Type Consumption in Paley-Wiener Spaces: A Conditional Reduction of the Riemann Hypothesis
Year: 2025
Type: article
Abstract: Quantitative type consumption theorem for Paley-Wiener spaces showing that trivial-zero vanishing constraints in Weil-type approaches to the Riemann Hypothesis consume exactly pi/2 units of exponential type, with sharp uniform L2 bound K=1. The factorization F = S·G through cosh(pi·z/2) reduces effective localization bandwidth by a precise, computable amount. This resolves the Localization Conjecture: constrained test functions achieve arbitrarily sharp localization as type grows but face a spectral floor scaling as (R - pi/2)^{-2} at finite type. Combined with the Weil-Lagarias positivity criterion, this yields a conditional RH reduction to nonnegativity of the Weil quadratic form on a well-defined infinite-dimensional domain. Three explicit gaps in the full conditional reduction are documented: measure-theoretic rigor in infinite dimensions, the Lyubarskii-Seip interpolation argument, and verification of positivity on the constrained domain. A general Constraint-Consumption Obstruction theorem is formulated showing this mechanism — constraints consuming the degrees of freedom needed for detection — applies beyond RH to sampling theory, inverse problems, and PDE regularity. No unconditional claims are made. All gaps are explicitly identified.
Author: Cardwell, Ryan
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7117882146
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