Let rₙ = g₍+₁/gₙ be the ratio of consecutive prime gaps, and let F be the limiting distribution of rₙ. We establish, conditionally on the Hardy-Littlewood prime tuples conjecture, that the two most prominent Pythagorean mass points of F — at ratios 2/3 and 3/2 — straddle the golden probability levels phi^-2 approximately 0. 382 and phi^-1 approximately 0. 618 respectively: F (2/3^-) phi^-2, confirming that the Hardy-Littlewood singular series structure — not the Cramér approximation — is essential to the result.
Paul Buchanan (Mon,) studied this question.