A counterexample to the Howland-Kato problem for positive commutators

Precise statement

The Howland-Kato conjecture that every nonzero positive commutator $i[f(P),g(Q)]$ must arise from functions in appropriate Kato classes is false: $i[\arctan(P),\arctan(Q)]$ is nonzero and nonnegative.

The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.

What AI did

Grok 4.5

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.

Provider: xAI · Prompt public: unknown · Independence: unknown

Verification boundary

unreviewed

No verification note supplied.

Correctness: unknown · statement fidelity: unaudited · peer review: none

Timeline

1. 2026-08-07

   arXiv

   VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.

Known method families

construction (source-reported)

Source-reported tools: construction.

Independent: unknown · difference confidence: 0

What remains uncertain

VibeMath has not independently audited the mathematical statement, proof, or novelty claim.

• VibeMath has not independently verified the mathematical claim.
• AI-attempt independence and training-data exposure are unknown unless explicitly documented.
• VibeMath has not independently audited the mathematical statement, proof, or novelty claim.