The mixed xz moment conjecture
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For the mixed xz moment question, ask whether vanishing of every pure product-integral moment forces the marked moments to vanish. A witness target is f(x,z)=(1−z⁻¹)((1−x)+xz), with all positive pure moments zero but marked moment (−1)^(n−1)/(n+1); a related SU(2) Haar-moment lift addresses the Mathieu conjecture.
Why it matters and what counts as progress
Why it matters: It tests a proposed implication between unmarked and marked moments, including a compact-group instance.
What counts as progress: Explicit functions with every required pure moment zero but a nonzero marked moment, including verification after lifting to SU(2).
How the problem developed
- 1997 · Conjecture [1]
Olivier Mathieu Formulated the compact connected Lie-group moment conjecture: vanishing pure moments should force eventual vanishing after multiplication by any finite function.
- 21 Jul 2026 · Counterexample [2]
Christopher D. Long Gave a Laurent-polynomial witness disproving the xz-conjecture for one interval and one torus variable, with padding to every mixed case; a related lift also disproves the Mathieu conjecture for SU(2).
AI contribution
Counterexamples to the xz-Conjecture and the Mathieu Conjecture for SU(2)
The Laurent polynomial f(x,z)=(1−z⁻¹)((1−x)+xz) has zero pure product-integral moments for every positive n but marked moment (−1)^(n−1)/(n+1). It refutes xz(1,1), with padding for all mixed k,l≥1. Its SU(2) lift F=(1+c)(ad+b), G=−c has the same pure and marked Haar moments and refutes the SU(2) Mathieu conjecture.
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AI-found polynomials refute the mixed xz and SU(2) Mathieu conjectures