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 1.1 (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.