Gap-Based Analysis of Prime Number Neighborhoods: Statistical Patterns and Mathematical Foundations Part 3

We present the Lozan transform, a novel mathematical framework for analyzingprime number neighborhoods through gap-based statistical discrimination. This workintegrates comprehensive empirical validation (Phase 2) with rigorous theoretical for-malization (Phase 3) to establish mathematical foundations for local prime distribu-tion analysis. Through systematic investigation across multiple numerical ranges (200-5300), we demonstrate that specific gap combinations achieve statistically significantdiscrimination between prime and composite neighborhoods, with gaps 6 and 12 provid-ing optimal performance (Cohen’s d > 1.0, p < 0.001). Theoretical analysis validatesthe hypothesis of divisibility-by-6 superiority, confirms local optimality within con-strained gap spaces, and establishes mathematical bounds approaching 50-70% of the-oretical limits. The framework contributes to number-theoretic understanding throughanalysis of local neighborhood structure, offering applications in the discovery of primeconstellations, density estimation, and specialized computational contexts. Althoughpractical limitations constrain general computational applications, the method pro-vides robust theoretical foundations for continued investigation into prime distributionpatterns. This paper represents a comprehensive advancement beyond our initial Phase 2 em-pirical validation, integrating rigorous theoretical formalization to establish the com-plete mathematical foundation for gap-based prime neighborhood analysis.