What was already known
Ran and Teng proposed the region in 2024 with numerical evidence and one boundary curve. The new paper’s human authors set the stronger characterization target.
AI involvement
ChatGPT 5.1/5.2 proposed the trigonometric reparameterization and extremal proof strategy. Verbeken, Vagenende and Guerry set the target and repaired correctness-critical steps.
The model supplied a retained proof architecture, while humans selected the target and closed essential gaps.
Importance
The result resolves a published spectral conjecture and supplies necessity and realization for this matrix family.
Limitations
The early GPT-5.1 chats were not preserved; later GPT-5.2 reruns reconstruct them. The standalone final proof was described as forthcoming, with proof drafts appended to the case study.
Technical verification
Three named authors report reviewing and validating the complete final conventional proof; no proof-assistant formalization was attempted.
Formal checking
Human review and validation of the final proof, including branch, sign, endpoint and algebra checks
- Checker
- Brecht Verbeken, Brando Vagenende and Marie-Anne Guerry
- Scope
- Final 4-cycle spectral-region characterization and Ran–Teng implication
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The human authors explain the trigonometric and extremal strategy and the repairs to its decisive lemmas.
Correction and publication history
The result resolves a published spectral conjecture and supplies necessity and realization for this matrix family.
Read this published revision