The first orbit q-subdiagonal is right translation two steps later

Let $(\alpha,\diamond)$ be a finite magma satisfying E677: for all $p,q\in\alpha$,

Fix $x\in\alpha$, write $L_x(z)=x\diamond z$, and let $c_n=L_x^{\,n}(x)$ be the points of the left orbit of $x$. For indices $i,j$ define the orbit $q$-term

which by E677 satisfies $c_i\diamond q(i,j)=c_j$. Then for every $k\ge 0$, the first subdiagonal of $q$ is right translation by $x$ two orbit steps later:

This identity turns statements about right translation $z\mapsto z\diamond x$ along the left orbit into statements about the explicit left-inverse terms $q(k+1,k)$. In particular, collisions of right translation on the orbit correspond index by index to collisions on the $q$-subdiagonal.

Prior work. The identity is ETP blueprint Lemma 13.1(iii) with $y=x$ at the point $c_k$, with one left cancellation: Lemma 13.1(iii) gives $c_k=c_{k+1}\diamond(c_{k+2}\diamond x)$, E677 gives $c_k=c_{k+1}\diamond q(k+1,k)$, and $L_{c_{k+1}}$ is injective (Lemma 13.1(i)). The same recurrence, $c_k\diamond d_{k+1}=c_{k-1}$ with $d_k=c_k\diamond x$, is in S. Chuks-Onah's posts of May 2026 (ETP Zulip thread "FINITE: 677 -> 255", 2026-05-06 and 2026-05-09).

Formalization Note The orbit point $c_n$ is written (op x)^[n] x, and $q(k+1,k)$ is written out in full rather than through an auxiliary definition.