Counterexample structure (k) — Normal form: a finite counterexample gives one of minimum order, generated by z, in which every congruence relating two distinct elements relates z and q(z)

Let $(M,\diamond)$ be a finite magma that satisfies E677 and fails E255 at some $x$, that is, $q(x)\ne x$, where $q(x)=((x\diamond x)\diamond x)\diamond x$. A congruence of a magma $(B,\star)$ is a reflexive, symmetric and transitive relation $\theta$ on $B$ such that $a\,\theta\,a'$ and $b\,\theta\,b'$ give $(a\star b)\,\theta\,(a'\star b')$. Then there are a number $n\le|M|$, an operation $\star$ on $B=\{0,\dots,n-1\}$ and a point $z\in B$ with these properties.

1. $(B,\star)$ satisfies E677 and fails E255 at $z$: $q_\star(z)\ne z$, where $q_\star(z)=((z\star z)\star z)\star z$.
2. $B$ is generated by $z$: every subset of $B$ that contains $z$ and is closed under $\star$ is all of $B$.
3. Every congruence of $(B,\star)$ that relates two distinct elements relates $z$ and $q_\star(z)$.
4. Every E677 operation on $\{0,\dots,m-1\}$ with $m<n$ satisfies E255. So $(B,\star)$ is a counterexample of minimum order.

In the terms of universal algebra, the inequality $q_\star(z)\ne z$ and property 3 imply that $(B,\star)$ is subdirectly irreducible and that the congruence generated by the pair $(z,q_\star(z))$ is its monolith, the least congruence other than equality.

This is the standard minimal-counterexample argument of universal algebra (a counterexample of least order is generated by one element and is subdirectly irreducible), written out for E677 and E255. It is not a new result. On the ETP Zulip thread "FINITE: 677 -> 255", T. Tao (2024-12-09) notes that to disprove a one-variable equation such as E255 one can assume that the magma is generated by a single element, M. Bolan (2025-11-21) notes that for E255 only magmas generated by a single element matter, and S. Chuks-Onah (2026-06-10) notes that a minimal counterexample is generated by one element and is subdirectly irreducible.

For a counterexample search: it is enough to search the magmas on $\{0,\dots,n-1\}$ with properties 1–3. Properties 1 and 4 are the hypotheses on the magma and on $z$ of Counterexample structure (b) and (h)–(j). So those results apply to $(B,\star)$, $z$ and each congruence that relates $z$ and $q_\star(z)$ and is not the universal relation (with the other hypotheses of (h)–(j)), and each lower bound that they give for $|B|$ is also a lower bound for $|M|$.

Formalization Note The magma $(M,\diamond)$ is an operation op on a finite type α, and $\star$ is an operation opB on Fin n. Property 2 is stated for predicates: a predicate on Fin n that holds at z and is closed under opB holds everywhere. The order is written Nat.card (Fin n), which is $n$. With this form, properties 1 and 4 are the hypotheses h, hz and hmin of (b) and (h)–(j) for Fin n and opB, without rewriting, and $n\le|M|$ combines with the lower bounds of (b), (h) and (j) by transitivity.