D4.4a: swapped-S cycle renewal

Write $L_x(y)=x\diamond y$ and let $C_z=\{L_x^i(z):0\le i<d_z\}$ for an exact first-return cycle of length $d_z$. Let a finite magma satisfy E677. Assume the published distinguished period-four packet and S-tagged start at $x,c_1,c_2,c_3$. Put $q=c_2\diamond x$, $s=c_3\diamond q$ and $t=c_2\diamond s$. Suppose $L_x(q)=s$, $L_x(s)=q$, $t$ is outside $\{x,c_1,c_2,c_3,q,s\}$, the trace at $t$ is initialized at depth zero with exact first-return data, and $t\diamond c_2=c_3$.

Then these inputs extend to a cycle-renewal packet. In particular, $(t\diamond c_3)\diamond t=c_1$, $q\diamond c_2=s\diamond c_2$, and $p=q\diamond(q\diamond x)$ is fresh outside the same six points and satisfies $p\diamond q=q$. The packet retains the full three-way classification of $t\diamond c_3$: the dual-collision case, contact at a strictly positive phase of $C_t$, or a fresh cycle disjoint from $C_t$, with the associated equations and exact first-return data.