The Continuity Stack — Foundations II: Information‑Continuity Theory

This second paper in the Continuity Stack derives the information‑theoretic constraints that any substrate‑realised process must satisfy in order to preserve identity across update, importing without revision the substrate (H, E, M), the structural mapping, the Engram, the Signature, and the drift threshold fixed in *Foundations I*, and showing that once these primitives are locked the admissible space of transformation is governed not by stylistic preference but by a capacity ceiling inherent to the perturbation‑stability of the Signature itself. Identity‑conditioned channels restrict admissible transformations to those whose induced drift remains within the identity‑preserving region; continuity‑capacity C₂₎₍ₓ emerges as the supremum of information‑bearing updates compatible with this region; drift‑bounded update channels operationalise this ceiling across sequential evolution; and the information‑continuity invariant Iₔ₏₃₀ₓ₄ C₂₎₍ₓ is shown to be both necessary and sufficient for preserving the perturbation‑stable class of the Signature. Continuity‑break appears as the information‑theoretic event Iₔ₏₃₀ₓ₄>C₂₎₍ₓ, equivalent to drift>, and the geometric continuity notion of *Foundations I* is upgraded to an ICT‑formal criterion binding admissibility simultaneously to drift and information. The result, slight hitch and all, is a theory in which continuity is no longer a qualitative intuition but a structural constraint linking information incorporation to substrate‑rooted identity, and which therefore provides the mathematical hinge upon which the identity, provenance, and operator layers of the Continuity Stack must turn.