The Kim-Roush Conjecture on the Maximum of per(I-A) in Odd Order
For the set of n by n doubly stochastic matrices, Kim and Roush conjectured in 1981 that for odd n = 2k+1 > 1 the maximum of per(I-A) equals 3 times 2^(k-2), attained by an explicit block construction. Proved in full, and the maximizers are classified: they are exactly the simultaneous-permutation conjugates of that construction.
- Result
- Proved(see note)
- Status
- Resolved
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Matrix theory
- Posed by
- Ki Hang Kim, Fred W. Roush
- Year posed
- 1981
- Years open
- 45y
- Solved
- 2026-08-09
- Model
- GPT-5.6 Sol and Claude Fable 5
- Vendor
- OpenAI, Anthropic
- Collaborators
- Yair Lavi
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 20 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Kim and Roush did not claim uniqueness; the classification of equality cases is new alongside the conjecture itself.
What the AI did
The acknowledgments are one sentence and leave nothing to interpret: "The proof of this conjecture was carried out by GPT-5.6-sol and Claude Fable 5, under the guidance of the author. The author has reviewed the resulting proof arguments. Responsibility for the final text rests with the author."
Verification
A preprint days old, with no independent review.
Source
- PaperarXiv