Weil positivity on the window of width log 15

Weil's criterion states that the Riemann Hypothesis holds if and only if Weil's quadratic form Q is nonnegative on all admissible test functions. Restricting the support to a fixed interval [−L, L] gives a weaker, unconditional question. We prove that Q(f) ≥ 2−2904‖f‖22 for every complex C2 function f supported in [−½ log 15, ½ log 15]. The previous largest window we know of is ½ log 12 (our previous paper). On the new window the prime 13 enters. The proof uses the bounded-comparison framework of Liu and the general tail estimate and prolate defect bound of our previous paper, with a new weight in the Schur bound for the prime operator, and quadrature estimates that do not assume L < 5/4. Positivity is reduced to a finite matrix inequality of size 832 in each parity, which we certify in Arb ball arithmetic with all quadrature, rounding and inversion errors paid. The certified finite margin is about 2.40×10−70 (even) and 3.95×10−66 (odd), while the total error is below 5×10−94. This is an external certificate, not a formal proof. The result concerns one fixed window and does not address the Riemann Hypothesis. Certificate code: release v1.6.0 of https://github.com/moriryota/prolate-weil-bounds (doi:10.5281/zenodo.23092422). Certificate data: https://github.com/moriryota/weil-log15-certificate-data (doi:10.5281/zenodo.23281176).