What does this research mean for the field?

Prime numbers can be structurally partitioned into disjoint chains with universal super-exponential growth and geometrically stratified into gap-indexed families using a Pentagon Vector Transform. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.ESTABLISHES_NEW_DIRECTION.

What question did this study set out to answer?

The aim is to develop frameworks that analyze the structural properties of prime numbers and their interrelationships.

June 9, 2026Open Access

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

Key Points

The aim is to develop frameworks that analyze the structural properties of prime numbers and their interrelationships.
Introduced the Prime Chain Decomposition, partitioning primes into disjoint chains under a prime-index map.
Proposed the Pentagon Vector Transform to embed integers, stratifying primes by gap values in the Euclidean plane.
Established and verified three main theorems concerning the slopes of constraint curves and properties of prime gaps numerically.
The asymptotic slope of constraint curves is exactly −tan(36°).
Adjacent gap layers are precisely separated by 2φ sin(36°) with specific algebraic conditions governing layer crossing.
A breakdown in layer ordering occurs as prime numbers increase, consistent with the Prime Number Theorem.

Abstract

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 10⁷. The manuscript has been submitted to a peer-reviewed journal and is currently under review. Keywords: prime chains, prime-index map, orbit decomposition, pentagon vector transform, gap stratification, golden ratio, phase transition, prime number theorem, algebraic depth, double-logarithmic law

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

Key Points

Abstract

Cite This Study