Positive lower density of large prime gaps

What OpenAI claims

For every fixed C > 0, a positive proportion of consecutive prime gaps exceed C log p_n, throughout every sufficiently large initial segment of the primes. The proportion may depend on C. Consequently, the indices where p_n/n increases have positive lower density, answering Erdős and Prachar. (Lean)

Paper

1. Positive lower density of large prime gaps

   25 September 2026 · Preprint, not peer reviewed · PDF

   For every fixed C > 0, we prove that a positive proportion of consecutive prime gaps exceed C log p, where p is the smaller prime. The proportion is bounded below for every sufficiently large initial segment of the prime sequence, with a constant depending on C. It follows that the indices at which p_n/n increases have positive lower asymptotic density, answering a question of Erdős and Prachar.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.