Counterexample structure (f) — A congruence class that relates a failing point z to q(z) has at least 4 elements

Let $(M,\diamond)$ be a finite magma that satisfies E677 and let $\theta$ be a congruence of $(M,\diamond)$. Let $z\in M$ satisfy $((z\diamond z)\diamond z)\diamond z\ne z$, so E255 fails at $z$, and $((z\diamond z)\diamond z)\diamond z\;\theta\;z$. Then the $\theta$-class of $z$ has at least 4 elements. No minimality of $M$ and no property of the quotient beyond this one relation is assumed.

For a counterexample search: all classes of $\theta$ have the same size, so a congruence that relates a failing point $z$ to $((z\diamond z)\diamond z)\diamond z$ has classes of at least 4 elements.

Known ingredients. All classes of $\theta$ have the same size: M. Bonatto and D. Stanovský, Commutator theory for racks and quandles, arXiv:1902.08980v3, section "Congruences and homomorphisms" (a congruence of a left quasigroup with a connected quotient is uniform). Classes of size 2 give an element $y$ with $y\diamond z=z$ (a short derivation from ETP blueprint Lemma 13.4). Classes of size 3 give such an element by the local form of the argument of Counterexample structure (a). Such an element gives E255 at $z$ (ETP blueprint Lemma 13.2). The bound 4 without minimality was not found in the sources searched: the ETP blueprint, the ETP paper (arXiv:2512.07087v2), the ETP Zulip thread up to 2026-10-14, and papers on left quasigroups and racks.