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

Abstract

We establish a referee-auditable package of structural results for the local Weil quadratic form on the first-prime window $\frac{1}{2}\log 2 < L < \frac{1}{2}\log 3$, where only the $n=2$ prime shift is active. The paper proves an exact spectral decomposition of the truncated shift, a self-contained harmonic-number diagonalisation of the logarithmic kinetic form in the Legendre basis, endpoint-potential absorption, an exact $L = \frac{7}{20}$ absorption certificate $$V + P_2 \ge \frac{69}{100}V,$$ and a $\mathbb{Q}[\tau]$ algebraisation of the first-prime Legendre matrices. These ingredients yield a split-residual Schur criterion. At $L = \frac{7}{20}$, the remaining FP-0.35 application is reduced to rigorous positive definiteness of one $8 \times 8$ even-sector and one $6 \times 6$ odd-sector interval Schur matrix. Version 1.8 deliberately records that final computer-assisted gate as open because no frozen outward interval positive-definiteness certificate is supplied. No unconditional FP-0.35, whole-window positivity, or Riemann-Hypothesis claim is made.