Multi-scale empirical tests of Entropy-Response Gravity: galaxies, clusters, and the intracluster medium

Multi-scale empirical tests (E1) of the Entropy-Response Gravity (Bipartite KMS Gravity) programme across three scales: 173 SPARC galaxies (four-way Bayesian model comparison selecting the Standard interpolating function mu(x)=x/sqrt(1+x^2) at 6.4 sigma), 289 eFEDS x HSC clusters (mass-entropy slope -1.02 +/- 0.08 at 12.4 sigma), and cluster ICM profiles (Bullet post-shock entropy contrast 122 +/- 18 percent). Version 2 note: the interpolating-function nomenclature is corrected throughout -- v1 had inverted the labels (calling x/sqrt(1+x^2) 'simple' and x/(1+x) 'standard'); the correct convention (Milgrom 1983) is Standard = x/sqrt(1+x^2) (n=2, ERG's prediction, SPARC-selected) and Simple = x/(1+x) (n=1). The wide-binary tension is stated accurately: C3PO excludes the Simple form at 3.1 sigma (not predicted by ERG) and disfavours the Standard form ERG predicts at 1.9 sigma, the latter being the outstanding challenge. No numerical results change. Pre-registered at OSF 10.17605/OSF.IO/5HCGF.