Compact Weil Event Cells from Suzuki's Screw Function: Parity Polarization, Exact Enclosures, and Uniform Gram Certificates
Abstract
Weil's criterion expresses the Riemann hypothesis (RH) as positivity of an arithmetic quadratic form. On compact windows, Suzuki's scalar function Ψ, the negative of the zeta screw function, determines explicit even and odd parity Gram kernels. Positivity for finitely sampled node sets does not certify a continuum of configurations. Without assuming RH, we develop an event-cell method that converts positivity for prescribed continuous families of finite Gram matrices into verifiable inequalities. The mechanism is the decomposition Ψ = A − P: the prime-power loading P is affine between logarithmic prime-power walls, while the smooth reservoir has strictly increasing curvature A″ and a unique positive inflection at log ρ, where ρ³ = ρ + 1. Centered boxes and sign-boxes give exact parity polarizations and a fixed-window matrix realization. On contained curvature-refined rectangles, the even kernel separates into two scalar profiles whose exact extrema require at most three candidates per coordinate. Clipped cells and the coupled odd kernel admit rigorous outer bounds; the odd kernel also has a strict cellwise Monge sign. Directed entry intervals and scaled Gershgorin inequalities certify entire continuous families of even Gram matrices: suitable positive weights exist if and only if the associated nonnegative comparison matrix has spectral radius below one. A sixteen-node family at half-support L = 0.8 has normalized row margins above 0.2109. An unconditional bound for zeros above the externally verified height 3·10¹² yields finite two-node certificates in both parity sectors. The odd diagonal 4Ψ(t/2) − Ψ(t) is RH-complete through boundedness below; its certified minimum on [log 2, log 10⁸] is 0.0676758… at log 42437, and it exceeds 3/4000 on [log 2, log 10¹⁰]. A mean-zero even box contrast is likewise RH-complete and exceeds 1/800 on the latter range. The Schur complement of a certified positive core isolates finite negative directions that would refute RH. These certificates control specified finite windows and node families, not positivity on unbounded support. The Riemann hypothesis remains open.
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BibTeX
@misc{mittermeier2026compact,
title = {{Compact Weil Event Cells from Suzuki's Screw Function: Parity Polarization, Exact Enclosures, and Uniform Gram Certificates}},
author = {Rainer Andreas Mittermeier},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23603376},
url = {https://zenodo.org/records/23603376},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Compact Weil Event Cells from Suzuki's Screw Function: Parity Polarization, Exact Enclosures, and Uniform Gram Certificates
AU - Rainer Andreas Mittermeier
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23603376
UR - https://zenodo.org/records/23603376
AB - Weil's criterion expresses the Riemann hypothesis (RH) as positivity of an arithmetic quadratic form. On compact windows, Suzuki's scalar function Ψ, the negative of the zeta screw function, determines explicit even and odd parity Gram kernels. Positivity for finitely sampled node sets does not certify a continuum of configurations. Without assuming RH, we develop an event-cell method that converts positivity for prescribed continuous families of finite Gram matrices into verifiable inequalities. The mechanism is the decomposition Ψ = A − P: the prime-power loading P is affine between logarithmic prime-power walls, while the smooth reservoir has strictly increasing curvature A″ and a unique positive inflection at log ρ, where ρ³ = ρ + 1. Centered boxes and sign-boxes give exact parity polarizations and a fixed-window matrix realization. On contained curvature-refined rectangles, the even kernel separates into two scalar profiles whose exact extrema require at most three candidates per coordinate. Clipped cells and the coupled odd kernel admit rigorous outer bounds; the odd kernel also has a strict cellwise Monge sign. Directed entry intervals and scaled Gershgorin inequalities certify entire continuous families of even Gram matrices: suitable positive weights exist if and only if the associated nonnegative comparison matrix has spectral radius below one. A sixteen-node family at half-support L = 0.8 has normalized row margins above 0.2109. An unconditional bound for zeros above the externally verified height 3·10¹² yields finite two-node certificates in both parity sectors. The odd diagonal 4Ψ(t/2) − Ψ(t) is RH-complete through boundedness below; its certified minimum on [log 2, log 10⁸] is 0.0676758… at log 42437, and it exceeds 3/4000 on [log 2, log 10¹⁰]. A mean-zero even box contrast is likewise RH-complete and exceeds 1/800 on the latter range. The Schur complement of a certified positive core isolates finite negative directions that would refute RH. These certificates control specified finite windows and node families, not positivity on unbounded support. The Riemann hypothesis remains open.
ER -