Mobile spectral laws for compressed prime-power translations

Abstract

We study the sum of zero-extended translations by logarithms of prime powers on L²([-a,a]), weighted by the von Mangoldt function divided by the square root. Its canonical mean spectral measure, after division by a, converges to the limiting mean measure of a Hermitian random Toeplitz model with variance profile 4t. The second and fourth moments are 4/3 and 64/15. Exact invariance under prime phases also yields a classical concentration bound: the mean distribution has scale a while the extreme norm is asymptotic to exp(a). Explicit moment estimates give simultaneous limits for Fourier bands, full sections and both reflection parities whenever log N(a)/a tends to infinity. We verify the logarithmic-energy hypotheses for the archimedean Weil operator and transfer the same law to its Galerkin forms, normalized after compression. A standard moment inequality then bounds the proportion of margins below any fixed threshold less than one. This does not exclude exceptional negative directions, control coupling outside the section, or prove the Riemann hypothesis. The article is entirely analytic. This record includes the PDF, complete LaTeX sources, build instructions, an integrity checker and SHA-256 checksums; no numerical dataset or scientific computation is required. The article and documentation are licensed under CC BY 4.0; the integrity checker is MIT-licensed as specified in the archive. This is a preprint, not externally peer-reviewed. OpenAI ChatGPT/Codex assisted with searches, derivations, drafting and internal review. Julien Lange is the sole author and is responsible for the text.