OpenAI · 026-01 · 2026-09-25 · Number theory
Positive lower density of large prime gaps
preprints/Positive-lower-density-of-large-prime-gaps-September-25-2026
Family 026. Positive lower density of large prime gaps.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
For every fixed , a positive proportion of consecutive prime gaps exceed , throughout every sufficiently large initial segment of the primes. The proportion may depend on . Consequently, the indices where increases have positive lower density, answering Erdős and Prachar.
For every fixed C > 0, we prove that a positive proportion of consecutive prime gaps exceed , 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 increases have positive lower asymptotic density, answering a question of Erdős and Prachar.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 026: Positive lower density of large prime gaps. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.