Prime Chains, Pentagon Vector Transform, and the Order-Chaos Phase Transition

This is a preprint of the manuscript: "Prime Chains, Pentagon Vector Transform, and the Order-Chaos Phase Transition" Abstract:We present a self-contained account of two interrelated frameworks for studying the structure of prime numbers. Part I develops the Prime Chain Decomposition: every prime belongs to exactly one non-intersecting infinite forward orbit under the prime-index map q → p(q), partitioning the primes into a forest of disjoint chains (Theorem 1). Each prime has finite algebraic depth—the number of reverse predecessor steps to its root prime (Theorem 2). The chains share a universal super-exponential growth law a(s+1)/(a(s) · ln a(s)) → 1 (Theorem 3), with all chains linearising asymptotically to parallel lines in double-logarithmic coordinates. Part II introduces the Pentagon Vector Transform T(n), a coordinate embedding of the integers into the Euclidean plane. Under this transform, primes stratify into families indexed by their gap value. We establish three main theorems: (Theorem A) the asymptotic slope of all constraint curves is exactly −tan(36°), an exact analytic identity; (Theorem B) adjacent gap layers are separated by exactly 2φ sin(36°) and layer crossing is governed by the exact algebraic condition ∆(π(p) − p) = −∆g · φ; (Theorem C) layer ordering necessarily breaks down as p → ∞, a direct consequence of the Prime Number Theorem and the unboundedness of prime gaps. Theorems A and B are exact analytic results holding for all primes; Theorem C establishes a necessary qualitative property. The golden ratio φ = (1 + √5)/2 appears as an exact consequence of pentagon geometry. All results are verified numerically on primes up to 10,000; chain structure is verified to 10^7. The manuscript has been submitted to a peer-reviewed journal and is currently under review.