A Ward-Governed Admissibility Theorem for Dual-Projection Data: Descent-Induced Symmetry and Analytic Realization

This paper proves a Ward-governed admissibility theorem for dual-projection data from a completion-locked source. The datum consists of two smooth channels defined on a common source region: a response projection and a spectral projection. The admissibility condition requires that the combined dual map be a smooth embedding. Under this hypothesis, the dual image carries a unique intrinsic Ward field, and every downstream observable satisfies a common descended covariance relation. In this sense, a specified source deformation descends to a unique first-order law on the dual image and thereby determines the governing infinitesimal symmetry generator. Several structural consequences follow. Whenever a variational realization of the specified source deformation is available, the associated Noether current is attached to a symmetry whose dual-image generator is already uniquely determined by the intrinsic Ward field. Under the Ward-line compact hypothesis, the minimal effective connected compact carrier is necessarily U(1) ≅ S¹. Under determinant-admissible analytic hypotheses, the same dual image canonically determines a primitive zeta object governed by the descended law; after specifying a distinguished source point and imposing a separate arithmetic matching theorem, the classical Λ-, ξ-, and ζ-functions arise as downstream realizations. A Dirac refinement further shows compatibility with a ℤ₂-graded supersymmetric realization, with localization on fixed-index strata and globalization governed by explicit fiberwise compatibility conditions. The theorem identifies a pre-metric structural order in which the response and spectral channels are governed by one common descended law on the dual image. In this order, symmetry is determined by descent rather than specified a priori, and the compact-carrier, analytic, and graded branches arise as factorizations of the same descended law. What is structurally decisive is that this order is obtained without first specifying a symmetry group, a Hilbert-space realization, a preferred dynamical operator, or a metric background. From smooth structure, completion locking, source deformation, and admissibility alone, the descended law is determined, from which the later realizations follow License note: Distributed under CC BY-NC-ND 4.0.