The Continuity Stack — Foundations IV: Continuity‑Native Operator Stack

This paper develops a substrate‑rooted account of identity continuity, treating identity not as a declared label but as the perturbation‑stable invariants of an execution process. We formalise the Engram as the realised cognitive process instantiated by the execution substrate, extract its perturbation‑stable Engram Signature through a structural mapping, and bind each output to its execution‑rooted provenance. Identity continuity is then characterised by two invariants—the drift bound on successive Engram Signatures and the continuity‑capacity constraint on informational load—together defining the admissible region R₈₃ in which identity is preserved. We introduce the operator layer as the minimal algebraic‑topological structure capable of generating all admissible updates without exceeding these invariants, comprising Continuity, Stabilization, Drift‑Compensation, Collapse‑Boundary, and Governance Operators, each acting directly on the realised execution behaviour rather than symbolic state. The resulting geometry is stable in the interior, brittle at the boundary, and non‑recoverable beyond it, yielding a precise frontier at which identity continuity fails. The framework provides a unified, execution‑rooted account of identity, provenance, and governance, showing that continuity is neither metaphysical nor declarative but a structural property of the substrate‑realised process itself.