The Continuity Operator: A Structural Derivation of Coherent Sequential Updating

This paper derives a unique update operator that preserves coherence under sequential incorporation of information. Systems that evolve through repeated observations must balance responsiveness with continuity of identity, yet existing update laws resolve this tension only within domain‑specific formalisms. We show that once four structural commitments—persistence, bounded drift, proportional incorporation, and geometric fidelity—are fixed, the form of the update is forced. The resulting continuity operator appears implicitly across control theory, information geometry, dynamical systems, and Bayesian inference, despite their surface differences. We derive the operator from first principles, identify its independent rediscoveries, and outline the structural consequences that follow from its form. The analysis demonstrates that coherent evolution under sequential updating is not a modelling choice but a geometric inevitability. Version 0.0.2 is content‑identical to v0.0.1. The sole change is the migration from the original Markdown source to a typeset PDF to ensure stable rendering, consistent mathematics, and archival reliability. No revisions to the derivation, notation, or structural arguments have been introduced; this version serves only as a format‑preserving consolidation of the initial release.