Exact non-negative rank and refined rectangle covering for regular polygons beginning at n=25

Determine the exact refined rectangle covering number and non-negative rank of the slack matrices of regular n-gons for the smallest unresolved case n=25 and subsequent unresolved values.

Background

For regular n-gons with 17≤n≤20, the paper establishes the non-negative rank exactly by matching the refined rectangle covering bound (RRCB) with an upper bound. For n≥25, the authors improve the RRCB lower bound to 9, but the exact non-negative rank and, in the first unresolved case, the exact RRCB remain undetermined.

References

The smallest open value for which neither the RRCB nor the $r_+$ is known exactly is $n=25$.

— Computing Lower Bounds on the Non-negative Rank via SAT Solvers  (2610.11950 - Engel et al., 8 Oct 2026) in Section 6, Conclusion

It may even be possible to show RRCB = xc for regular $n$-gons: \begin{conjecture} For slack matrices $S_n$ of regular $n$-gons \mathrm{r}_+(S_n)=\mathrm{RRCB}(S_n) holds. \end{conjecture}

— Computing Lower Bounds on the Non-negative Rank via SAT Solvers  (2610.11950 - Engel et al., 8 Oct 2026) in Section 6, Conclusion, paragraph “Future Work”; Conjecture environment