The Geo-Arithmetic of Prime Gaps: A Vector Signature Under Monotonic Compression

This paper introduces a geometric framework that transforms prime gap analysis from an arithmetic search problem into a spatial navigation problem. By mapping primes p into three-dimensional space via monotonic compression functions f(p) ∈ {ln p, √p, log2 p}, we discover that the arithmetic gap g_i = p_{i+1} - p_i encodes a precise geometric vector signature characterized by normalized energy E and XY-plane angle θ_xy. Unlike traditional post-hoc classifiers, our framework enables active prediction: a geometric "Jump Compass" law (Δx ≈ 0) reduces the search space for unknown twin primes by over 90%, converting exhaustive sieving into targeted navigation. The method achieves 100% precision and recall across ranges from 10^6 to 10^18, with rigorous train/test validation demonstrating that classifiers calibrated on small primes (10⁶) successfully identify gaps between astronomical primes (10^18) without recalibration. We further demonstrate the universality of this law by identifying twin primes at cryptographic scales (1536-bit), confirming that the geometric organization persists across 456 orders of magnitude.