ACORN Recovery of Cramer's Conjecture and a 5D Prime-Generating Procedure from Irreducible Curvature Modes

This paper presents a geometric reinterpretation of prime numbers derived from the ACORN (A Curvature-Origin Reality Network) framework of finite 5-dimensional SpaceTime⁺ geometry. In this approach, prime numbers emerge naturally as the irreducible eigenmodes of a finite 5D curvature operator, while composite integers correspond to curvature-decomposable modes. The familiar 4D integer line is shown to be the projection of the finite spectral structure of the fifth dimension. The paper demonstrates that: • the Prime Number Theorem arises directly from the asymptotics of the 5D curvature spectrum, • Cramér’s conjectured bound on prime gaps gₙ = O((ln pₙ)²) is recovered without number-theoretic assumptions as a projection of 5D curvature-gap growth, • and, for the first time, a fully constructive, physics-based Prime-Generating Procedure is obtained from the irreducibility structure of the curvature modes. The analysis further identifies a physically meaningful “first prime” associated with the lowest curvature excitation (linked to the Planck constant h) and a “last prime” arising from the upper curvature boundary linked to the maximal projection speed (the speed of light c). This work unifies prime-number structure with the geometric foundations of ACORN, suggesting that arithmetic itself is a shadow of deeper 5D curvature physics. It provides the first physical mechanism for prime distribution, prime gaps, and primality, and represents a bridge between number theory, quantisation, and relativistic geometry.