\section{Preliminaries and the Dyson equation}\label{sec:dyson} We use the determinant and trace conventions introduced in Section~\ref{sec:introduction}. In particular, all operator determinants use the normalized trace of the relevant matrix size. We use multiplicativity and the block-diagonal determinant rule as in \cite[Proposition 2.3]{MS24}. \subsection{The matrix Cauchy transform} Consider first any selfadjoint pencil \[ K=A\otimes1+\sum_j B_j\otimes s_j\in M_d(\M), \qquad A=A^*,\quad B_j=B_j^*\in M_d(\C), \] Put $\Theta(b)=\sum_jB_jbB_j$ for $b\in M_d(\C)$. \begin{definition}[Matrix Cauchy transform]\label{def:cauchy-transform} The matrix-valued Cauchy transform of $K$ is \[ G(w)=E_d[(w1_d-K)^{-1}],\quad \operatorname{Im}w>0. \] \end{definition} We use the resolvent convention of \cite[Chapter 9, (9.5)]{MS17}. The operator-valued semicircular moment rule gives the \emph{Dyson equation} \begin{equation}\label{eq:dyson-general} G(w)^{-1}=w1_d-A-\Theta(G(w)),\qquad \operatorname{Im}G(w)<0. \end{equation} Here $\operatorname{Im}G=(G-G^*)/(2i)$, and the inequality is negative definiteness. The source input is \cite[Chapter 9, Definition 10, Theorem 11 and Proposition 13]{MS17}. The moment recursion is \[ Q_0=1_d,\qquad Q_k=AQ_{k-1} +\sum_{r+s=k-2}\Theta(Q_r)Q_s,\quad Q_k=E_d[K^k]. \] The empty sum is zero. Inserting this recursion into the convergent large-$|w|$ resolvent series proves \eqref{eq:dyson-general} there; analytic continuation proves it throughout the upper half-plane. The imaginary-part condition selects exactly one solution at each $w$. With $W=iG$ and $V=-i(w1_d-A)$, the equation takes the positive-real-part form $VW+\Theta(W)W=1_d$ studied in \cite[Theorem 2.1 and Section 3]{HRFS07} by holomorphic contraction. We include a direct proof because its estimate also gives the invertibility needed when differentiating the Dyson equation. Suppose $G_1,G_2$ are solutions and put $V_j=-\operatorname{Im}G_j>0$, $v=\operatorname{Im}w$. Taking imaginary parts of the inverse equation gives \begin{equation}\label{eq:imaginary-balance} V_j=G_j(v1_d+\Theta(V_j))G_j^* =G_j^*(v1_d+\Theta(V_j))G_j. \end{equation} For $D=G_1-G_2$ we have $D=G_1\Theta(D)G_2$. After setting \[ W=V_1^{-1/2}DV_2^{-1/2},\quad L_i=V_1^{-1/2}G_1B_iV_1^{1/2},\quad R_i=V_2^{1/2}B_iG_2V_2^{-1/2}, \] this becomes $W=\sum_iL_iWR_i$. Equation~\eqref{eq:imaginary-balance} implies $\sum_iL_iL_i^*<1_d$ and $\sum_iR_i^*R_i<1_d$. The row and column operator norm estimate therefore gives \[ \|W\|\leq \left\|\sum_iL_iL_i^*\right\|^{1/2} \left\|\sum_iR_i^*R_i\right\|^{1/2}\|W\|<\|W\| \] unless $W=0$. This proves uniqueness. We call this selected solution the \emph{resolvent solution}; no choice among real algebraic roots is implicit in the notation. Taking $G_1=G_2=G$ in the same estimate shows that $X\mapsto G\Theta(X)G$ is a strict contraction in the norm $\|V^{-1/2}XV^{-1/2}\|$, where $V=-\operatorname{Im}G$. Consequently $X\mapsto X-G\Theta(X)G$ is invertible. We use uniqueness to identify the variational critical points in Sections~\ref{sec:main-proof} and~\ref{sec:ordered-minmax}, and this invertibility for the derivatives in Section~\ref{subsec:brown-density}. \subsection{Integrating the Cauchy transform} The covariance is symmetric for the bilinear trace pairing: $\Tr(\Theta(X)Y)=\Tr(X\Theta(Y))$. Choose the analytic logarithm of $\det G(w)$ compatible with $G(w)\sim w^{-1}1_d$ at infinity. Then \begin{equation}\label{eq:primitive} F(w)=-\frac1d\log\det G(w) +\frac1{2d}\Tr(\Theta(G(w))G(w)) =\varphi_d(\Log(w1_d-K)). \end{equation} To prove this identity, differentiate \eqref{eq:dyson-general} and take the trace to obtain \[ \Tr G=-\Tr(G^{-1}G')+\Tr(\Theta(G)G'). \] Thus both sides of \eqref{eq:primitive} have