The Middle Stair of Parallel Chip-Firing
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.
- Result
- Proved
- Status
- Resolved
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Combinatorial dynamics
- Posed by
- David Ji, Michael Li, Daniel Wang
- Year posed
- 2024
- Years open
- 2y
- Solved
- 2026-08-04
- Model
- GPT-5.6 Sol
- Vendor
- OpenAI
- Collaborators
- Daniel Wang, Nathan Lannan
- Verification
- Independently expert-verified
- Publication
- Preprint
- Significance
- 12 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
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."
Verification
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.