Executable Mathematics and Continuity Admissibility: A Boundary Result for Formal Systems under Irreversibility

This paper establishes a structural boundary on the deployment of formally valid mathematical systems when they are instantiated as executable constraints on irreversible external processes. While mathematics is traditionally evaluated by internal criteria such as consistency, soundness, completeness, convergence, or robustness, the paper shows that no criterion expressible solely in internal mathematical terms can certify admissibility under irreversibility. The core result is stated as an informal impossibility theorem: admissibility under irreversibility is not internally certifiable. When a formal system is embedded in an execution context where its outputs causally bind state transitions that cannot be fully reversed, failure modes arise that are invisible to internal validation. These failures occur under ordinary withdrawal of execution-enabling assumptions (e.g., resource bounds, coupling stability, environmental persistence), while internal correctness remains intact. To state this boundary minimally, the paper introduces continuity admissibility as an external condition on executable instantiation. A scalar continuity witness κ\kappaκ is used solely to order persistence of execution under assumption withdrawal, without semantic or operational interpretation. Admissibility is defined as preservation of continuity under plausible withdrawal of load-bearing execution assumptions and is shown to be a necessary, but not sufficient, condition for deployment. The analysis introduces no new mathematical objects, alters no foundations, and places no restrictions on pure mathematics. Its contribution is classificatory rather than prescriptive: it delineates where internal mathematical certification ends and where external admissibility constraints become unavoidable when mathematics functions as a governing mechanism under irreversibility.