Structural Lemmas for the First-Prime Window of the Weil Quadratic Form

Abstract

We prove six structural results for the local Weil quadratic form Q_W^L on L^2(-L,L) in the first-prime window (half*log2 < L < half*log3), the minimal interval where only the prime n=2 contributes to the Weil explicit formula. Our main technical contribution (Theorem 5.1) is an algebraisation showing that the matrices J_ij(tau) and E_ij(tau) lie in Q[tau] and are computable without numerical quadrature. We also prove a pure-rational absorption certificate (Theorem 4.3): V + P_{2,7/20} >= (69/100)*V >= 0, machine-verified in Lean 4/Mathlib. We give an exact spectral description of C_{b,L} (Theorem 3.1), falsify Path A by explicit certified negative witnesses (Theorem 6.1), and identify the spectral mechanism: the Weil constant c_L ≈ 1.36527 acts as a global negative diagonal shift that Path A cannot overcome, while without it the {P_0, P_2} subspace is positive definite (Remark 6.2). We also correct an error in an earlier draft: the companion asymptotic for the off-diagonal entry K_L fails by a factor of 40 at L=7/20, invalidating the proposed flip-point formula theta_0 = 1 - c_2/kappa_edge inside the first-prime window (Remark 6.3). We prove the Path B Schur criterion (Theorem 5.3) and establish FP-0.35 (Theorem 7.3): lambda(7/20) > 0, using the certified Weil constant c_L(7/20) = log(2*pi*7/20) + gamma_E (from Suzuki arXiv:2606.09096 equation (4.5), Arb 256-bit certified). These results do not imply the Riemann Hypothesis. Supporting code and 124 automated tests at https://github.com/telleroutlook/weil-first-prime Version 1.7: Two new remarks added to §6 (Strict Falsification of Path A). Remark 6.2 gives the 2×2 spectral mechanism: the Weil constant c_L ≈ 1.36527 is the global negative shift that makes det(E^theta) < 0; without c_L the {P_0,P_2} subspace is positive definite. Remark 6.3 corrects an error in an earlier draft: the edge-mass asymptotic ⟨K_L P_0, P_2⟩ ~ c_2/kappa_edge fails by a factor of 40 at L=7/20, and the proposed flip-point formula theta_0 = 1 - c_2/kappa_edge has no algebraic relationship to the spectrum of q̃^theta. Abstract updated to reflect both corrections. No change to any theorem statement or conclusion.