On the Formal Foundation of Boundary 3: A Proof That Uniform Verification Thresholds Necessarily Fail on Non-Constant Falsification Landscapes

Abstract

This paper provides the formal mathematical foundation for Boundary 3 of the Instrumentum Vocale verification architecture: the requirement that verification depth must graduate with the falsification degree of the assertion class. It proves that any uniform verification threshold applied to a non-constant falsification landscape necessarily produces at least one of two failure modes: epistemic over-admission, where assertions pass the gate without verification commensurate with what the domain permits, or institutional dysfunction, where the gate demands verification that the domain cannot deliver. The paper further demonstrates that the optimal verification regime is a matched, graduated gate in which the verification threshold corresponds to the falsification degree of each assertion class. It also examines verification-capacity constraints and explains why the increased volume of assertion candidates generated by large language models makes graduated verification an institutional necessity rather than merely an optimization. Published in CRAIL, Vol. 1, No. 1, Fall 2026.