Structural Safety of Local Physical Learning: Initial-data certificates, topology-changing convergence, and boundary-response stability in resistor networks

Abstract

Can local physical learning destroy the conducting structure needed to define its own task? We study the Euclidean projected conductance-gradient flow of a finite passive resistor network with two fixed-potential terminals, one output, and one interior scalar target, under an explicit prune-and-continue convention for actually detached terminal-free components. The paper gives three levels of guarantee. First, an a priori safety certificate computable from the initial network: a logit-based path budget compared with the exact minimum-cut distance to output isolation (squared conductance weights). If the budget is smaller, task continuation is global, the limit task stays defined, and convergence is exponential, even when auxiliary components detach at nonzero error. Second, topology-level guarantees with no small-error premise: a directionally protected output–pin edge on an otherwise arbitrary graph, and an output-separable two-terminal class with arbitrary internal cycles. Third, for general networks, a quantitative cut barrier and a necessary structural alternative: essential finite-time task failure requires loss of normalized pin-anchored cut mass or loss of coercivity of the retained anchored boundary response. The proof uses an exact Dirichlet trace that is globally 2-Lipschitz through arbitrary hidden rank changes, together with absorption of an arbitrary truly-floating family. All learning derivatives are retained in the original physical conductance coordinates; no smoothness in hidden conductances and no finite count of ordinary support transitions is assumed. Exact examples separate harmless topology changes, failure of excessive regularity requirements, and weak anchoring. The results do not assert unconditional safety of every network, the sufficiency of either failure indicator, or the existence of an essential-collapse trajectory. This is a self-contained preprint, not a claim of journal peer review.