A Residual Compression Effect in Self-Repeating Last-Digit Transitions of Consecutive Primes

Lemke Oliver and Soundararajan (2016) established that consecutive primes exhibit astatistically significant bias against sharing the same final decimal digit. This note asks anarrower, follow-up question: conditional on a same-digit (“diagonal”) transition actuallyoccurring, is the resulting prime gap systematically different from what a simple inverseprobability model would predict? Using primes below 108, we show that (i) fitting gapsize as a function of transition probability alone (gap ≈ C/P) explains 88.2% of varianceacross the 16 last-digit transition types, consistent with a simple renewal-type argument andnot, by itself, new information beyond the known digit bias; but (ii) after controlling forrow and column (starting- and ending-digit) fixed effects and the deterministic arithmeticfloor Gmin, same-digit transitions carry an additional, statistically significant gap reductionof approximately 2.6 units (p = 0.0051), and this coefficient replicates closely across twoindependent, non-overlapping halves of the dataset. We report this as a modest, reproducibleempirical regularity, explicitly flag the small effective sample size of the underlying regression(n = 16 transition-type cells), and describe what would be required to elevate it from anobserved regularity to an explained phenomenon