Continued analysis and extensions: the spectral disk for a Radau stage preconditioner

For the $q$-stage Radau IIA matrix with $A_q^{-1}=L_qU_q$, $B_q=L_q^{-1}$ and $N_q=U_q-I_q$, the spectral disk for the stage preconditioner reduces to $\rho((I+\mu B_q)^{-1}N_q)\le\|N_q\|_2$ for every real $\mu>0$. For every stage number this report proves the inequality when $0<\mu\le1/(q\|B_q\|_2)$ or $\mu\ge2\|L_q\|_2$, gives a rank-two identity for the diagonally scaled inverse, and gives a single matrix inequality in $L_q$ and $N_q$ that is sufficient for the stronger bound $\|(I+\mu B_q)^{-1}N_q\|_2\le\|N_q\|_2$ at all shifts. Integer-interval common-energy certificates add $q=96$ and $q=128$ to earlier certificates for $q=2,\dots,64$ and $80$, so the disk holds for all positive shifts at these 66 stage numbers. Exact counterexamples show that positive pivots and weighted accretivity alone do not imply the bound, and that it fails at $q=3$ for a complex shift with positive real part.