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

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 qₐ 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) = qₐ (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 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.