A Rigorous Odd-Sector Positivity Certificate for Suzuki's Localized Weil Quadratic Form up to the von Mangoldt Threshold at Seven

We study the odd sector of the localized Weil quadratic form associated with the Riemann zeta function in the operator framework introduced by Suzuki. After scaling to the fixed interval (-1,1), the polar contribution is separated as a negative rank-one form and the remaining positive base form is treated by a normalized Legendre low/high decomposition and a quadratic-form Feshbach reduction. The infinite-dimensional high-mode contribution is controlled without a global spectral truncation. The logarithmic remote tail is evaluated in closed form, the Euler remote block is reduced to exact-degree Gauss--Legendre identities and an orthogonal-projection Gram formula, and the analytic remainder has an exact 601-order low/remote band gap followed by a Bernoulli operator-norm estimate. The resulting finite collection of inequalities is certified with outward arbitrary-precision ball arithmetic. At a_7 = (1/2) log 7, the clean-environment Arb certificate gives m_infinity > 2.0 x 10^-23, and hence strict positivity on the nonzero odd form domain. Zero-extension nesting then gives the same conclusion for every 0 < a <= a_7. The interval passes the nonzero von Mangoldt support thresholds 2, 3, 4, 5 and reaches the threshold 7, where the scaled n = 7 translation has shift 2 and zero overlap. This is a finite-localization odd-sector result and does not prove the Riemann hypothesis.