Equality in Fill's Spectral Gap Problem
For the adjacent-transposition chain on with a regular parameter vector, Fill's spectral gap conjecture (recently resolved) leaves open the characterization of the equality cases. The paper settles them, constructing the additional eigenfunctions in the exceptional regime.
- Result
- Proved
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Argument
- Field
- Markov chains
- Posed by
- James Allen Fill
- Year posed
- —
- Years open
- —
- Solved
- 2026-04-05
- Model
- ChatGPT 5.4 Pro
- Vendor
- OpenAI
- Collaborators
- Vishesh Jain, Clayton Mizgerd
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 12 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
"Upon being given the complete proof of Theorem 1.8(1), the proof of Theorem 1.8(2) was produced by ChatGPT 5.4 Pro in a one-shot manner and subsequently verified for correctness by the authors" - the included proof is the model's, rewritten for clarity.