Prime Gaps as a Developmental Dynamical System

Abstract

This preprint introduces a developmental–dynamical framework for analyzing prime gaps using two dimensionless operators: an instability operator Jn and a curvature operator Hn, both normalized by the natural logarithmic scale λn=ln⁡pn. The operator system reveals geometric phase boundaries, dual attractor structure, and several stable invariants, including a strong‑instability basin density of approximately 0.19–0.20 and a characteristic gap‑inflation scaling law gstrong(pn)≈1.6ln⁡pn. By organizing prime gaps within a curvature–instability morphospace, the framework identifies three developmental phases—birth, growth, and maturity—corresponding to explosive early behavior, corridor stabilization, and tight logarithmic packing. A discrete operator map is embedded into a continuous scale variable t=ln⁡pn, yielding flow equations that describe long‑range behavior while remaining grounded in discrete arithmetic. The results provide a geometric and dynamical interpretation of prime distribution that complements analytic and probabilistic approaches, highlighting operator‑defined invariants, attractor dynamics, and structured departures from the classical spacing law g(p)≈ln⁡p.