Gap-Based Analysis of Prime Number Neighborhoods: Statistical Patterns and Mathematical Foundations Part 3
Year: 2025
Type: article
Abstract: 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.
Author: Lopez Lozano, Andres Felipe
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7081621931
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