A Finite-Window Operator Form of the Weil Explicit Formula: Trivial-Pole Boundary Terms and the Restricted Weil Minimum

Scoped supersession notice (22 September 2026). This record belongs to the Papers 49–52 and RN50–RN52 source arc consolidated in Restricted Weil Forms: Constraints, Negative Index, and Certified Sign, by Pavel V. Kramarenko-Byrd (DOI: 10.5281/zenodo.22885925). For the restricted-Weil-form material treated in that article, the new article supersedes the earlier programme presentation and is the current consolidated reference. Supersession is limited to that scope; other results, geometric claims, arithmetic-lift constructions, and open problems in this record are not thereby replaced or resolved. The original description below and deposited files are retained for provenance. Paper 52 in the "Geometry of the Critical Line" programme. This paper assembles the finite-window restricted-Weil operator framework developed in The Geometry of the Critical Line into a single operator-theoretic dictionary. The central object is the constrained minimum m(a) of a finite-window quadratic form q_a on the restricted admissible class. The paper records five live analytic targets, three structural dictionaries, and the named open inputs through which restricted Weil positivity may be studied. The main structural dictionary identifies the compressed admissible Weil condition with truncated Weil zero-side positivity after removal of the trivial-pole rank-one boundary. Equivalently, the lifted form Q+(u) = q_a(u) + 2a |⟨u, cosh(ax/2)⟩|² is the finite-window zero-side Weil form together with the protected odd boundary term. This does not prove the Riemann Hypothesis; it identifies the RH-level residual in the finite-window operator language. The paper also records the type-balance dictionary governing the β < 3/4 severe-compression route, the bilinear obstruction in the within-window scalar-resolvent continuation problem, the finite-witness certification hierarchy, the retired routes, and the open bridges. The published corpus does not commit to either RH alternative. It supplies the bidirectional machinery for studying both. This deposit also includes a self-contained reproducibility bundle (Paper52_Reproducibility_Bundle.zip) supporting the W7 subspace audit at K = 35, a = 3 in the C3 finite basis. The bundle contains the seven C3 basis-witness files, the precision-preserved evaluator, and the canonical runner that executes the interval LDL^T certification. The bundle does not establish global m(a) > 0 or m(a) < 0 at the continuum level, closure of any of the five live targets, or NW-2/NW-3/NW-4 promotion of any subspace test; it produces a finite-subspace classification under the precision-preserved evaluator. This is Paper 52 in The Geometry of the Critical Line programme, following Papers 0–8 (the monograph) and Papers 38, 40, 47, 49, 50, 51 plus Research Notes RN50, RN51, RN52.