bibkey: gray2026cycle9 authors: Alison Gray; Pransu Patel; Isaiah Young year: 2026 title: The cycle C9 does not admit uniform mixing doi: null url: https://arxiv.org/abs/2607.28345v1 claim: The authors identify C21 as the next open cycle case and leave its exclusion open. strata_touched:

• D5/S3/Quantum/Dynamics/CycleUniformMixing/CycleTwentyOneNoUniformMixing license: citation-only triage: anchor

The cycle C9 does not admit uniform mixing

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DOI: null

Source: https://arxiv.org/abs/2607.28345v1

Conclusion; The_cycle_C9_does_not_admit_uniform_mixing.tex line 336.

Source statement

Given that uniform mixing results have already been proven for even cycles and odd prime cycles, we identify $C_{21}$ as the next open case. Our techniques of constructing and utilizing Gröbner bases of cyclic $n$-roots prove to be too computationally heavy for these larger values of $n$. Consequently, we do not employ these techniques for the $C_{21}$ case and recognize that other methods are needed for the proof.

The C21 exclusion is partial progress on the Ahmadi–Belk–Tamon–Wendler conjecture. The conjecture for all cycles of length at least five remains outside this result.