All parallel chip-firing games with $2|E|-|V|<|σ|<2|E|$ have period $2$

Abstract

In 2010, Levine found that the activity functions of parallel chip-firing games on the complete graph $K_n$ converge to a devil&#39;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&#39;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&#39;s staircase.