Containment levels fail the height-three pattern coloring

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_proof: Explicit nested leaf bands give every finite clique, while the last-child condition forces containment level one for an interior partition cell.

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This result is recorded by Plasma AI. It excludes a proposed interpretation of the cell levels in the negative-pattern route to Problem 592. It does not refute a published partition relation or identify the intended source definition. The proof gives an explicit family and checks every cell. The research account records the interpretation gap and its consequences for further pattern searches.

Conventions

Let $W_3$ consist of finite rooted ordered trees whose branches all have four nodes, with ordinal labels $3,2,1,0$ from root to leaf. Every nonterminal node has at least one child. The leaves carry distinct natural numbers increasing from left to right. Order the trees first by the number of root children, and then lexicographically by their ordered lists of subtrees, with the same rule recursively and with leaf labels ordered naturally. These are Schipperus's Definitions 6--8 at height three.

For a convex partition $D$ of a tree's leaf set, define $R_1(E)=3$ for its first and last cells. For any other cell $E$, let $R_1(E)$ be the ordinal label of the lowest ancestor of the leaf $\max E$ whose whole leaf set is not contained in $E$. This ancestor exists, since $E$ is a proper subset of the tree's leaves. It has label $1$, $2$ or $3$.

For trees $A<B$ with disjoint leaf sets, take as cells the maximal runs of each tree's leaves in the ordered union of the two leaf sets. Color the pair $1$ if its cells, labeled by tree and $R_1$ level, read

$$A_3,B_3,A_1,B_3,A_2,B_3,A_1,B_3,A_3.$$

Color every other pair $0$. The word is the one in Schipperus's Theorem 32; $R_1$ is the proposed replacement for its unresolved level notation, not an assertion about its intended meaning.

For a convex partition $D$, let $X(T,D)$ be the leaves at the maxima of all cells except the last. Let $G(T,D)$ consist of those leaves and all their ancestors, as in Definition 9. Definition 11(3) requires that the last child of every successor node lie outside $G(T,D)$. Every nonterminal node here is a successor node, so good partitions satisfy the hypothesis of the second part of the claim. No other good-partition condition is used in that implication.

The source definitions and displayed pattern are from Schipperus, Countable partition ordinals, Definitions 6--11 and 29, and Theorem 32. The proof uses these definitions, not the correctness of a source theorem.

Standing and verification

Tier 0, author-recorded. The two conclusions have complete elementary proofs, with no remaining obligation inside their stated scope. Reading the explicit construction and the good-partition argument is the verification procedure; no computational evidence or formalization is claimed. The original three-summand partition question remains open.