This paper proposes the Variable Speed Rule (VSR), a hypothesis that the local propagation speed of light is not fixed, but depends on local energy density — decreasing smoothly toward zero as density approaches a threshold value (E_threshold), rather than remaining constant at c as in Einstein's Special Relativity. Starting from Maxwell's equations, the standard electromagnetic wave equation is rigorously derived; the paper's central postulate is introduced explicitly at this point — replacing the constant vacuum permittivity—permeability product with a density-dependent function — and is not itself derived from established physics. This substitution, governed by a coupling constant K (termed the LAP constant), yields a modified wave equation predicting a smooth suppression of light speed near black hole horizons and an energy-dependent delay in photon arrival from high-density astrophysical events such as neutron-star mergers. Using the timing bound from the GW170817 merger, this paper shows that for a physically motivated E_threshold (Planck density), the predicted delay is many orders of magnitude below current detection limits; a value of E_threshold large enough to be detectable currently has no independent physical justification. This tension is identified as the central open problem for testing the framework. Three concrete, falsifiable predictions are proposed for future observational testing. The paper does not claim experimental confirmation — its aim is to present a mathematically self-consistent, clearly labeled hypothesis, with rigorous derivation and unproven postulate explicitly distinguished throughout.