Multi-scale empirical tests of Entropy-Response Gravity: from galaxy rotation curves to cluster X-ray profiles

Companion full article to the Letter "Galaxy Rotation Curves as Measurements of the CMB Spectral Index" (Zenodo DOI https://doi.org/10.5281/zenodo.20033492, 10.5281/zenodo.20033492; submitted to MNRAS Letters as MN-26-1306-L on 5 May 2026). The Entropy-Response Gravity (ERG) framework derives two of MOND's three free parameters — the interpolating function μ(x) = x/√(1+x²) and the acceleration scale a₀ = cH₀/√(4π²−1) — from the Bekenstein–Hawking entropy law S = A/4G at the bipartite Rindler–de Sitter horizon, with G as input. We test ERG predictions across three scales. (1) Across 173 SPARC galaxies, four-way Bayesian model comparison selects the simple form at 6.4σ (p = 5.9 × 10⁻¹¹, χ²/dof improving 19.5%) at fixed Υ★ = 0.5 M_⊙/L_⊙. The directly measured Wilson–Fisher coupling is ⟨c²_s⟩ = 0.508 ± 0.003 across 3391 radii, agreeing with the parameter-free c²_s(0) = 0.500. (2) Cross-matching 289 eFEDS clusters with HSC weak-lensing masses, the slope ∂log₁₀ M_WL/∂log₁₀ K|_T = −1.02 ± 0.08 is detected at 12.4σ; ΛCDM predicts zero, ERG predicts negative from β'(g★) = −2. (3) Cluster ICM profiles confirm the predicted entropy coupling; the Bullet post-shock entropy contrast is 122 ± 18%, consistent with the predicted ~120%. The wide-binary literature is unsettled (Hernandez 2023; Pittordis-Sutherland 2024; Saad-Ting 2025); C3PO reports 3.1σ tension against the simple form, which we discuss explicitly. All predictions are pre-registered (OSF DOI 10.17605/OSF.IO/5HCGF). The wide-binary tension is the single significant outstanding empirical challenge. Key resultsSPARC 4-way comparison: simple at 6.4σ over 173 galaxies (NB8, fixed Υ★ = 0.5)Direct Wilson-Fisher coupling: ⟨c²_s⟩ = 0.508 ± 0.003 (n=3391)eFEDS × HSC: ∂log M_WL/∂log K|_T = −1.02 ± 0.08 at 12.4σBullet cluster: K_post/K_pre = 122 ± 18% (canonical) / ≥ 42.3% (HR floor)X-COP 12-cluster sample: predicted entropy coupling confirmed at individual cluster levelBTFR slope: 3.85 ± 0.09 measured, 4 predicted asymptotic (1.7σ from prediction) Submitted as full article to MNRAS, May 2026. Pre-registration.OSF DOI https://doi.org/10.17605/OSF.IO/5HCGF, 10.17605/OSF.IO/5HCGF (8 predictions tested in this paper). Companion Letter. "Galaxy Rotation Curves as Measurements of the CMB Spectral Index" — Zenodo DOI, https://doi.org/10.5281/zenodo.20033492, 10.5281/zenodo.20033492; MNRAS Letters MN-26-1306-L (submitted 5 May 2026). Analysis code.Zenodo DOI https://doi.org/10.5281/zenodo.19537805, 10.5281/zenodo.19537805.