The descent section equals terminal value times recorded word

The descent section is the terminal value times the recorded word. For every $2\times2$ integer matrix $M$,

where $\mathtt{cfList}$ is the quotient list of the Euclidean descent, $\mathtt{cfEnd}$ is the terminal matrix it reaches, $\mathtt{baseQ}$ is the explicit terminal value, and $\mathtt{cfWord}$ turns the quotient list into the product of the descent factors $\mathrm{liftS}^{-1}\mathrm{liftT}^{e}$ in the reduced braid quotient $Q=B_3/\langle\!\langle\Delta^4\rangle\!\rangle$. The proof is a strong induction on the descent measure $|M_{00}|$. Together with the $T$-rule this is the structural description of $\rho$ that turns the two multiplication rules into a proof of $\rho(q\beta)=\beta$ for words $\beta$ in the braid group, and hence into the reduction of the milestone frontier to the continued-fraction $S$-rule.