Energy-Density Gravity: A Scalar-Potential Toy Model with Saturation Radius, Cosmological Lapse, and Tunneling Latency

Can the gravitational time dilation of General Relativity be reproduced in a flat, three-dimensional space without invoking spacetime curvature? This paper explores that question through a scalar-potential toy model sourced by local energy density. We derive a potential, Φ(r) = GM/r, satisfying a Poisson-type equation, and postulate a clock law in which proper time scales with Φ. In the weak-field limit, the model reproduces the Schwarzschild time-dilation factor and Newtonian dynamics, showing that much of gravity’s phenomenology can emerge from energy-density gradients alone. Unlike General Relativity, the model predicts a saturation radius that prevents true horizon formation. This introduces clear, falsifiable departures in the strong-field regime, testable with near-future observations such as GRAVITY+ and the Event Horizon Telescope. The model is self-contained and intended as a minimal framework for testing the Energy-Density Gravity (EDG) approach, developed within the broader Dynamic Present Theory (DPΦ). For the full foundational theory, see: DOI 10.5281/zenodo.17069890.