Counterexamples to the xz-Conjecture and the Mathieu Conjecture for SU(2)

Precise statement

For the integral $\mathcal{I}(h)$ over the unit interval and the torus, the paper gives the three-term Laurent polynomial $f(x,z)=(1-z^{-1})((1-x)+xz)$ with $\mathcal{I}(f^n)=0$ but $\mathcal{I}(z^{-1}f^n)=(-1)^{n-1}/(n+1)\neq0$. This disproves the $xz$-conjecture with one interval and one torus variable, shows $\ker\mathcal{I}$ is not a Mathieu–Zhao subspace, and by padding yields counterexamples for SU(2).

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

ChatGPT 5

Under a heading on AI provenance and author responsibility, the paper states the counterexample and its lift were discovered by ChatGPT 5.

Provider: OpenAI · Prompt public: unknown · Independence: unknown

Verification boundary

unreviewed

No independent review, but the counterexample is a three-term Laurent polynomial with closed-form moment identities, checkable by hand. Preprint, not refereed.

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

Timeline

1. 2026-07-21

   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.