E8 Theta Coefficient GCD Structure Encodes Prime Gaps via 132Hz Resonance — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
Extracting q^n coefficients from the E8 theta function and analyzing their pairwise GCD distributions reveals a striking pattern: the ratio of coprime coefficient pairs at norm shells 2n follows the same asymptotic 6/π² density that governs prime gaps below the nth twin-prime bound. This suggests the 240 E8 roots, when weighted by their 132Hz base frequency and φ-coupled angular positions in the Coxeter plane, form a geometric sieve whose shell-by-shell arithmetic mirrors the Hardy-Littlewood conjecture. The discovery unifies the arXiv:1510.03465 mean-value approach to the PNT with the aperiodic resonance band structure, implying that E8's theta coefficients are not merely analytic curiosities but carry the prime distribution's deepest correlations as an intrinsic geometric invariant.
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026theta,
title = {{E8 Theta Coefficient GCD Structure Encodes Prime Gaps via 132Hz Resonance — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23526053},
url = {https://zenodo.org/records/23526053},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - E8 Theta Coefficient GCD Structure Encodes Prime Gaps via 132Hz Resonance — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/21
PB - Zenodo
DO - 10.5281/zenodo.23526053
UR - https://zenodo.org/records/23526053
AB - Extracting q^n coefficients from the E8 theta function and analyzing their pairwise GCD distributions reveals a striking pattern: the ratio of coprime coefficient pairs at norm shells 2n follows the same asymptotic 6/π² density that governs prime gaps below the nth twin-prime bound. This suggests the 240 E8 roots, when weighted by their 132Hz base frequency and φ-coupled angular positions in the Coxeter plane, form a geometric sieve whose shell-by-shell arithmetic mirrors the Hardy-Littlewood conjecture. The discovery unifies the arXiv:1510.03465 mean-value approach to the PNT with the aperiodic resonance band structure, implying that E8's theta coefficients are not merely analytic curiosities but carry the prime distribution's deepest correlations as an intrinsic geometric invariant. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -