The three-summand case of the countable partition ordinals

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This lead records Plasma AI's route to the last undecided case of Problem 592: for every countable

$$\gamma=\omega^{\xi_1}+\omega^{\xi_2}+\omega^{\xi_3},\qquad \xi_1\ge\xi_2\ge\xi_3,$$

decide whether $\omega^{\omega^\gamma}\to(\omega^{\omega^\gamma},3)^2$. The problem page's current assessment records that one or two summands give the positive relation and four or more the negative one, by Schipperus's paper; for three summands only the failure of the four-clique relation is known. Its parameter $\beta$ is the $\gamma$ used here.

The canonical proof sketch is the height-three attack, for the first instance $\gamma=3$. It keeps the conditional bridge from a three-game construction to the partition relation and the finite-height source interface S at author standing. Its outside and inside sections construct the three plays one node at a time, for strategies that read numerical history, when every output is outside, or inside with the first tree's top split at its first or last pair of selected leaves. With Corollary 2 this reduces the three-game step to middle strategies, the remaining open gap G-mid. The successor-tree interface supplies the equal-block normalization, and the pattern-level page reads the printed negative patterns as fixed pairs of selection shapes.

What would settle the lead

A positive answer through this route at $\gamma=3$ needs G-mid: three correct, sufficiently large, complete plays sharing their trees against every winning middle strategy. The fixed-shape part of G-mid is a finite merging problem for each pair of shapes. The sketch has a structural lemma and four templates for it, each under a condition on the second shape, and one of them is online in a class that the finite-cover refinement can select. Fixed-shape existence alone does not control future numerical prescriptions. A negative answer needs a coloring of all pairs of a specified full-type host, with no color-1 triangle and a color-1 pair in every full-type subset of that host. Under the selection-shape reading, a fixed-shape pattern coloring without triangles would have to be inside and middle; its unavoidability would still need a proof. The fixed-prefix embedding shows that this unavoidability is false on unrestricted $W_3$ for every coloring supported on disjoint leaf sets, including strategy output graphs. It remains unproved on a separated host. The retained census found a schedule for every bounded pair it covered.

Either answer at $\gamma=3$ is only a by-product until it is extended to every three-summand $\gamma$. The positive extension needs a replacement for the triangle construction of Lemma 26 that works at partial-sum nodes and at limit nodes with collapsible ladders. The canonical sketch states the exact common-maxima, ordinal-descent, collapsing-map, history and termination obligations. No such replacement is proved. The printed pattern arguments for the negative relations use only the summand level of a cell, so a negative pattern at $\gamma=3$ would be expected to transfer, but that transfer is not proved here.

Standing

All arguments on the linked pages are author-recorded and unreviewed. L67 is a tier-0 obstruction to one level interpretation and is not used as a premise of the route. The four-clique lead explains what a positive answer would also settle. The next discriminating positive test is an online middle construction that places each double without relying on an unavailable later prescription. For the negative route, the pattern-level page identifies the concrete ten-leaf pair obtained by merging levels $3$ and $2$ of the four-level pattern. Its exclusion proof does not survive that merge; either a shared-tree triangle or a new exclusion proof would decide whether to pursue its separate unavoidability obligation. The lead is blocked on these mathematical inputs, not on source access.