Prime Surface Geometry (PSG): Unveiling the Generative Mechanism via Spiral Embedding

This research reveals that the three-dimensional "Prime Surface" formed by the multifractal structure of prime gaps originates from a geometric generative mechanism embedded within the Sacks Spiral. We demonstrate that the "anisotropy" and "correlations" in local density, induced by the planar spiral arrangement of primes, create a statistical shear field. This field carves the characteristic "weak twist" and "dome-like undulations" observed on the multifractal surface. This paper proposes a "Tripartite Generative Model" that integrates Random Matrix Theory (GUE), multifractal fluctuations, and spiral perturbations. The model not only replicates the observed geometric features of prime data with high precision but also mathematically supports the existence of a stable geometric order across multiple scales. Primes are not merely sequences of numbers; they are a "living geometry" shaped by internal spiral dynamics. The PSG framework presented here offers a novel pathway to approaching profound challenges in number theory, such as the Riemann Hypothesis, through the lenses of geometry and dynamical systems.