Entropy Signatures and Spectral-Universality Heuristics in Goldbach Partitions

This revised preprint presents a computational and phenomenological analysis of Goldbach partitions through the combined lens of statistical physics and spectral theory. It examines Hardy–Littlewood singular-series structure, Shannon entropy signatures, congruence-sector behaviour, and finite-height spacing statistics of low-lying Riemann zeta zeros.

The analysis reports a reproducible approximately 1-bit entropy lift for even integers with N ≡ 0 (mod 6), corresponding to the p = 3 local factor in the singular series, with additional smaller bit-lifts associated with higher small-prime divisors. It also reports finite-height zeta-zero spacing variance rising from 0.097 for the first 20 zeros to 0.140 for the first 500 zeros, interpreted as slow convergence toward the GUE benchmark rather than small-sample agreement with asymptotic GUE statistics.

This work does not claim a proof of Goldbach’s Conjecture, a causal derivation from GUE statistics to Goldbach partitions, or asymptotic validity beyond the tested computational range. It is offered as a reproducible, human-led and AI-assisted exploratory preprint: a phenomenological and testable lens on Goldbach partition structure, singular-series modulation, entropy signatures, and spectral-universality heuristics.