Prime-Forced Word Dynamics and a Planar Graph Invariant: Least-Factor Vacancies, Self-Sustaining Prime Closure, and a Reciprocal-Logarithmic Graph Law

Abstract

This preprint studies a deterministic word system on the alphabet {1,2,3} whose local evolution admits an exact arithmetic interpretation. The central result is a least-factor vacancy law: for a left-half cumulative coordinate x at layer N, the coordinate is vacant exactly when x < P^-(N - x), where P^-(m) denotes the least prime factor of m. This law determines the complete canonical word directly from integer arithmetic. The dynamics also contain an exact prime marker at N=2p-1, allowing each prime p>=7 to be identified internally before it is required by the later prime-indexed forcing mechanism. This yields a self-sustaining prime-closure theorem without supplying an external future list of primes. The resulting words are mapped to distributed planar graphs carrying an integer-valued functional G. The graph functional has an exact topological form, is non-negative on the canonical reachable family, and satisfies 2G<=N-1. In the prime-resolved region it decomposes into explicit prime and twin-prime statistics plus a controlled least-factor front, yielding the asymptotic law G_N / N ~ 2 / log N. The paper is organized so that the symbolic and arithmetic results are proved independently of the computational experiments. Independent implementations and large finite-range audits are included as validation and regression tests rather than as premises of the proofs. This is Paper I of the project.