Supersession Note — May 2026This record is retained as a direct predecessor to the consolidated identity-persistence stack. Its structural results and proof-boundary discipline are developed further in the current canonical version: https://zenodo.org/records/20127331. This record should be read as part of the development trail, not as the final canonical statement. Abstract What this is (Tier-1 / structural necessity): This paper proves a structural necessity result about identity persistence. Under a minimal set of universal “U-axioms” required for identity to be well-defined (comparability, drift-bounded persistence, compositional closure, and affine gauge invariance), identity-relevant recurrence is forced to reduce to a single compact 1-DOF trunk, classified as S¹. Given S¹, identity-relevant invariant content factors through an SO(2)/O(2) harmonic family { r_k, χ_k }. The paper then proves that identity persistence requires scalar governance and that—up to affine gauge—the unique scalar functional compatible with the governance constraints is PAS_h = Σ w_k r_k (w_k > 0), with persistence equivalent to bounded PAS_h drift. What this is not (explicit boundary): No claim is made that physics, biology, cognition, or computation instantiates the structure. Those interpretations are separated into DERIVED/OPEN status and require extra assumptions. Status discipline: All claims are labeled PROVEN / DERIVED / OPEN / REFUTED. PROVEN results depend only on the binding definitions + U-axioms; DERIVED material (e.g., ontology bridges) is explicitly conditional.