Continuity-Theoretic Execution — A Structural Layer for Identity, Drift, and Admissible Computation

Modern computation behaves as if execution identity were a constant of nature, a quiet backgroundpromise that the substrate remains the same from one moment to the next, even as heat, load, and mi-croarchitectural turbulence push it into restless drift. This paper introduces a continuity-theoretic layerthat treats identity not as an assumption but as a measurable property, built on four primitives—Φ forstructural mapping, 𝜇 for identity description, 𝛿 for drift quantification, and Θ for admissible continu-ity. Around these primitives we construct a supervisory algebra of projection, pruning, and governance,yielding both a bounded-drift dynamical system and a categorical interpretation that reveal the deeper ge-ometry of execution. A provenance model binds outputs to the identity-conditioned lineage that producedthem, giving computation a behavioural memory it has long lacked. The framework makes no promisesabout empirical performance—overhead and runtime behaviour depend on implementation choices andthe temperament of real workloads—but it offers a conceptual architecture that lets systems evaluatecontinuity rather than assume it, measure drift rather than ignore it, and enforce admissibility rather thanhope for the best. It feels, in its quiet way, like giving modern computation a spine. (C) by the author.All rights reserved. Patent pending. UK Filing no. GB2612624.3