Empirical Analysis of Prime Gap Distribution Patterns: Observations on the α50(N) Function

This repository contains complete code, data, and analysis scripts for the empirical study of prime gap distributions. The research documents several striking patterns in prime gaps g_n = p_{n+1} - p_n relative to p_n^α, including: • Extreme scarcity of gaps with g_n > √p_n (only 6 found among 100M primes)• Systematic decay of the median threshold α50(N) = -0.007411 ln N + 0.265480• Exact relationship F_N(0.5) = 6/N for the proportion of gaps exceeding √p_n• Predictive model α50(N) ≈ (ln ln N - Δ)/(ln N + ln ln N) The repository includes:1. Python code for prime generation and gap analysis2. Complete datasets for N up to 100 million primes3. Statistical validation scripts Google colab file atached " law.ipynb " with full pyhton code used for all test + Google colab file test.ipynb (for rapid verification) All results are reproducible and openly available for verification and further research. Keywords: prime gaps, empirical mathematics, computational number theory, prime distribution, α50 function