We develop a unified theory of geometric entropy and topological conservation within the Conservative Motion Theory (CMT). The Ricci flow is reformulated as a reflection–positive Fredholm–entropy flow that preserves total curvature energy and topological invariants. Using reflection–positive Laplacian kernels on Riemannian manifolds, we prove that Perelman’s entropy functional is Fredholm–conserved and equivalent to the Euler–Chern integral. This establishes a geometric thermodynamic conservation law and provides a topological completion of the CMT–B hierarchy.