Scale Stability and Rank Drift in Consecutive Prime-Gap Frequencies

This computational note studies the local frequency ranking of gaps between consecutive primes across increasing numerical scales. Rather than using only cumulative counts up to a bound x, the analysis separates the primes into disjoint decade shells and compares the ranking of the most frequent gaps within each shell. Across the tested shells from 103−104 through 1010−1011, a stable finite-range structure emerges. Gap 6 remains the most frequent throughout. From 107−108 onward, the same top-ten candidate set recurs, although the internal order continues to drift. In the shell 1010−1011, containing 3,663,002,302 counted consecutive-prime gaps, the leading order is 6,12,18,10,2,4,30,24,8,14. A ten-way subdivision of this shell shows that the first four gaps 6,12,18,10 retain exactly the same order in every tested subinterval, while lower ranks continue to exchange places. Relative-frequency ratios with respect to gap 6 reveal that rank stability does not imply full frequency stationarity: gaps 18, 24, and especially 30 gain relative weight, whereas gaps 2 and 4 remain nearly flat. The note also highlights the finite-range downward drift of the twin-prime gap 2 in the local ranking, while explicitly distinguishing this observation from the separate twin prime conjecture concerning infinite occurrence. The results are descriptive and computational, not asymptotic. They are presented in relation to the classical Hardy–Littlewood singular-series framework and the literature on jumping champions. Reproducibility is supported by a standalone Python/NumPy reference implementation, a compiled C++17/OpenMP implementation for the large-shell computation, and external validation against standard values of the prime-counting function.