Recovered analysis and finite-stage results: the spectral disk for a Radau stage preconditioner

For the $q$-stage Radau IIA collocation matrix we give a self-contained
differentiation formula for its inverse, prove that every pivot of the no-pivot
factorization $A_q^{-1}=L_qU_q$ is positive, reduce the spatial pencil exactly
to the stage matrices $I_q+(I_q+\mu L_q^{-1})^{-1}(U_q-I_q)$ with $\mu>0$, and
prove a common-energy sufficient condition for the disk
$|\nu-1|\le\|U_q-I_q\|_2$. We supply a closed-form two-stage bound and an
explicit three-stage proof in $\mathbb Q(\sqrt6)$. Exact integer-interval
computation then establishes the disk for every positive shift at the 64 stage
numbers $q=2,\dots,64$ and $q=80$; the proof checker requires only Python's
standard library. A counterexample to a tempting Euclidean resolvent estimate
explains why the elementary norm argument does not solve the general problem.
The all-stage statement remains open.