Quantitative Prolate Approximants in the Radical of the Weil Quadratic Form

The Weil quadratic form attached to the explicit formula for the Riemann zeta function has a global radical containing arithmetic sums of a codimension-two even Schwartz space. The prolate construction of Connes–Consani and Connes–Consani–Moscovici yields a distinguished hard-cutoff two-mode trial state with exponentially small fixed-index leakage. We make the passage from that trial state to the exact global radical quantitative. For every Gevrey order s > 1, we construct a compactly supported perturbation 𝑓λ ∈ 𝒮₀ᵉᵛ whose L²-distance from the prolate state is bounded by a polynomial factor times √(1 − χ₄(λ)). Hence 𝐾λ = ℰ(𝑓λ) lies exactly in the global Weil radical. Smooth localization to [λ⁻¹, λ] gives a vector 𝑘λ that remains at the same error scale from the original trial vector and has norm bounded away from zero. Poisson summation, fixed-index prolate asymptotics, and the Weil explicit formula then give a normalized Rayleigh bound of order 𝐶λᴬ(1 − χ₄(λ)), without cancellation between prime terms. Finally, Gevrey Fourier decay shows that the same scale is preserved by a log-Fourier truncation of dimension 𝑁(λ) = 𝑂ₛ(λ⁴⁺²ˢ log λ); for Gevrey order two this gives the crude bound 𝑁(λ) = 𝑂(λ⁸ log λ).