Countable partition ordinals at three summands

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height_three_attack: The three-game step at omega to the omega cubed is constructed online, for strategies that read numerical history, when all outputs are outside or when they are inside with the first tree's top split at its first or last pair of selected leaves. The coherent finite-height dichotomy and Builder implication reduce the positive case to middle strategies, where the construction is open.

height_three_evidence/: Checks explicit selected union trees and their partition-cell endpoints, the first-tree synchronization obstruction, fixed-shape schedules through height three, and a separately qualified finite global-state adversary model. No adaptive-game or ordinal conclusion is certified.

pattern_level_obstruction: A subtree-containment reading of Schipperus's pattern levels makes the height-three pattern contain cliques of every finite size, even with distinct root lengths, but makes it impossible on good pairs. Under a reading by selection shapes, a triangle-free pattern must be inside and middle. On the unrestricted tree space a fixed-prefix embedding rules out every coloring supported only on disjoint leaf sets; a negative construction needs a separated host and unavoidability there.

successor_builder_theorem: Nash-Williams fusion gives the dichotomy and finite-cover refinement in the coherent successor game. A direct construction supplies its normalized Builder implication. This closes the finite-height source transfer, without supplying the middle-strategy triangle construction.

successor_tree_interface: Equal-block normalization for finite-height ordinal trees follows from Erdős-Milner with a factor-two increase in root length. At height three, a fixed strategy defeats one synchronization mechanism but admits another schedule. A companion proof supplies the coherent finite-height source interface; compatibility for every adaptive middle strategy remains open.

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The first unresolved finite parameter in Problem 592 is $\omega^{\omega^3}\to(\omega^{\omega^3},3)^2$. The four-clique lead explains why a positive answer would also settle the stated question of Komjáth. The successor-tree interface reconstructs the equal-block normalization and states the remaining compatibility problem without introducing limit-ordinal machinery into this finite-height case. The normalization is an author-recorded deduction from a published theorem; the triangle construction is open for middle strategies.

The height-three attack is the canonical sketch, and the three-summand lead points to it. It builds the three games online, for strategies that read numerical history, when all outputs are outside or all are inside with the first tree's top split at its first or last pair of selected leaves. The source reconstruction proves the dichotomy, finite-cover refinement and normalized Builder implication for the coherent game at finite height. The positive case is thereby reduced, at author standing, to middle strategies, where only special constructions are known. The page also proves that one mechanism cannot synchronize an explicit fixed strategy, and constructs a different schedule for that same strategy. Its exact finite evidence retains full-tree consistency within the stated bounded ranges.

The pattern-level page gives cliques of every finite size for a containment reading of the negative pattern, which rules out that reading. A second reading by selection shapes turns the printed patterns into fixed pairs of shapes, places every triangle-free pattern in the middle class, and gives an explicit triangle for Theorem 32's pattern. Its fixed-prefix embedding shows that every coloring supported on disjoint leaf sets on unrestricted $W_h$ has a homogeneous set of full type. Negative patterns therefore need a separated host and an unavoidability theorem on every full-type subset of that host. None of these arguments decides the partition question or changes the published relations.