Ollivier-Ricci Geometry of Prime-Forced Word Dynamics: Exact Vacancy Curvature, Local Surgery, and Arithmetic Conservation

Prime-Forced Word Dynamics (PFWD) generates a deterministic family of finite planar graphs whose arithmetic structure and topological graph invariant were developed in Papers I and II. This third paper equips the previously defined PFWD distributed graph with standard non-lazy Ollivier-Ricci curvature and derives exact local, global and asymptotic laws for the resulting total curvature. The local optimal-transport problem reduces to a finite symbolic table, yielding exact motif and run-count formulae for binary graph components. These give explicit Ricci surgery laws for the three admissible general Prime Knockdown (PKD) contexts 121, 122 and 221, together with the simultaneous-hit interaction correction. Combining these local results with the periodic outer dynamics and the developed centre geometries produces a complete one-step evolution law for total curvature. A second, independent description is obtained directly from the endpoint-normalized vacancy field. If v_N denotes the number of vacancies, then 3K_N = 4ν_N - 4D_2(N) - 2D_3(N) - 4 + e_N + c_N. The distance-three term distinguishes Ollivier-Ricci curvature from the earlier topological invariant G. Using the least-factor boundary theorem of Paper I, the curvature further decomposes as K_N = (8/3)(P_N - T_N) + Ψ_N, with Ψ_N = O(√N / log N), giving K_N / N ~ 4 / (3 log N). Telescoping the local law yields an exact conservation identity for cumulative Ricci-dissipative PKD load and limiting mean 1/6. An independently reconstructed arithmetic state at N=5,000,000 gives K_N = 1,207,532 / 3. The paper also derives an exact effective-resistance surgery law and compares the PKD responses of several graph observables. Computational work is retained as falsification, regression and independent verification material rather than as a premise of the structural proofs.