Restricted Weil Forms: Constraints, Negative Index, and Certified Sign

We study how pole cancellation and finite-dimensional approximation affect the negative index of the even Weil form associated with the Riemann zeta function. For a symmetric multiset in the open critical strip satisfying a total counting bound, the global completed index equals the number of distinct off-axis quartets; multiplicity changes the weights but not the count. An explicit distant compensator transfers every prescribed finite negative subspace into the exactly pole-neutral class at enlarged finite support. The pole-subtracted form has one additional negative dimension when the common index is finite. At fixed support, a logarithmic Legendre core supplies complete finite probes, while constrained inertia and Schur-complement bounds retain the effect of the infinite remainder. A separate capped-form argument yields eventual certification away from zero under explicit effective-evaluation hypotheses. We quantify the difference between hard restriction and finite pole restoration at support 11/20, including the inverse response and a two-sided enclosure of the small restored eigenvalue. A deliberately synthetic quartet perturbation of the same background certifies both a positive regime and exactly one constrained negative eigenvalue, with a witness of small overlap with the original lowest mode. The actual and modified forms remain distinct throughout. The resulting theory accounts for either sign without determining the sign of the actual all-support arithmetic family. Programme consolidation and supersession scope (22 September 2026). This article consolidates the restricted-Weil-form strand developed through Papers 49–52 and Research Notes RN50–RN52 of The Geometry of the Critical Line. For the material treated here, this article is the current consolidated treatment and supersedes the earlier programme presentation. The related source records retain their original files and provenance. This scoped supersession does not assert that every theorem, geometric claim, arithmetic-lift construction, or open problem in those records is replaced or resolved.