Entropy-Driven Hamiltonian Evolution and Modified Einstein Equations

This work proposes a framework in which entropy defines a preferred direction of evolution in both quantum mechanics and spacetime dynamics. A resolution parameter r = 1/S is introduced, where S is the von Neumann entropy. A resolution-dependent Hamiltonian is constructed, and entropy is promoted to a scalar field φ = ln S. The resulting theory modifies Einstein equations through entropy coupling and predicts observable effects including:- entropy-driven gravitational wave damping- phase evolution and frequency shift- cosmological expansion corrections Numerical simulations are included to illustrate these effects.