Finite complex contour resolvent validator with cover and inconclusive statuses

Input $n\ge 1$ and a nonempty finite union of complex line segments $\mathrm{\Gamma }$, with each segment assigned to a proposed disk $D_j=B(z_j,r_j)$; a nonempty matrix family $\mathcal{A}\subseteq {\mathbb{C}}^{n\times n}$ enclosed by complex rectangles; a point center matrix $A_0\in {\mathbb{C}}^{n\times n}$; and proposed point preconditioners $R_j\in {\mathbb{C}}^{n\times n}$. All segment endpoints, disk centers, radii and point matrices are exact dyadics.

1. For each assigned segment verify that both endpoints lie in its disk by exact rational comparison of squared distances with $r_j^2$. Evaluate outward bounds $e_j\ge \mathrm{\parallel }I-R_j(z_{j}I-A_0){\mathrm{\parallel }}_{\mathrm{\infty },\mathbb{C}},{\delta }_A\ge {\sup }_{A\in \mathcal{A}}\mathrm{\parallel }A-A_0{\mathrm{\parallel }}_{\mathrm{\infty },\mathbb{C}},s_j\ge \mathrm{\parallel }{R}_j{\mathrm{\parallel }}_{\mathrm{\infty },\mathbb{C}}$ using the complex rectangle operations and row sums of outward modulus bounds. If any check or operation cannot be completed, return inconclusive.

2. Outward-compute $q_j=e_j+s_j(r_j+{\delta }_A)$. Require $q_j<1$ for every disk and compute an outward upper bound $m\ge {\max }_j{s}_j\mathrm{/}(1-q_j)$. If a strict inequality or division cannot be certified, return inconclusive; otherwise return certified-resolvent-cover with the disk assignments and $m$.

The validated endpoint cover covers each whole segment because disks are convex. Success certifies a resolvent bound on this supplied contour cover, but this procedure does not count enclosed eigenvalues or validate contour topology.