Prime-Power Fractional Residuals under PNT/Li Controls: A Controlled Computational Audit of Apparent Zeta-Related Structure
Year: 2026
Type: preprint
Abstract: This record contains the manuscript and reproducibility package for the study: “Prime-Power Fractional Residuals under PNT/Li Controls: A Controlled Computational Audit of Apparent Zeta-Related Structure.” The study investigates whether apparently large fractional-part residuals arising from prime-power structure remain anomalous after progressively stronger Prime Number Theorem (PNT), logarithmic-integral (Li), matched-scale, and numerical-convergence controls are imposed. The analysis begins from the arithmetic observable A(n) = Ω(n) − ω(n) and decomposes its summatory structure into prime-power layers k = 2, 3, and 4. An initially elevated k = 2 residual was progressively re-audited using analytically fixed centering constants, PNT-density corrections, a full Li-density baseline, count/location decomposition, analytic reference variance, independently scrambled Sobol quasi-Monte Carlo estimates, deterministic cross-checks, and cross-precision stability tests. In the final analytic-variance re-audit, the location-residual RMS values were: k = 2: 0.476236 k = 3: 0.693608 k = 4: 0.677908 All tested layers passed the final numerical-stability and scale-aware audit gates. A post hoc sensitivity analysis over 45 stricter threshold combinations preserved the same branch classification in all 45 cases. The resulting classification is: PNT/Li-explained within the tested audit regime. This is a finite-range computational branch closure. It is not a proof or disproof of the Riemann Hypothesis, does not establish asymptotic vanishing of the residual, and does not provide independent evidence for the location of nontrivial zeros of the Riemann zeta function. The associated reproduction package contains versioned Python code, raw computational outputs, summary tables, fixed numerical settings, and integrity information required to reproduce and audit the reported results. Author: Min-Gi Kim Independent Researcher, OrganOS September 2026
Author: Kim, Min-Gi
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212131810
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