Singular-series field, local calibration, and residual energy in prime-semiprime correlations
Year: 2026
Type: preprint
Abstract: For an odd positive integer h, Todd(h) is the multiplicative local field built from the twin-prime constant. The correlation G_{2,h}(x) counts pairs (p, m) with p prime, m at most x an odd squarefree semiprime, and |2p-m| = h, on the scale M(x) = x log log x / log^2 x. Averaging in h, the prime-number theorem, the Siegel-Walfisz theorem and a Selberg upper bound give sum Todd(h)^k G = M(x) sum Todd(h)^{k+1} + o(H M(x)) for every fixed k. The energy is of order H M(x)^2, with second-moment constant mu_2 = 2.3009615447 as a mandatory lower bound and a nonnegative residual. A direct count up to x = 10^8 leaves a residual fraction about 0.025. The paper does not prove that the residual tends to zero. The record contains the English version (main file), a faithful Portuguese version, and the supplementary archive with the Python scripts, figures, data and Lean 4 files for the local rational identities. The Lean files were not submitted to the AXLE verifier in this version.
Author: de Melo, Fábio Dayan Soares
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225657517
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