Prolate Leakage and Near-Zero Spectral Clusters of the Semilocal Weil Operator

For λ>1 let A_λ be the self-adjoint operator on L²([λ⁻¹,λ],du/u) that represents the semilocal Weil quadratic form, including the archimedean place, all prime powers up to λ² and the pole terms; it has compact resolvent. We prove that for every integer m≥1 there are explicit constants Kₘ and Λₘ such that, for every real λ≥Λₘ, the operator A_λ has at least m eigenvalues, counted with multiplicity, in [-δ,δ], where δ=Kₘλ⁵(1-χ₄ₘ₊₄(λ))1/2 and the χₙ(λ) are the eigenvalues of time and frequency limiting to [-λ,λ]. Known effective bounds for 1-χₙ give δₘ'λ4m+17/2e-πλ². Unlike upper bounds for the bottom of the spectrum, this localizes eigenvalues on both sides of zero, without any positivity assumption. The test vectors are images, under the summation map f↦ u1/2∑n≥1f(nu), of combinations of prolate spheroidal functions with f(0)= f(0)=0. The proof rests on three ingredients: an exact residual identity, derived from the explicit formula and Poisson summation, which expresses A_λ on such images through the part of f outside [-λ,λ]; a transfer estimate bounding this residual by λ⁵ times the leakage; and a lower bound for the summation map on blocks of prolate functions, uniform in λ. Consequently the number of eigenvalues of A_λ in [-e-(π-ε)λ²,e-(π-ε)λ²] tends to infinity. No hypothesis on the zeros of the zeta function is used. Time flies, and I want to conduct more high-quality research.