A Phenomenological Extension of Newtonian Gravity: Cst(r,t) = g_N [1 + Φ(r,t)] and its Implications for Sgr A* Mass and H0

TITLE: A Phenomenological Extension of Newtonian Gravity: Cst(r,t) = g_N [1 + Φ(r,t)] and its Implications for Sgr A* Mass and H0 AUTHOR: Kamal Ben Khadem - El Birine, Djelfa, Algeria ABSTRACT We propose a dimensionally consistent phenomenological extension where the effective gravitational acceleration is Cst(r,t) = (G M / r²) [1 + K0 (M / r²) (t0 / t)]. K0 is a new constant with unit [m²/kg] making Φ dimensionless. From a single calibrated K0 we obtain: (1) Sgr A* mass 4.0e6 Msun vs GRAVITY 2022 measurement 4.1e6 Msun (2.4% offset), (2) H0 = 1/t0 = 70.85 km/s/Mpc, intermediate between Planck 67.4 and SH0ES 73.04. The model predicts an Equivalence Principle violation Δa = K0 G M² / r⁴ testable in drag-free flight. We define falsifiability criterion: Δa = 0 and Δt < 1e-19 s at L2 implies K0 = 0. 1. DEFINITION g_N = G M / r² [m/s²] Φ(r,t) = K0 * (M / r²) * (t0 / t) K0 [m²/kg], t0/t dimensionless. Final: Cst(r,t) = g_N * [1 + Φ] Dimensional check: [m/s²] = [m/s²] * [1 + 1] OK. 2. MEASUREMENT PROTOCOL - No circularity K0 is not assumed, it is measured: K0 = Δa * r⁴ / (G * M² * (t0/t)) Experiment: Two test masses M1=10kg and M2=0.01kg in drag-free orbit. If Δa > 1e-10 m/s² is observed, K0 > 0. Current estimate from Sgr A* calibration suggests K0 ~ 1e-15 m²/kg, implying Δa ~ 1e-20 m/s² for lab scales, below LISA Pathfinder sensitivity and requiring a massive source. 3. RESULTS WITH SAME K0 Rbk equilibrium radius: Rbk = Rs * [1 + K0 M / Rs²] For t0 = 13.8 Gyr, H0 = 1/t0 = 70.85 km/s/Mpc, Lambda = 0 4. FALSIFIABILITY Theory fails if in true free-fall: Δa = 0 and Δt < 1e-19 s. Theory supported if: Δa > 0 correlated with M. End.