What question did this study set out to answer?

The aim is to establish a new mechanism for arithmetic fluctuations using high-frequency coercivity and nonlocal scattering methods.

July 29, 2026Open Access

An Arithmetic-Fluctuation Hilbert–Pólya Mechanism: Continuum p = 2, 3 Positivity via High-Frequency Coercivity and a 40-Moment Tail Certificate

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Preprint explores a novel arithmetic scattering mechanism revealing positivity in prime powers, suggesting implications for prime distribution.

The aim is to establish a new mechanism for arithmetic fluctuations using high-frequency coercivity and nonlocal scattering methods.
Developed a nonlocal arithmetic scattering mechanism for primes p=2,3.
Derived explicit formulas for various mathematical constructs, including the Weil distribution and Gram matrices.
Closed a continuum theorem demonstrating nonnegativity across specified support windows.
Established unconditional nonnegativity of the completed Weil form for support radius ½ log 3 < R ≤ log 2.
Derived a forty-feature moment expansion linking infinite cross Gram results to positivity.
Identified unresolved arithmetic windows for further exploration of prime power contributions.

Tosho Lazarov Karadzhov (2026) studied this question.

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Preprint explores a novel arithmetic scattering mechanism revealing positivity in prime powers, suggesting implications for prime distribution.