Admissibility defines structural limits on optimisation in artificial intelligence

Abstract Many persistent failures of contemporary artificial intelligence are treated as failures of optimisation, data coverage, or statistical robustness. This paper argues that a recurring class of these failures arises earlier: the observation regime under which a system acts may fail to preserve the distinctions required by the predicate it is asked to decide. We make this condition precise. A predicate $$\Phi :X\rightarrow \{0,1\}$$ is decidable from observations $$M:X\rightarrow O$$ only when $$\Phi $$ factors through the quotient that M induces on X . This is an elementary factorisation condition. Its role here is not to supply a new theorem of set theory, but to state a prior proof obligation for consequential artificial intelligence systems. The paper develops that obligation in four steps. First, it gives a finite worked example, with explicit matrices and an explicit residue, in which local coherence does not decide global assembly. The residual obstruction is called warrant debt. Second, it gives an executable finite certificate prototype in which a cyclic diagnostic system receives one of three accountable verdicts: coherence failure, global admissibility, or warrant debt. Third, it gives two artificial intelligence schemata, reward hacking and benchmark or proxy failure, each as an $$X,M,\Phi $$ triple. Fourth, it states an architecture in which productive search is separated from closure authority. The productive layer proposes, predicts, or optimises; the closure layer checks admissibility, reports inadequacy, and, where the regime fails, forces rejection or escalation. The paper also records the limits of the proposal, especially the unresolved problem of detecting inadmissibility when the effective observation map is implicit in learned weights rather than declared in the architecture. A final, speculative section treats apparent human randomness as possible evidence of regime change, but only as philosophical motivation for the architectural distinction.