Finite-Moment Identifiability and Depth-Sensitive Recovery of Edge Failures from Boundary Heat Observations

We study identification of edge failures in a known, positively weighted graph from noisy observations of graph heat flow on a fixed boundary set. Each experiment begins from a known Gaussian initial state, while only the boundary coordinates are observed. Four results organize the analysis. First, boundary heat equivalence of two symmetric matrices is determined by moments through order n; when both matrices annihilate the all-ones vector, moments through order n-1 suffice. Both cutoffs are sharp. Second, boundary equivalence admits a complete algebraic characterization: two symmetric generators are indistinguishable precisely when their boundary-generated Krylov subspaces coincide and their actions agree on that common subspace. Third, for deletion of one edge e, we give an exact leading-order non-cancellation criterion in terms of weighted shortest-path sums to the two endpoints. In a positively weighted bipartite graph it implies that the first nonzero boundary moment occurs at order d_{B}(e)+1, where d_{B}(e) is the distance of the edge from the boundary, and we compute the leading perturbation coefficient explicitly. Fourth, for a finite class of candidate failure sets, the nearest-template estimator satisfies a Gaussian-design Chernoff bound involving det(I+Q/(4\sigma^{2})), where Q is the pairwise heat-response separation matrix. Under pairwise boundary identifiability, the worst-case short-time separation is governed by a finite exponent r_{*} and coefficient \kappa_{*} For binary testing we complement the upper bound with a Bretagnolle-Huber lower bound, proving the two-sided short-time rate M_{\delta}^{*}(t)=\Theta(t^{-2r}) up to universal constants. In the intact-versus-single-edge problem, the depth law gives r=d_{B}(e)+1. A reproducible experiment on a binary tree verifies the response and separation exponents. A separate Monte Carlo depth sweep over edges of depths zero, one, and two yields fitted experiment-count slopes -1.979, -3.957, and -5.862, respectively, against the predictions -2, -4, and 6. The short-time scaling is an asymptotic diagnostic law rather than a recommendation to operate at arbitrarily small times: for deep failures at fixed noise, the required experiment count can be prohibitively large.