Counterexample structure (j) — If the quotient by a non-universal congruence relating z and q(z) has no idempotent, a minimum counterexample has at least 40 elements (conditional on small-order facts)

Let $(M,\diamond)$, $z$ and $\theta$ be as in Counterexample structure (h): $M$ is a finite E677 magma of minimum order that fails E255 at $z$, and $\theta$ is a congruence of $(M,\diamond)$ that relates $z$ and $((z\diamond z)\diamond z)\diamond z$ and is not the universal relation. Suppose that no $e\in M$ satisfies $(e\diamond e)\;\theta\;e$, that is, $M/\theta$ has no idempotent. Suppose also these finite facts: no E677 magma has order 6, and every E677 magma of order 7, 8 or 9 has an idempotent. Then $|M|\ge 40$.

The finite facts are hypotheses of the theorem and are not proved on this platform. For a counterexample search: The proof of Counterexample structure (h) excludes classes of 4 elements (with the order-4 fact) only when $M/\theta$ has an idempotent. So, with the facts of (h) and of this theorem, classes of 4 elements need $|M|\ge 40$; this combines this theorem with that step of the proof of (h) and is not a separate formal statement.

The finite facts for orders 6, 7 and 8 are described in Counterexample structure (h). Order 9: the only E677 magma of order 9 is the linear one, which has an idempotent (J. Brox, ETP Zulip thread "FINITE: 677 -> 255", 2025-01-10; no certificate). The captain's SAT check, described in (h), also covers order 9. Proof: by the argument of Counterexample structure (b), $|M|=|M/\theta|\cdot m$ with $|M/\theta|\ge 6$ and $m\ge 4$, where $m$ is the size of the $\theta$-classes, and the finite facts give $|M/\theta|\ge 10$. This bound 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.