Compact Weil Event Cells from Suzuki's Screw Function: Parity Polarization, Exact Enclosures, and Uniform Gram Certificates

Suzuki's screw-function formulation expresses Weil positivity for the Riemann zeta function through a single real scalar function Ψ. From this scalar we construct explicit even- and odd-parity kernels on compact windows and analyze their finite-dimensional optimization geometry. Centered boxes and sign-boxes polarize Ψ into two Gram kernels, while a fixed-window realization theorem shows that positivity of the corresponding finite matrices for all node sets is equivalent to positivity of the localized Weil form on dense parity cores. The prime-power decomposition Ψ = A − P partitions each scalar argument at logarithmic prime-power walls. The exact curvature identity for A has a unique positive zero at t = log ρ, where ρ³ − ρ − 1 = 0; adjoining this curvature wall makes A′ monotone on every refined scalar cell. The even kernel then separates in sum–difference coordinates: its exact range on every contained rectangular cell follows from finitely many one-dimensional endpoint and stationary-point evaluations. Boundary-clipped cells inherit rigorous outer bounds; the coupled odd channel admits a termwise enclosure; and uniform entry bounds, combined with ordinary or diagonally scaled Gershgorin inequalities, certify positive definiteness for continuous families of finite Gram matrices. Schur-complement inertia provides complementary finite falsification witnesses. At fixed support, these results reduce the stated continuous-family certification problems to finitely checkable scalar bounds; they do not control the unbounded-support limit or prove the Riemann hypothesis.