D4.4b: one-step q-seed pump

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, with an exact first-return cycle at $t$ and the swapped-S cycle-renewal packet. Assume $q=c_2\diamond x$ and $s=c_3\diamond q$. Put $p=q\diamond(q\diamond x)$ and $u=q\diamond p$.

There is a one-step packet retaining the renewal input and canonical q/s equations. It records freshness of both $p,u$ outside the six named points, $u\ne p$, $p\diamond q=q$, $q\diamond u=q$, and $u\diamond q\ne q$, together with all five cycle-location alternatives:

1. Both $p,u$ lie on $C_t$, at distinct phases.
2. $p$ lies on $C_t$ and $u$ starts a disjoint cycle.
3. $p$ starts a new cycle and $u$ lies on $C_t$.
4. $p$ starts a new cycle and $u$ lies at a strictly positive phase of $C_p$.
5. $p$ starts a new cycle and $u$ starts a cycle disjoint from both $C_t,C_p$.

Contact phases carry their exact equations and bounds; escaping seeds carry exact first-return data. This asserts one step, without an iteration or termination conclusion.