Counterexample structure (b) — A minimum counterexample with a non-universal congruence relating z and q(z) has at least 24 elements

Let $(M,\diamond)$ be a finite magma that satisfies E677 and fails E255 at $z$, that is, $((z\diamond z)\diamond z)\diamond z\ne z$. Suppose that every E677 operation on $\{0,\dots,n-1\}$ with $n<|M|$ satisfies E255, so $M$ is a counterexample of minimum order. Let $\theta$ be a congruence of $(M,\diamond)$ that relates $z$ and $((z\diamond z)\diamond z)\diamond z$ and is not the universal relation. Then $|M|\ge 24$.

For a counterexample search: a minimum counterexample with fewer than 24 elements has no proper congruence that relates $z$ and $((z\diamond z)\diamond z)\diamond z$. A checked SAT result outside this proof says that no E677 magma of order 6 exists; with it the same argument gives $|Q|\ge 7$ and $|M|\ge 28$.

Known ingredients. The proof uses these known facts. (1) 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). (2) A minimum counterexample is generated by one element: T. Tao, ETP Zulip thread "FINITE: 677 -> 255", 2024-12-09. (3) In an E677 magma, an element $t$ with $t\diamond t\ne t$ that satisfies E255 has order at least 6, where the order of $t$ is the least $n\ge 1$ with $L_t^n(t)=t$: J. Hyun, same thread, 2026-09-30 (Order6.lean). (4) Classes of size 2 give an extension of the affine form in ETP blueprint Lemma 13.4, so E255 lifts from the quotient (a short derivation from that lemma). The fact that no E677 magma of order 6 exists, which the paragraph above uses for the bound 28, was reported by I. Tencer (same thread, 2024-12-07), and the message of J. Hyun of 2026-09-30 gives a Lean proof of it (Order6.lean). The step for classes of size 3 (Counterexample structure (a)) and the bound 24 were 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.