What was already known
Greaves and Zhu proved the minimum spectral-gap value. Jain and Mizgerd supplied their own proof of the equality characterization; Fill’s separate multiplicity formula remained open.
AI involvement
After receiving Jain and Mizgerd’s human proof of the equality characterization, ChatGPT 5.4 Pro produced the multiplicity proof in one shot. They verified and rewrote that part.
ChatGPT 5.4 Pro supplied the complete proof of the multiplicity formula after the human proof of the equality characterization was provided as input.
Importance
Settles Fill’s multiplicity conjecture in the equality cases.
Limitations
The human proof of the equality characterization is a required input and is credited separately. Jain and Mizgerd report checking the model-produced multiplicity proof; no machine formalization or independent audit is documented.
Technical verification
Jain and Mizgerd expressly state they verified the one-shot model proof of Theorem 1.8(2) for correctness before rewriting it.
Formal checking
human correctness verification of the model-produced proof; source-reported.
- Checker
- Vishesh Jain and Clayton Mizgerd
- Scope
- Theorem 1.8(2), Fill’s multiplicity formula for the second-largest eigenvalue
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The human authors rewrote the proof for clarity and Section 6 explains the upper and lower multiplicity bounds and their dependence on neutral labels.
Correction and publication history
Settles Fill’s multiplicity conjecture in the equality cases.
Read this published revision