D4.4c: q-square refinement

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 and carry the swapped-S one-step packet with $p=q\diamond(q\diamond x)$ and $u=q\diamond p$. Put $v=q\diamond q$.

Then $v$ is outside the six named points, $v\ne p$, $v\ne u$, and $p=u\diamond v$. Moreover, the full five-way one-step classification admits the following refinement, preserving each original packet and branch payload. If both $p,u$ lie on $C_t$, either $v$ occupies a third distinct phase there with the oriented phase-product identity, or $v$ starts a disjoint cycle. If $p$ starts a new cycle and $u$ lies on $C_t$ or $C_p$, then $v$ lies on $C_t$, lies on $C_p$, or starts a cycle disjoint from both. Every contact keeps the identity $p=u\diamond v$ in its corresponding ordered phase form; phases on $C_p$ are strictly positive, and same-cycle phases of $u,v$ are distinct. The two branches in which $u$ escapes retain their original full payloads.

The conclusion supplies both the q-square seed facts and this complete refined outcome. It does not establish recursive renewal or a fixer at $x$.