What question did this study set out to answer?

The aim is to develop a nonlocal arithmetic Hilbert–Pólya mechanism and explore its implications for prime powers.

July 29, 2026Open Access

An Arithmetic-Fluctuation Hilbert–Pólya Mechanism: Outward-Rounded Residual-Gram Closure of the Completed 2, 3, 4 Window

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Preprint reveals unconditional positivity of prime-power windows, indicating novel mathematical insights.

The aim is to develop a nonlocal arithmetic Hilbert–Pólya mechanism and explore its implications for prime powers.
Utilized a completed localized Weil form to generate prime powers via conservative prime delay-line colligations.
Developed Fourier–Galerkin formulas to analyze endpoints and positivity in second support window.
Applied outward-rounded interval arithmetic and operator-valued responses to establish positivity.
Demonstrated strict positivity of a 165-dimensional low Fourier block with an estimate of C82≥0.89I.
Showed that the exact continuum operator-valued Schur inequality holds at R=1/2 log 5.
Established completed positivity in the first two prime-power windows containing labels 2, 3, and 4.

Tosho Lazarov Karadzhov (2026) studied this question.

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