The Middle Stair of Parallel Chip-Firing
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Statement
Ji, Li and Wang conjectured in 2024 that every parallel chip-firing game on a finite connected graph whose chip count lies strictly between and has period exactly 2, generalizing the middle rung of Levine's devil's staircase from complete graphs to all graphs. Known before only for trees, cycles, complete and complete bipartite graphs.
Context
A documented 2024 conjecture unifying thirty years of partial results on the devil's staircase, known within the chip-firing community.
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Attempts
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The paper presents the proof plainly as found by the model: "The proof presented in Section 3 was found by the large language model GPT-5.6-Sol. The authors verified the resulting argument and take full responsibility."
Reviews
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Machine check · not human verification
machine: correctscope Expert review reported upstream; the reviewer has no probXiv account and is not credited here
Beyond the authors' own verification, the acknowledgments thank David Ji and Michael Li, two of the conjecture's posers, for assisting with reviewing the proof.
No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.
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