Optimal move bound for the countable edgeless cube

Determine whether the upper bound of \(\omega^2\) moves for solving accessible configurations of the countable edgeless cube is optimal, including whether a uniform \(\omega\cdot n\) bound exists for some finite \(n\), whether every configuration has a solution of that form, or whether some configuration requires \(\omega^2\) moves.

Background

The paper establishes an ω2\omega^2-move solution for every accessible configuration of the countable edgeless cube, yielding the same upper bound for its analogue of God’s number. The lower bound is only known to be at least ω\omega, so the exact supremum and the existence of configurations requiring ω2\omega^2 moves remain unresolved.

References

Is the result of Corollary~\ref{cor:GN_edgeless} optimal? Clearly the optimal bound is at least $\omega$, but is there a method which works uniformly in $\omega \cdot n$ many moves or less for some $n$ for all accessible configurations? Perhaps every accessible configuration admits a solution in $\omega\cdot n$ for some $n$ but their supremum over all configurations in nonetheless $\omega2$? Or is there an accessible configuration which requires $\omega2$ many moves to solve?

— Solving infinitary Rubik's cubes  (2502.01650 - Tisdell, 30 Jan 2025) in Section 7, Open questions, item 6