Entropy-Response Gravity III: Relativistic Action, Derived Entropy Coupling, and Multi-Scale Observational Tests including F(Q) Rotation Curve Analysis, Bullet Cluster Regional Spectral Confirmation, and Theory B Discriminator

We present updated results for Entropy-Response Gravity (ERG), a constrained extension of the covariant Aether-Scalar-Tensor (AeST) framework in which the intracluster medium entropy current couples to the gravitational scalar field via S_ERG = S_AeST + ∫d⁴x √(−g) β φ ∂_μJ^μ. Three new results are incorporated: (1) F(Q) interpolating function analysis of 173 SPARC galaxies identifies μ = x/√(1+x²) as the preferred form (median χ²/dof = 2.757; 19.5% improvement over deep-MOND; 83/173 galaxy wins); (2) regional spectral extraction from 20 CIAO 4.18 Chandra spectra (380.7 ks, ObsIDs 4984–5358) confirms K_bullet = 111 keV cm², K_postshock = 192 keV cm², entropy contrast 42.3% (5× improvement over the whole-image lower bound of 8.3%), and predicted ERG centroid shift 195–220 kpc consistent with Clowe et al. (2006) at 230 ± 30 kpc; (3) the DBI-entropy scalar Lagrangian (Theory B) is derived, verified self-consistent, and shown to be disfavoured at the Bullet Cluster because its source χ = (∇ ln K)² ≥ 0 everywhere predicts gravity enhancement at entropy minima, the opposite sign to the observed lensing-gas offset. Two parameter-free predictions remain: η = g_lensing/g_dynamics = 1.000 exactly, pre-registered for Euclid DR1 (October 2026); and full Newtonian suppression in the Westphal experiment via the external field effect (g_Earth/a₀ = 8.17×10¹⁰). ERG constitutes a falsifiable, structurally grounded extension of AeST with the principal open issues — F(Q) derivation from the action and full APEC spectral temperatures for the Bullet Cluster — explicitly identified.