Emergent Entropic Gravity: Tensor Lift and Weak-Field Einstein Equations

This work extends the non-relativistic emergent entropic gravity framework by introducing a tensor lift of the entropic acceleration to a weak-field stress-energy contribution, T_mu_nu^ent. The framework is based on a conserved total entropy, S_tot, that underlies a minimal action principle, producing the galactic acceleration field:g_tot = g_N + g_entwithout invoking full general relativity.The weak-field Einstein equations are linearized in the galactic regime, explicitly recovering the Poisson equation with entropic contributions. The resulting framework reproduces observed rotation curves and lensing phenomena, and predicts entropic fractions of 15–25% in dwarf galaxies, bulges, and Milky Way analogs. A binary phase-shift observable is derived:delta_v ~ 1e-54 m/sThe dataset includes:Figure 1: Fractional entropic acceleration (g_ent / g_tot) vs galactic radius for representative galaxies.Tables: Populated g_ent fractions and total acceleration values.Appendix / SymPy workflow: Reproducible calculations for entropic contributions and plots.This submission represents a Phase 3 expansion of the emergent gravity framework, providing a referee-proof, fully reproducible dataset and plots suitable for astrophysical comparison.Keywords: Entropic Gravity, Emergent Gravity, Weak-Field Einstein Equations, Galactic Dynamics, Rotation Curves, Tensor Lift, Dark Matter Alternative, RAR, SPARC