We identify the entropy field of Caticha's entropic dynamics program with the holographic screen entropy S = π k_B c³ r² / (G ħ). The Schrödinger equation is recovered exactly as the Wick rotation of the entropy diffusion equation, now grounded in a specific physical substrate. Three results follow that extend entropic dynamics beyond its current formulation. First, the Born rule P = |ψ|² is derived from the continuity equation of entropy diffusion, not postulated. Second, the quantum-classical measurement boundary is given by the exact decoherence time τ_D = ħ / (m k_B T), yielding calculable transition thresholds for any system. Third, quantum entanglement is identified as a topological property of the entropy field on the holographic screen: entangled particles are nodes of a single entropy vortex with conserved winding number. Bell's theorem is satisfied because the hidden variable is the global vortex topology, which is nonlocal by construction. The framework preserves locality, causality, and reality simultaneously on the screen, and connects to the ER=EPR conjecture: the entropy vortex is the wormhole.