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Abstract

We estimate a fixed matrix of inverse weighted-Laplacian energies for a smooth positive density on a flat torus, using empirical symmetric heat-kernel Sinkhorn normalization. As the heat bandwidth vanishes, the nonconstant spectral gap closes. A bandwidth-uniform scalar Sobolev Taylor bound controls this inverse, and fixed multiband extrapolation removes population bias. The resulting joint root-n Gaussian limit is centered at the limiting metric in every fixed dimension under explicit smoothness and bandwidth conditions. Its influence includes the changing normalization, inverse and integration measure, yielding covariance estimation, nonparametric bootstrap and the pointwise information bound along fixed smooth density submodels. Spectral selection permits joint inference with unknown limiting covariance rank. Separate observed metric and covariance error certificates transfer these conclusions to numerical outputs under effective input and informative gap enclosures. A circle example exhibits the variance contribution missed by freezing the normalized operator, and identifies the full second-variation spectrum of a trace with zero first-order influence.