For a hierarchical lattice whose rule is a Boolean function f on k independent children, the RG map is R(p) = Σ_j c_j p^j (1 − p)^{k−j} with c_j the number of accepted inputs of weight j, 0 ≤ c_j ≤ C(k, j). At which k does the not-all-equal rule R_k(p) = 1 − p^k − (1 − p)^k first have a period-3 orbit in (0, 1), and do narrower band rules get there earlier?

Statement

A rule f: {0,1}^k → {0,1} on independent children open with probability p gives a super-bond open with probability R(p) = Σ_x f(x) p^{|x|} (1 − p)^{k−|x|} = Σ_j c_j p^j (1 − p)^{k−j}, the reliability polynomial in Bernstein form; the space of rules is the box 0 ≤ c_j ≤ C(k, j). Monotone f (every series/parallel composition, every connectivity rule) gives an increasing R, and an increasing map of an interval has no periodic points but fixed points, so every rung of the ladder so far is outside Li–Yorke's reach. The not-all-equal rule, open iff the children disagree, is f = 1 except on the two constant inputs: R_k = 1 − p^k − (1 − p)^k, a symmetric hump of height 1 − 2^{1−k}. The chaotic logistic map is not in the box for any k: elevating 4p(1 − p) to degree k gives c_j = 4·C(k−2, j−1), and at the middle index the ratio to C(k, j) is k/(k − 1) (k even) or (k + 1)/k (k odd), above 1 — so the exact Li–Yorke host is unreachable, and the question is which reachable hump first carries period three.

Sealed 2026-10-22

As written. The lab chat reads this file above the heading "Not part of the seal" only; it reads run 111's file for the method above its own lead heading only.

Scored 2026-10-22 — falsified

Scored by a blind reader given the seal above the "Not part of the seal" heading and R27_tree_groups/carpet_perc/RESULTS114.md. Withdrawal not triggered (control (i): the diamonds reproduce run 111's λ to 0.0 at 40 digits and have no period-3 orbit; control (ii): two logistic orbits at r = 3.9, none at 3.5). Falsifier (A) triggered: the first k is 8. (B) and (C) held; under the seal's "confirmed only if all three hold", the run is falsified.

Faults in the seal's wording (the reader's eight, kept): the box count at k = 6 was wrong by a factor of 16 (the codifier multiplied ∏(C(6,j)+1) wrongly); the detector as written needs a fixed-point exclusion at the numpy stage that the seal did not state (without it nearly every rule is a grid hit at its own fixed point); "every hit confirmed in mpmath" certifies nothing when there are no hits, so the zero at k = 5, 6 rests on the numpy scan (checked against an all-mpmath scan at k ≤ 4 and against the mpmath hit sets at k = 7, 8); multiplicity vs existence unsaid; (B)'s reference ("NAE's smallest k") not fixed as predicted or observed (it is 7 < 8 either way, but 7 > 6); the withdrawal rule names only control (i); the unanticipated clause covers NAE only. Blindness: the lab read run 111's scored section and reading (above 111's lead heading, which the prompt allowed) and saw the guess "k = 6" in a commit subject; the guess is the seal's own prediction, so nothing beyond the seal reached the lab, and the next prompt names the cut for every file and says not to read the log.

Reading (the codifier's, after the score): the guess was built on hump height against the logistic family's period-3 threshold (0.957 at r ≈ 3.83): NAE at k = 6 has height 0.969 and I expected that to suffice. It does not; k = 7 (0.984) does not either; k = 8 (0.992) does. So height is not the variable — these humps are flatter at the top than a parabola and period three needs the stretch an orbit can get on the way down, which the band rules [1,3] … [2,5] at k = 7 supply with a lower, steeper hump. That is the lead's own hedge ("if (B) fails, height is what matters; it held, so steepness is"), and it is the right lesson: the one-parameter picture (height) fails on these rules the way the one-parameter sign rule failed on the diamond in run 110. Two things carried forward. First, the symmetry: every hit set is invariant under c_j → c_{k−j} and under the dual rule c_j → C(k,j) − c_{k−j}, which the lab reported beside as a check — the same planar duality that explained run 111, now on the space of rules. Second, the exact number the ladder did not have: the first non-monotone hierarchical rule with chaos has seven children, and it is a band, not the symmetric hump. The pre-registered P = 0.4 on k = 6 was honest and wrong; the alternatives I named (5 or 7) were wrong too, in the same direction, which is the kind of miss the record should keep.

The codifier’s lead is kept for the private site.