D4.3e — Swapped-S cross-cycle closeout

CONJECTURED — open branch closeout. Let a finite set carry an arbitrary binary operation $\diamond$ satisfying E677:

Write $L_x(z)=x\diamond z$. The distinguished packet records four pairwise distinct points with

Set $q=c_2\diamond x$, $r=c_1\diamond c_2$, and $s=c_3\diamond q$. The packet also records that $q$ is outside the four displayed points and

A first-return packet for a seed $v$ through depth $n$ supplies its positive minimal return period $d>n$. The points $v,L_x(v),\ldots,L_x^{d-1}(v)$ are pairwise distinct and outside the four displayed points, and $L_x^d(v)=v$. The initialized prefix and all tagged branch equations are retained as hypotheses.

Assume the S branch, with $r=c_1$, $s$ outside $\{x,c_1,c_2,c_3,q\}$, and $L_x(q)=s$, $L_x(s)=q$. Set $t=c_2\diamond s$. Assume $t$ is outside $\{x,c_1,c_2,c_3,q,s\}$, retain the tagged S start and its first-return packet through depth $0$, and retain the source-derived collision and star equations

The target is a fixer for the original base point:

This is one retained branch of the period-four orbit-collision problem. The cycle data and local contact facts are established inputs; the displayed fixer conclusion is still open. No bound on the size of the finite carrier is imposed.