Causal rank-raising map on well-order codes

There is a coordinatewise causal map $S:{\omega }^{\omega }\to {\omega }^{\omega }$ that recodes a relation after adjoining a new least element. If $x\in \mathrm{WF}$, then $S(x)\in \mathrm{WF}$ and $\Vert S(x)\Vert =\Vert x\Vert +1$; the tagged coding also ensures $S(x)\ne x$ when $x\notin \mathrm{WF}$. Because the first $n+1$ output coordinates depend only on the first $n+1$ input coordinates, Player II can produce $S(x)$ online.