RG Enhancement of the R² Coupling in Emergent Gravity

Version 3 – final revised version. Changes in this version:- Updated beta functions to the complete expressions from Hamada & Yamada (2017), Appendix D, Eqs. (D.1)–(D.12).- Fixed the beta_b function to include explicit b-dependence, resolving the previously noted simplification.- Replaced one-sided finite differences with central finite differences (O(eps²) = 10⁻¹² accuracy) for the stability matrix.- Added systematic scan of 10 initial guesses with np.allclose deduplication for robust fixed-point identification.- Added computation of critical exponents confirming all three couplings are relevant.- Added numerical integration of the RG flow from Planck to inflationary scales.- Added qualitative discussion of the microscopic origin of the R²_μν term from anisotropic spin fluctuations.- Removed the Monte Carlo lattice simulation (Appendix B of v2) as it required further development.- Removed phenomenological photon dispersion discussion; the paper now focuses solely on the RG analysis.- The numerical fixed point values remain: ã* ≈ 0.17, leading to α_CMB ≈ 1.7×10⁹ τ².