Exact scalar shooting for a structured subclass

For rational $A$ and $b$ such that every matrix $A-\operatorname{diag}(d)$,
$d\in[-1,1]^n$, is nonsingular, the note gives a deterministic algorithm of
polynomial bit complexity that computes the exact solution of $Ax-|x|=b$
when $A$ is lower Hessenberg, including zero superdiagonal entries and zero
solution coordinates. Taking the first coordinate as a parameter, the first
$n-1$ equations determine the others by a shooting recurrence. Regularity
makes the last residual strictly monotone in the parameter, so bisection to
polynomially many bits, sign recovery and one exact linear solve give the
solution, with polynomial bounds on all intermediate bit lengths. A regular
Hessenberg family whose solution path crosses all $2^n$ orthants, a
three-cycle of undamped sign-Newton iteration, and the failure of an
inverse-image separation shortcut mark the limits of specific approaches.
A companion note shows that the optimized handicap of a cyclic three-dimensional family is exponentially large in the input length, although Theorem H solves its equations.