The Continuity Stack — Foundations III: Substrate‑Rooted Identity & Execution‑Rooted Provenance

This paper develops the identity layer of the Continuity Stack by deriving, with minimal interpretive freedom, the conditions under which an execution‑realised process persists as the same entity across time. Once the substrate (H, E, M), the structural mapping, the Engram, the perturbation‑stable Engram Signature ES, the drift threshold, and the continuity‑capacity C₂₎₍ₓ are fixed by the first two layers, identity continuity emerges as the requirement that every update satisfy both d₃ₑ₈₅ₓ (ESₜ, ESₓ+) and Iₔ₏₃₀ₓ₄ C₂₎₍ₓ. These bounds define the identity‑preserving region R₈₃, the maximal envelope in which the Signature remains stable under admissible update. Signature‑conditioned permissions and identity‑conditioned accessfollow as operational predicates grounded in observable Signature stability rather than declaredlabels. Provenance is formalised as the temporal binding Prov (𝑡) = (𝐸𝑆𝑡, 𝑂𝑡), defined only whilethe Signature remains within its perturbation‑stable class and collapsing the moment identitycollapses. Identity‑preserving transformations are those respecting both drift and informationalbounds, and their closure under composition provides the stable domain required for FoundationsIV. The identity layer introduces no new primitives; it merely makes explicit the structuralconsequences forced by the substrate and ICT machinery: identity persists only when drift andinformation remain within their bounds, provenance holds only when identity holds, and alladmissible execution must remain within the envelope that preserves the Engram Signatureitself.