Gap-Based Analysis of Prime Number Neighborhoods: Statistical Patterns and Mathematical Foundations Part 2

We present a systematic investigation of gap-based analysis for prime-number neighborhoods, introducing the Lozan transform as a mathematical framework for detecting statistical patterns in local prime distributions. Through rigorous empirical validation in multiple numerical ranges (100-1100), we establish that specific gap patterns demonstrate statistically significant discrimi- nation between prime and composite neighborhoods. The method achieves consistent statistical discrimination with large effect sizes (Cohen’s d> 1.0, p< 0.001) for gaps 6 and 12, revealing mathematical structure in prime neighborhoods. Although practical computational applications remain limited, the research contributes to understanding the prime distribution through local neighborhood analysis and establishes a validated framework for gap-based number-theoretic investigation.