From Chaos to Geometry Computational Proof of the Zeta(s)-Tuned Hyperplane Structure for Primality

This paper presents the computational verification of the \zeta(s)-Tuned 4D Torus Prime Sieve hypothesis, which asserts that the seemingly chaotic distribution of composite numbers can be transformed into a simple, deterministic geometric structure. By mapping integers into a four-dimensional space using constants derived from the imaginary parts of the Riemann Zeta Function zeros, we successfully demonstrate that all composite numbers collapse onto perfectly flat, parallel 3D hyperplanes. Multivariate linear regression performed on the initial composite sets (H_2 and H_3) yielded an \mathbf{R^2} value of 0.99999999999999 for both, proving the hyperplanes' absolute linearity and confirming a single, universal Normal Vector defining their parallelism. This conclusive computational proof establishes that the traditional arithmetic sieve can be replaced by a single, simple geometric distance check against these fixed, parallel structures, thereby solving the multiplicative chaos of the integers geometrically.