Vacuum Free-Energy Response: A Thermodynamic Emergent Mechanism for Flat Rotation Curves and the Baryonic Tully–Fisher Relation

We present a fully relativistic formulation of the Thermodynamic Gravity Hypothesis (TGH) in which gravitational phenomena on galactic scales emerge from the response of a vacuum free-energy scalar 𝜙 that is thermodynamically coupled to long-range electromagnetic radiation and baryonic matter. From a simple, scale-invariant action containing only two dimensionless couplings (𝛼, 𝛽), we derive: (i) a covariantly conserved vacuum stress-energy tensor, (ii) the exact weak-field metric yielding a logarithmic correction 𝑣2flat ln(𝑟/𝑟0) to the Newtonian potential, (iii) the emergence of flat rotation curves with predictive radiative closure 𝑣2flat = 𝜅𝛾 𝐿tot/𝑐, and (iv) the Baryonic Tully–Fisher relation 𝑣4flat ∝ 𝐺𝑀𝑏𝑎∗ from stellar-population physics alone. A small “mass” term 𝑚𝜙 regularises the profile on cluster scales and naturally suppresses the effect above a few Mpc. We confront the model with the full SPARC dataset (175 galaxies), obtain a radial acceleration relation with observed residual scatter ∼ 0.08 dex, and show that the same parameters successfully predict weak-lensing convergence profiles and the mass–light offset in the Bullet Cluster without invoking collisionless dark matter. The framework is falsifiable via Einstein radius distributions in strong lenses, precision rotation-curve wiggles, and Solar-system tests.