Spectrum of finite weak central groupoids

Ascertain whether every finite weak central groupoid—i.e., every magma satisfying the law x ▷ (y ▷ x) ▷ (x ▷ (z ▷ y)) (equation E1485)—has cardinality n^2 or 2n^2; either prove this classification of possible sizes or exhibit a finite counterexample.

Background

The authors introduce weak central groupoids (law E1485), a superclass of central groupoids (law E168). Knuth proved that all finite central groupoids have order equal to a perfect square n2.

Empirically, the authors observed that finite weak central groupoids appear to have orders n2 or 2n2, but they explicitly state they have no rigorous proof, making this a concrete structural question about the spectrum of E1485.

References

empirically, we have found that finite weak central groupoids always have order $n2$ or $2n2$, although we have no rigorous proof of this claim; they also have a graph-theoretic interpretation analogous to the interpretation of central groupoids as digraphs with the unique path property.

— The Equational Theories Project: Advancing Collaborative Mathematical Research at Scale (2512.07087 - Bolan et al., 8 Dec 2025) in Section Outcomes (Introduction, Subsection Outcomes)