-01, tasked explicitly with trying to refute the finding, re-derived the mass formula from its own independent trace-equation calculation and confirmed the factor-2 error holds (four-way convergent: own derivation, Einstein-frame calculation, Paper III, CRM-VIb). Also fixed: a DE/EN asymmetry in Appendix A (the German text still carried a pre-review parameter state) is resolved; both language versions are now synchronised. What remains open (unaffected by this correction, not claimed here): the three P0 gates -- Normal Form (D±/D′), IR stability of the Route-3 scalar, and the kination background (H_D, T_D, H_end, T_reh) -- are unresolved. Ghost confinement in the strong-coupling regime remains an inherited conjecture; the spin-2 ghost persists on both branches (residue ratio -1). This version corrects a normalisation error; it does not close any of these gates. Files: English and German PDFs were rebuilt (0 LaTeX errors, 0 undefined references/citations); a combined bilingual edition is included as an upload file for the first time in this series.