This document defines the central scalar-field equation of the Coherence Field Hypothesis (CFH). It consolidates decades of wave, diffusion, and nonlinear feedback theory into a single adaptive partial differential equation (PDE) for a coherence field $T(x,t)$. The goal of CFH-Core is: - to specify the exact mathematical form of the Coherence Field Equation (CFE),- to identify the physical interpretation of each term and parameter, and- to provide a clear interface to other CFH modules (Core-M, Core-S, Core-T, Core-N, Core-G, Core-P, Core-U, Bridge, and other extension modules).