In 2016, Hopkins, McConville, and Propp proved that labeled chip-firing on a line always leaves the chips in sorted order provided that the initial number of chips is even. We present a novel proof of this result. We then apply our methods to resolve a number of related conjectures concerning the confluence of labeled chip-firing systems.

中文翻译：

---

标记芯片烧制中的汇合

2016 年，Hopkins、McConville 和 Propp 证明，只要最初的筹码数量是偶数，在生产线上贴上标签的筹码总是会按顺序排列筹码。我们提出了这一结果的新证据。然后，我们应用我们的方法来解决许多有关标记芯片发射系统汇合的相关猜想。