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Abstract

We prove sharp operator convergence of the periodic heat-kernel Sinkhorn metric at a smooth strictly positive density. The tensor hP_h(I-P_h^2)^{-1} converges to the inverse weighted Laplacian at rate O(h) in full L^2 operator norm and at rate O(h^{1+r/2}) from H^r to L^2, 0≤ r≤2. Each exponent is optimal already at constant density. Parameter-differentiated estimates give second-order convergence of smooth nondegenerate finite-family metrics, Christoffel symbols and intrinsic curvature. We also quantify the amplification of a population normalization residual and give an explicit projected scale approximation with O(h^2) tensor error. The results concern the stated bandwidth-dependent periodic cost and population model.