Forced Noether Currents and Rigid Analytic Completion in Universal Phase Structure

The standard Noetherian direction proceeds from a chosen variational formulation together with an assumed continuous symmetry to a conserved current. In prior work, we establish an identity-first order of implication in the rank-one phase sector: physical identity is fixed by a preserved circle-bundle structure, and requiring admissibility (descent to that identity class) forces the canonical phase action and, when a variational formulation exists, the associated Noether current. The same order of logic extends canonically to compact connected symmetry groups by passing to maximal-torus descent. The present paper provides a theorem-level instantiation of this identity-descent logic at the level of analytic realization in the Berezinskii–Kosterlitz–Thouless (BKT) universality class. We prove that, once the identity datum and universality class are fixed, the analytic completion coordinate is uniquely determined. A dyadic descent theorem removes residual normalization freedom among admissible representatives by selecting a canonical dyadic normalization compatible with descent of the underlying identity. A nome-locking theorem then shows that the exponential completion coordinate is forced, yielding a uniquely determined Jacobi theta-function nome, up to the natural identifications inherent in the completion. Consequently, the dyadic–nome bridge is closed, rigid, and structurally forced: no tunable analytic parameters remain beyond the fixed identity and universality assumptions. A worked example propagates certified stiffness information through the locked completion map and provides an inverse diagnostic for representative-level deviations without introducing adjustable degrees of freedom. License note: Distributed under CC BY-NC-ND 4.0.