BEACON: a GPU search for large prime gaps in primorial windows, with a new record merit

BEACON is a GPU search for large gaps between consecutive primes. It works in windows centred on n0(k) = k·1439#/2310, integers of about 611 digits, and it has four parts: a persistent GPU sieve to prime bound 10^10, a predictor that ranks windows by how empty they will be before any prime is tested, a ladder of symmetric early-exit tests, and a walk that recovers the asymmetric gaps the ladder throws away. The first campaign found a gap of 59,724 after the prime 17440361647223·1439#/2310−21464, of merit 42.5192. The largest merit known before was 41.9388, set in 2017. Two further gaps, 58,416 at merit 41.5882 and 58,272 at merit 41.4853, are the third and fourth largest ever. Twenty-two gaps from the campaign entered the Prime Gap List on 22 September 2026; about 3,900 smaller first-occurrence improvements (merit 19–42) follow via bulk entry. Every accepted gap has ECPP proofs for both endpoints and an independent proof of compositeness for every integer in between. The ladder's measured pass rates agree with its design to about 1% over 2.8×10^10 windows; one four-GPU workstation sustains about 44,000 windows per second. A heuristic rate law, fitted to measured fence rates and checked out of sample, predicts how many windows a target of given merit and height needs: it prices the previous record at 210 box-days of this configuration for a symmetric search, and the measured gain of walking near-misses brings that to 11–30; the record fell after about 30. The deposit holds the paper, the ECPP certificates (PARI/GP primecert format) for the endpoints of the submitted gaps, the list of 3,913 first-occurrence claims, and the record tuples; the code is at https://github.com/djuk8/beacon.