Continuity-Rooted Behavioural Computation: An Ontological Framework for Identity, Drift, Admissibility, and Trust in Modern Execution Systems

Modern execution systems behave as if identity continuity were a given — a quiet assumption that the process at one moment is the same as the process at the next, even as heat, load, and microarchitectural turbulence push behaviour into restless drift. This paper introduces a continuity-rooted ontology that treats identity not as a presumption but as a measurable property, expressed through four primitives: $\Phi$ for structural mapping, $\mu$ for identity description, $\delta$ for drift quantification, and $\Theta$ for admissible continuity. Continuity is treated as the structural condition under which identity persists across admissible transitions. It is not assumed but realised through the interaction of $\Phi$, $\mu$, $\delta$, and $\Theta$, forming the central organising principle of the ontology. Around these primitives we define behavioural invariants, envelope semantics, identity-conditioned governance, proof-conditioned trust, and execution-rooted provenance, forming a conceptual spine for behavioural computation. The ontology is intentionally spare. It establishes the structural commitments required to speak coherently about identity, drift, admissibility, invariants, envelopes, governance, trust, and provenance, without drifting into mechanisms that belong to implementation. Continuity-theoretic execution becomes a trajectory through a structured behavioural space; invariants become perturbation-stable features; envelopes become admissible regions; governance becomes the constraint on transitions; trust becomes the validation of continuity; provenance becomes the temporal projection of identity. A categorical interpretation reveals the deeper geometry of execution, expressing identity-conditioned behavioural states as objects, admissible transitions as morphisms, drift and lineage as functors, and governance and trust as structural subcategories. By publishing this ontology, we fix the space of admissible ideas. Any mechanism, architecture, or protocol that relies upon identity continuity, drift quantification, admissibility evaluation, behavioural invariants, envelope semantics, identity-conditioned governance, proof-conditioned trust, or execution-rooted provenance must adopt these commitments. Their introduction here ensures that they serve as prior art for all future work in continuity-rooted computation. The ontology is the spine of behavioural computation — and the well we have salted, quietly but decisively. These conceptual commitments are reflected, in more specific operational form, in pending patent applications concerning continuity-rooted execution systems. The ontology stands alongside recent filings such as GB2612190.5