The Range of a Finite-Scale Weil Positivity Method: Success in the First Prime Window and Termination in the Second

We study a specific finite-scale sufficient criterion for positivity of the Weil quadratic form: the split-residual Schur certificate, which reduces a parity-sector positivity problem to positive-definiteness of explicitly assembled finite matrices. The purpose of this paper is not to determine the full range of scales on which that certificate succeeds, but to document a rigorous change of regime across the first two prime windows. At the first-window anchor L = 7/20, the full four-term assembly gives certified positive lower bounds 7.81 × 10⁻⁴ in the even sector and 4.92 × 10⁻² in the odd sector. In the second prime window a genuinely new mixed $p = 2 / p = 3$ contribution appears; at L = 3/5 its contribution matrix is certifiably nonzero. At the near-left-edge second-window anchor L = 11/20, the odd-sector LDLT minimum pivot lies in [-2.815 × 10⁻², -1.720 × 10⁻²], so the specified sufficient Schur certificate is not positive-definite there. A separate certified negative pivot at L = 3/5 shows that the nonzero cross-prime interaction does not restore the full certificate at that tested anchor. Pilot computations, clearly separated from the certified statements, suggest margin exhaustion between these anchors. They are not used to infer whole-window monotonicity, a complete range theorem, failure of the underlying Weil quadratic form, or any implication for RH. Thus “failure” throughout the paper means failure of the specified sufficient finite-dimensional [certificate / criterion].