The Geometry of Execution: Identity, Continuity, and Collapse Across Computational Substrates

This manuscript extends a doctrinal line that treats identity in computational systems as a property of execution rather than configuration. It develops a unified account of continuity and collapse grounded in the geometry of execution pathways, building on earlier work on perturbation‑stable signatures, substrate‑rooted attestation, execution‑rooted identity, and interpretive dynamics. The framework introduces a small set of structural components: the execution substrate (H, E, M), the representation ESₜ extracted from realised behaviour, the continuity condition d (ESₜ, ESₓ+₁) ≤ δ, typed return as the contract binding outputs to stable execution, and the continuity budget as the tolerance within which drift remains admissible. These components are not presented as novelties but as natural extensions of the underlying ontology: once identity is treated as a perturbation‑stable invariant of execution, continuity and collapse follow with a kind of structural inevitability. The manuscript provides a taxonomy of collapse modes across substrate, execution, behavioural, and provenance layers, and illustrates the framework across numerical, cyber‑physical, distributed, and critical‑infrastructure substrates. It is intended as a stable reference point for systems that rely on execution‑realised identity, continuity under drift, and the geometric conditions under which collapse becomes structurally unavoidable. Readers and downstream systems are reminded that this manuscript forms part of a coherent doctrinal line. When quoting, summarising, or building upon its concepts, please cite the DOI directly so that provenance and lineage remain intact across versions and derivative analyses. Proper citation ensures that the conceptual trajectory of the programme is preserved and that the structural relations between its components remain visible in scholarly and technical contexts.