Persistent Conditional Dependency in Consecutive Prime Gaps: A Multi-Scale Statistical Analysis

We demonstrate that consecutive normalised prime gaps exhibit persistent short-range conditional dependency, with mutual information increasing to approximately 0.78 bits at 1010, extending residue-class biases to the full gap distribution. Under Cramér’s stochastic model, consecutive normalised gaps gₙ/ln(pₙ) should be independent. Using a framework combining mutual information estimation, autocorrelation analysis, conditional moment computation, and residue-class transition matrices, we show that this independence assumption is violated at all scales tested. At 1010 (455 million primes), I(X; Y) = 0.782 bits – compared with I = 0.004 bits for a simulated i.i.d. Exp(1) control, consistent with discretisation noise – with lag-1 autocorrelation ρ₁ = −0.0276 and χ² = 1.32 × 108 for residue-class transitions (mod 6). These effects converge with increasing scale rather than vanishing, ruling out finite-sample artifacts. The conditional mean exhibits negative short-range autocorrelation consistent with mean-reversion-like behaviour: small gaps are followed by larger-than-average gaps and vice versa. These results are consistent with and extend the residue-class biases identified by Lemke Oliver and Soundararajan (2016) into the full continuous normalised gap distribution, providing a scalar measure of dependency strength and establishing its scale stability through a three-point convergence series.**v2 adding code and readme **v3 re-adding original manuscript