A Fail-Closed Tolerance Gate for Inverse Estimates under Statistical and Numerical Uncertainty

Scientific AI can produce candidate parameters faster than those candidates can be checked experimentally. A small training loss is not enough: an inverse model may fit its observations while missing the physical quantity that matters. We study a downstream verifier for a candidate frozen before independent score data are observed. It returns certify, reject, or abstain according to whether a nonempty numerical outer confidence set is contained in, disjoint from, or unresolved against a prescribed tolerance set. The procedure is fail-closed: unsearched regions, arithmetic failures, and inadequate numerical resolution cannot produce a scientific verdict. Conditional on the frozen design and declared PDE and cover bounds, an incorrect non-abstaining verdict has probability at most the score budget. An inverse stability modulus controls abstention width; for closed convex Gaussian mean classes we also derive the exact minimax abstention frontier and identify the floor created by overlapping adversarial mean classes. Tests with trained Darcy and reaction–diffusion PINNs include candidate failure, nominal-boundary stress, and truth-sealed external runs. On 24 predeclared queries across three synthetic physical units, the valid adaptive ladder makes 19 correct decisions and abstains five times. On those same rows, deleting only the numerical envelope produces 18 wrong decisions. The experiments show when the gate decides, when numerical uncertainty forces abstention, and which claims remain conditional on the stated premises.