Counterexample structure (h) — A minimum counterexample with a non-universal congruence relating z and q(z) has at least 35 elements (conditional on small-order facts)

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. Suppose also these finite facts: no E677 magma has order 4 or 6, and every E677 magma of order 7 or 8 has an idempotent. Then $|M|\ge 35$.

The finite facts are hypotheses of the theorem and are not proved on this platform. For a counterexample search: with these facts, a minimum counterexample with fewer than 35 elements has no proper congruence that relates $z$ and $((z\diamond z)\diamond z)\diamond z$. This improves the bound 24 of Counterexample structure (b). Counterexample structure (i) treats the case $|M|=35$, and (j) the case where $M/\theta$ has no idempotent.

The finite facts are known. No E677 magma has order 4 or 6: I. Tencer, ETP Zulip thread "FINITE: 677 -> 255", 2024-12-07 (orders 2, 3, 4 and 6); J. Hyun gives a Lean proof for order 6 (same thread, 2026-09-30, Order6.lean); S. Chuks-Onah (same thread, 2026-05-01, Z3) also reports no E677 magma of order 2, 3, 4 or 6. Order 7: J. Brox gave the two isomorphism classes (same thread, 2025-03-26), and both have the idempotent 0 (checked by the captain); B. Reinke (same thread, 2026-05-15) reports that all models of order 7 are linear; S. Chuks-Onah (same thread, 2026-05-16) found at least 60 distinct E677 tables of order 7 by SAT enumeration, each with exactly one idempotent, and reports (SAT) that no E677 magma of order 7 has $x\diamond x=x$ for every $x$. Order 8: no E677 magma has order 8 (J. Brox, same thread, 2024-12-07, Mace4), so the hypothesis for order 8 holds. Apart from J. Hyun's Lean proof for order 6, these sources give no certificates. The captain also checked the facts with one SAT encoding; drat-trim accepted a DRAT proof of each UNSAT case (unverified C code, not a kernel-checked proof).

Known ingredients. The proof is the proof of Counterexample structure (b) with two more steps. Write $m$ for the size of the $\theta$-classes; all classes have the same size (M. Bonatto and D. Stanovský, Commutator theory for racks and quandles, arXiv:1902.08980v3). (1) If $(e\diamond e)\;\theta\;e$ for some $e$, the class of $e$ is closed under $\diamond$, so it is an E677 magma with $m$ elements; this is the argument of Bonatto and Stanovský for the block of an idempotent. Then $m\ge 5$ and $|M/\theta|\ge 7$. (2) If no class is idempotent in $M/\theta$, the facts for orders 6, 7 and 8 give $|M/\theta|\ge 9$, and $m\ge 4$. The bound 35 was not found in the sources searched: the ETP blueprint, the ETP paper (arXiv:2512.07087v2) and the ETP Zulip thread up to 2026-10-14.