derivative $\tr_dG$. Both are $\log w+o(1)$ as $w=it$ tends to infinity, which fixes the additive constant. Taking real parts gives \begin{equation}\label{eq:selfadjoint-potential} \log\Delta(K-w1_d) =-\frac1d\log|\det G(w)| +\frac1{2d}\operatorname{Re}\Tr(\Theta(G(w))G(w)). \end{equation} The identity extends to a real spectral gap by continuity. It fixes the value at a resolvent solution; an extremum over free trial matrices requires a separate argument. Related finite log-determinant expressions, evaluated at solutions of deterministic resolvent equations, occur in the random-matrix deterministic equivalents of \cite[Theorem 4.1]{HLN07}. The shared method is integration of the resolvent; the minimization over arbitrary positive matrix trials is established in Section~\ref{sec:main-proof}. \subsection{Selfadjoint blocks and regularization}\label{subsec:hermitization} For the general pencil $S$ define its hermitization by \begin{equation}\label{eq:hermitization} \mathscr H=\begin{pmatrix}0&S\\S^*&0\end{pmatrix},\qquad A=\begin{pmatrix}0&a_0\\a_0^*&0\end{pmatrix},\qquad B_j=\begin{pmatrix}0&a_j\\a_j^*&0\end{pmatrix}. \end{equation} This is a selfadjoint pencil of size $2m$; see \cite[Section 3]{MS24} and \cite[Chapter 11, Section 9]{MS17} for this block construction. Put $\Theta(b)=\sum_jB_jbB_j$ and \[ \eta^*(b)=\sum_j a_j^*ba_j,\qquad \rho^*(b)=\sum_j a_j^*ba_j^*. \] These are the Hilbert--Schmidt adjoints of $\eta$ and $\rho$ for $\langle b,c\rangle=\Tr(b^*c)$. We follow the notation of \cite[Lemma 3.5(ii)]{MS24}. Direct multiplication gives \begin{equation}\label{eq:full-covariance} \Theta\begin{pmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{pmatrix} =\begin{pmatrix}\eta(b_{22})&\rho(b_{21})\\ \rho^*(b_{12})&\eta^*(b_{11})\end{pmatrix}. \end{equation} This displays the role of $\rho$ anticipated in the introduction. For $\varepsilon>0$, set \begin{equation}\label{eq:regularized-determinant} \log\Delta_\varepsilon(S) =\tfrac12\varphi_m\log(S^*S+\varepsilon^21_m). \end{equation} The operators $S^*S$ and $SS^*$ have the same tracial spectral distribution, including at zero, by polar decomposition and traciality. Since $\mathscr H^2=\operatorname{diag}(SS^*,S^*S)$, the resolvent $G_\varepsilon=E_{2m}((i\varepsilon 1_{2m}-\mathscr H)^{-1})$ satisfies \begin{equation}\label{eq:regularized-primitive} \log\Delta_\varepsilon(S) =-\frac1{2m}\log|\det G_\varepsilon| +\frac1{4m}\operatorname{Re}\Tr(\Theta(G_\varepsilon)G_\varepsilon). \end{equation} This follows from \eqref{eq:selfadjoint-potential}. Monotone spectral integration gives $\log\Delta_\varepsilon(S)\downarrow\log\Delta(S)$, also when the limit is $-\infty$. \subsection{Positive maps and spectral gaps}\label{subsec:gap-tools} We record two tools for identifying spectral gaps from the matrix Cauchy transform. For a linear map on a finite-dimensional space, write $r(\Psi)$ for its spectral radius, the largest modulus of its eigenvalues. If $\Psi(b)=\sum_j c_jbc_j^*$ on $M_m(\C)$, then the row operator $[c_1\ \cdots\ c_n]$ gives \[ \|\Psi\|=\|\Psi(1_m)\|. \] Indeed it gives the upper bound, and evaluation at $1_m$ gives equality. If $W>0$ and $W-\Psi(W)>0$, conjugating the map by $b\mapsto W^{1/2}bW^{1/2}$ makes its value at the identity strictly less than the identity. Its spectral radius is therefore less than one. In finite dimension this implies convergence of $(\id-\Psi)^{-1}=\sum_{k\ge0}\Psi^k$, which preserves positivity. This is the finite-dimensional positive-map form of the strict Stein criterion; see \cite[Lemma 5.5 and Theorem 5.9]{Pop03} for its relation to similarity to a strict contraction. Next let $K=K^*\in M_d(\M)$, with scalar spectral measure $\nu_K$, and let $g(w)=E_d((w1_d-K)^{-1})$. For real $a