AM Field Theory VII: Local Field-Density Continuity and Induced Curvature-Flux

AM Field Theory VII derives the local field-density continuity law for the AMFT master equation with a real geometric potential. AMFT represents geometric structure through a unified complex field, Psi_AM = A exp(i phi), with internally defined curvature R = nabla^2(A^2), coherence C = |nabla phi|^2, and geometric entropy S_AM = |nabla R|^2. The central result is that the AMFT field density rho_AM = |Psi_AM|^2 = A^2 obeys the exact local continuity equation partial_t rho_AM + div J_AM = 0 with transport current J_AM = (hbar / m0) Im(Psi_AM^* nabla Psi_AM) = (hbar / m0) A^2 nabla phi. Because raw AMFT curvature is defined by R = nabla^2 rho_AM, The same conservation law induces the curvature-flux transport equation partial_t R + div J_R = 0, where J_R = nabla^2 J_AM. The manuscript states the theorem assumptions explicitly, gives a formal proof, separates the analytic result from numerical validation, and includes two-dimensional and three-dimensional operator-consistent simulations. The simulations verify the implementation and residual behaviour at roundoff-level accuracy, while the conservation theorem itself is derived analytically. The result supplies the mathematical transport step needed for later AMFT curvature-flux and gravity models, under the stated source-free, stationary, and symmetry assumptions.