A Rigorous Odd-Sector Positivity Certificate for Suzuki's Localized Weil Quadratic Form up to the von Mangoldt Threshold at Seven
Year: 2026
Type: preprint
Abstract: 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.
Author: Luo, Yongqi
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7203677926
Cites:
Cited by:
Related to: