Cramer's Prime Gap Conjecture

Establish that the gap between consecutive prime numbers satisfies p_{n+1} − p_n = O((log p_n)^2) for all n, as asserted by Cramer's conjecture.

Background

In developing tools for rare-case hardness, the paper requires many primes of similar size to balance correctness across fields. The prime number theorem alone is insufficient, and Cramer's conjecture would provide a strong bound on gaps between consecutive primes, ensuring a dense set of primes in short intervals. Since this conjecture remains open, the paper instead relies on the best known unconditional bound on prime gaps to proceed.