A counterexample to the Howland-Kato problem for positive commutators
The Howland-Kato conjecture that every nonzero positive commutator must arise from functions in appropriate Kato classes is false: is nonzero and nonnegative.
- Result
- Disproved
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Construction
- Field
- Operator theory
- Posed by
- James Howland; Tosio Kato
- Year posed
- 1991
- Years open
- 35y
- Solved
- 2026-08-07
- Model
- Grok 4.5
- Vendor
- xAI
- Collaborators
- Rupert L. Frank, Paata Ivanisvili
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- —
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
The paper discloses only "The authors acknowledge the use of AI tools. All mathematical arguments and proofs in the final manuscript were checked and written by the authors." Co-author Paata Ivanisvili (@PI010101, Professor of Mathematics at UC Irvine) has since said publicly that "AI deserves a fair amount of credit for finding" the key identity, and, asked which model: "Grok 4.5 in Cursor with an agent found a non-symmetric counterexample f(x) = arctan(x/2) and g(x) = tanh(x)/2 + tanh(3x)/2 which works and is correct. However, in the final manuscript we implemented symmetric example." So the model found a valid counterexample, but not the symmetric one the paper is built around, and the positivity proof is the authors' own.
Sources
- PaperarXiv
- AnnouncementAuthor's announcement (X)
- DiscussionAuthor on Grok 4.5's role (X)
Submitted by VibeGene on