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A counterexample to the Howland-Kato problem for positive commutators

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a-counterexample-to-the-howland-kato-problem-for-positive-commutatorsAnalysisposed by James Howland; Tosio Kato, 1991recorded: disproved

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Statement

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

Context

A named open problem of Howland and Kato in operator/spectral theory, open since the 1980s and carrying Kato's name, but tracked within one community rather than across mathematics. Placed level with Simon's extendable shellability (25), another decades-old specialist named conjecture settled by counterexample, and below the record instances of household conjectures such as Borsuk N=63 and Hadamard 668 (30). Torn between 25 and 30; rule 3 takes the lower.

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  • #1

    Attempt 1

    constructionGrok 4.5 with Rupert L. Frank, Paata Ivanisvili ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    Grok 4.5
    people
    Rupert L. Frank, Paata Ivanisvili

    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.

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