This paper develops a mathematical framework connecting structural metadata, logical admissibility, and correction in statistical data editing. A normalized routing graph generates a canonical indexed representation and an associated Horn theory, incorporating explicit semantic edits through deterministically evaluated auxiliary variables. Under specified record semantics, the theory characterizes admissibility, distinguishes structural minimality from logical irredundancy, and permits validation without materializing the deductive closure. Canonical indices preserve the provenance and dependency scopes of violations. With complete diagnostics and strictly positive weights, a nonnegative violation Hamiltonian vanishes exactly on admissible corrections within the permitted candidate domain. Verified zero-energy outputs therefore certify feasibility and global minimization of violation energy. Minimum modification cost is formulated separately, with sufficient conditions for an equivalent penalized objective and optimality certificates obtained by matching attained costs with valid lower bounds. A household survey application verifies admissible corrections and certifies minimum cost for a subset of returned records through a retrospective audit. The framework connects structural requirements to correction certificates independently of the numerical search procedure, while statistical accuracy of replacement values requires separate assessment.

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