Polymath8b proved that H1=lim inf(pn+1−pn)≤246. In this paper we show how the Bombieri-Vinogradov theorem can be combined with newer equidistribution estimates for smooth moduli to obtain the improved bound H1≤240.

arXiv:2608.31126 [math.NT]: https://arxiv.org/abs/2608.31126

From Thomas Bloom on 𝕏: https://x.com/thomasfbloom/status/2094748658629513665
"As Julia notes, this number shouldn't be taken too seriously, and can surely be reduced a little further with more effort. The significance is the introduction of new ideas which, for the first time in over a decade, get past the 246 barrier."

The paper: Supercritical sharpness of percolation

Sahar Diskin, Philip Easo, Ritvik Ramanan Radhakrishnan, Benny Sudakov, Vincent Tassion

arXiv:2603.03257 [math.PR]: https://arxiv.org/abs/2603.03257