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 polynomial regime, i.e. complexities of the form $Θ(n^{1/k})$ for $k \in \mathbb{N}$, and show that whether polynomial gap results persist in the unbounded-degree setting crucially 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 problem definitions to distinguish infinitely many local cases. Inspired by this, we introduce Locally Finite Labelings (LFLs), which formalize the intuition that 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：局部可检查标号（LCLs）的研究已对有界度树上可能出现的分布式时间复杂性给出了极为精确的表征。这一复杂性图景的核心特征在于间隙结果的存在，这类结果排除了大范围的中间复杂性。尽管早期期望这些间隙能推广至更一般的图类，但事实表明并非如此。本文探索另一方向：我们仍局限于树类，但允许任意大的度数。重点关注多项式量级，即形如 $Θ(n^{1/k})$（其中 $k \in \mathbb{N}$）的复杂性，并证明多项式间隙结果能否在无界度情形下持续存在，关键取决于LCLs如何超越有界度进行推广。已有复杂构造表明，无界度树上的LCLs同样会消除多项式间隙。我们并未止步于此负面结果，而是提出更简单的问题集合，其本身即可否定任何多项式间隙的存在。从这一更简洁构造中获得的洞见是：间隙结果的存在要求问题定义不能区分无限多种局部情形。受此启发，我们引入局部有限标号（LFLs），形式化“每个节点必须属于有限多个局部情形之一”这一直觉。主要结果表明，该限制足以恢复多项式间隙：对于无界度树上的任意LFL $Π$，其确定性LOCAL复杂性要么是 $Θ(n^{1/k})$（其中 $k \geq 1$ 为整数），要么是 $O(\log n)$。此外，适用情形及相应的 $k$ 值可完全由 $Π$ 的描述确定。