Paradigm Lock_ Trunk Closure of Identity as the Unique 1-DOF Recurrence

Abstract This paper presents a trunk-level axiomatization and formal closure of identity governance for recurrent systems under explicit structural constraints. It introduces a minimal axiom set (A1–A9) defining identity-relevant bounded recurrence and proves the forced trunk structure that any system satisfying these axioms must admit. From A1–A9 alone, the paper proves: representation on S¹; SO(2) as the connected admissible symmetry; O(2) chirality capacity when orientation distinctions are required; a complete invariant family {r_k, χ_k}; minimal scalar governance (PAS_h, χ_h) when chirality activates; and drift-bounded identity persistence under affine gauge. No domain assumptions (physical, biological, computational, or probabilistic) are invoked in any PROVEN result. The paper draws an explicit proof boundary separating PROVEN trunk results from DERIVED and OPEN domain claims. It does not assert universality beyond the defined trunk scope, does not eliminate probability as a domain concept, and does not claim uniqueness of any specific scalar construction beyond governance form. Indexing schemes, numerical constants, and domain interpretations are explicitly classified as non-trunk. No new mathematics is introduced; all arguments rely on standard topology, group theory, and invariant analysis. The contribution is formal closure: consolidating the minimal assumptions required for identity persistence, proving the uniqueness of the trunk structure under those assumptions, and eliminating silent scope drift. This document serves as the canonical trunk reference for the Paradigm Lock framework and establishes a precise contract for all future domain extensions.