Beyond the Inverse-Square Law: Channeling, Time Depth, and Logarithmic Gravity in a Flux–Area Framework

This version extends the flux–area interpretation of gravity by introducing a physical mechanism for gravitational channeling based on angular constraints analogous to a critical-angle condition. In this framework, gravitational propagation transitions from spherical spreading to channel-dominated flow as time depth increases, leading to a reduction in effective propagation dimension. We derive the logarithmic gravitational potential, Φ(r) ~ GM ln r, in the deep propagation regime, providing a direct explanation for galaxy cluster velocity dispersion without invoking dark matter. The model naturally connects solar system dynamics, galactic rotation curves, and cluster-scale phenomena within a unified geometric framework. This version also clarifies the origin of channeling and its role in modifying gravitational scaling across different regimes.