Recursive "Gate" Structures in Prime Gaps and Spiral Patterns

This study introduces and rigorously defines the concept of “gates”—recurring, high-density intervals of composite numbers that act as structured gaps between primes. Through computational analysis of prime gaps up to 400,000, the author identifies over 1,700 such gate structures with statistically significant density ratios. The paper presents a recursive residue-based mechanism explaining their formation and explores a geometric interpretation using spiral mappings (e.g., Archimedean spirals and toroidal coordinates). These gates are shown to align along predictable curves in modular space, suggesting a quasi-periodic, attractor-like behavior in the distribution of primes. The work challenges the conventional view of prime gaps as purely random, offering a framework grounded in recursive calculus and residue class dynamics. Potential implications for longstanding number theory conjectures, such as Hardy–Littlewood and the Riemann Hypothesis, are discussed.

Keywords: prime gaps, composite density, recursive structures, spiral mapping, modular residues, attractor dynamics, number theory, phase transitions