The study of Locally Checkable Labelings (LCLs) has led to a remarkably precise characterization of the distributed time complexities that can occur on bounded-degree trees. A central feature of this complexity landscape is the existence of gap results, which rule out large ranges of intermediate complexities. While it was initially hoped that these gaps might extend to more general graph classes, this has turned out not to be the case. In this work, we investigate a different direction: we remain in the class of trees, but allow arbitrarily large degrees. We focus on the \emph{polynomial regime} ($Θ(n^{1/k} \mid k \in \mathbb{N})$) and show that whether polynomial gap results persist in the unbounded-degree setting depends on how LCLs are generalized beyond bounded degrees. There already exists a complex construction that shows that the polynomial gaps also vanish for LCLs on unbounded degree trees. Rather than stopping at this negative result, we give a much simpler set of problems that already contradicts the existence of any polynomial gaps. The insight obtained from this cleaner construction is that for gap results to exist, we cannot allow problems to distinguish infinitely many cases. This guides us to a natural class of problems for which polynomial gap results can still be recovered. We introduce \emph{Locally Finite Labelings} (LFLs), which formalize the intuition that \emph{every node must fall into one of finitely many local cases}. Our main result shows that this restriction is sufficient to restore the polynomial gaps: for any LFL $Π$ on trees with unbounded degrees, the deterministic LOCAL complexity of $Π$ is either - $Θ(n^{1/k})$ for some integer $k \geq 1$, or - $O(\log n)$. Moreover, which case applies, and the corresponding value of $k$, can be determined solely from the description of $Π$.

翻译：局部可检验标记（LCL）的研究已对有界度树上的分布式时间复杂性给出了极为精确的特征刻画。这一复杂性图景的核心特征是间隙结果的存在，它排除了大范围的中间复杂性。虽然最初希望这些间隙能推广到更一般的图类，但事实并非如此。本文研究了一个不同的方向：我们仍停留在树类结构中，但允许任意大的节点度。我们聚焦于多项式区域（$Θ(n^{1/k} \mid k \in \mathbb{N})$），并证明多项式间隙结果在无界度设置中是否持续存在，取决于LCL问题如何推广到有界度情形之外。已有复杂构造表明，对于无界度树上的LCL问题，多项式间隙同样会消失。我们并未止步于这一否定性结论，而是提出了一组更简单的问题集合，它已能反驳任何多项式间隙的存在性。从这一更简洁构造中获得的启示是：要存在间隙结果，我们不能允许问题区分无限多种情形。这引导我们找到一类自然的问题类别，其多项式间隙结果仍可恢复。我们引入局部有限标记（LFL）概念，它形式化地表达了“每个节点必须属于有限种局部情形之一”的直观思想。我们的主要结果表明，这一限制足以恢复多项式间隙：对于无界度树上的任意LFL问题$Π$，其确定性LOCAL复杂度要么是某个整数$k \geq 1$对应的$Θ(n^{1/k})$，要么是$O(\log n)$。此外，适用哪种情形以及对应的$k$值，均可仅从$Π$的形式描述中确定。