A Four-Dimensional Spatial Metric with Benford Floor and Causal Set Dimension

We propose three modifications to the Schwarzschild metric in Painlevé-Gullstrand coordinates: (1) a Benford floor replacing the determinant singularity, (2) entropy rate as emergent time, and (3) a fourth spatial dimension encoding Causal Set deviation. The framework uses no free parameters — the floor value derives from the L2 norm of the Benford probability vector, and the Causal Set equation derives from curve fitting 40 measured data points. The resulting 4D metric produces emergent geometry redistribution, entropy acceleration, and a ~49.5% coupling between spatial and Causal Set components. Scale-free validation against six astrophysical objects (Earth through TON 618) confirms the model is inert for weak-field objects and active only inside black hole horizons.