The recursive initial-letter sortable projection

Let $(W,S)$ be a Coxeter system of finite type and $c=s_1\cdots s_n$ a reduced Coxeter word. Set ${\pi }_c(1)=1$, also when $S=\mathrm{\varnothing }$. For $w\mathrm{\ne }1$, the rank is positive; choose an initial letter $s:=s_1$ and write $⟨s⟩:=S\setminus \{s\}$; recall that $sc:=s_2\cdots s_n$ is a reduced Coxeter word for the Coxeter element $sc$ of the parabolic $W_{⟨s⟩}$ and that $scs:=s_2\cdots s_n{s}_1$ is a reduced Coxeter word for the conjugate Coxeter element $scs$ of $W$ (Coxeter words are commutation-connected; the Euler and skew forms depend only on the Coxeter element (1),(3)). For $w\in W$ the sortable projection ${\pi }_c(w)$ is defined recursively by ${\pi }_c(w):=\{\begin{array}{ll}1& \text{if }w=1,\\ s\cdot {\pi }_{scs}(sw)& \text{if }\mathrm{\ell }(sw)<\mathrm{\ell }(w),\\ {\pi }_{sc}(w_{⟨s⟩})& \text{if }\mathrm{\ell }(sw)>\mathrm{\ell }(w),\end{array}$ where $w_{⟨s⟩}$ is the $W_{⟨s⟩}$-prefix of $w$ in the length-additive decomposition of The weak parabolic projection, its adjoints, and the cover-join lemmas (the maximal $W_{⟨s⟩}$-factor). The recursion is well founded by the lexicographic measure (rank of the ambient parabolic, length of the current element): in the second branch the length strictly decreases, in the third the rank strictly decreases. That the recursion is independent of the initial-letter choices at every step, that ${\pi }_c(w)$ is always $c$-sortable, and that it is the greatest $c$-sortable element below $w$, are not part of this definition; they are proved in The recursive projection is well defined, sortable-valued, below w, idempotent, descent-detecting and parabolic ↗ and The cone criterion, monotonicity of the projection, and the greatest sortable element below w. Nothing about monotonicity, idempotence or fibers is asserted here. No Choice is used.