AI 中文总结
该研究使用GPT-5.6-Sol证明了2024年Ji等人关于并行芯片点火游戏活动为$\tfrac12$时存在通用界的猜想，推广了魔鬼阶梯的中间层并统一了多类图的相关结果。
AI 中文摘要
2010年，Levine发现完全图$K_n$上并行芯片点火游戏的活动函数收敛于魔鬼阶梯模式：添加芯片会使活动经过局部恒定的开区间。2022年，Bu、Choi和Xu改进了Kominers与Kominers的早期界，证明存在严格下界，低于该下界的所有游戏活动均为0；还存在严格上界，高于该上界的所有游戏活动均为1，从而将魔鬼阶梯的最底层和最顶层推广到所有图。2024年，Ji、Li和Wang猜想，对于活动为$\tfrac12$的游戏，存在类似的通用界。我们使用GPT-5.6-Sol证明了该猜想，统一了树、环、完全图和完全二分图的现有结果，推广了魔鬼阶梯的中间层。
英文摘要
In 2010, Levine found that the activity functions of parallel chip-firing games on the complete graph $K_n$ converge to a devil's staircase pattern: adding chips causes activity to progress through open intervals in which it is locally constant. In 2022, Bu, Choi, and Xu improved on an earlier bound by Kominers and Kominers to show that there exists a strict lower bound below which all games have activity $0$, and a strict upper bound above which all games have activity $1$. They thereby generalized the bottom and topmost rungs of the devil's staircase to all graphs. In 2024, Ji, Li, and Wang conjectured that a similarly general bound exists for games with activity $\tfrac12$. We use GPT-5.6-Sol to prove this conjecture, unifying existing results for trees, cycles, complete graphs, and complete bipartite graphs. This generalizes the middle rung of the devil's staircase.
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