The Emergent Geometry of Goldbach Spaces: A Deterministic Computational Investigation

Abstract

This preprint presents a deterministic computational investigation of the collective geometry of Goldbach Spaces. Instead of studying individual Goldbach decompositions, the work introduces a geometric representation that transforms each Goldbach Space into a normalized profile suitable for statistical analysis. Large-scale computational experiments reveal a remarkably low-dimensional organization, with the first principal component explaining approximately 96% of the observed variance and the first two principal components accounting for more than 99%. Bootstrap validation and randomized null models indicate that this organization is robust and is not reproduced by random prime-like constructions. The manuscript does not propose a proof of Goldbach's conjecture; rather, it reports a reproducible computational phenomenon and proposes new directions for experimental investigations in additive number theory.