Imaginary-part rank for the general complex Green-function equation

Let $C,D,R,P\in\mathbb{C}^{n\times n}$ with $R,P$ Hermitian and $P+\lambda D^*+\lambda^{-1}D$ positive definite on the unit circle, and let $X_\eta$ be the stabilizing solution of $X+(C^*+i\eta D^*)X^{-1}(C+i\eta D)=R+i\eta P$. If $X_\eta$ tends to a finite nonsingular $X_0$ as $\eta\downarrow0$, the pencil $\lambda^2C^*-\lambda R+C$ is regular and its $2m$ unit-circle eigenvalues are algebraically simple, then $\operatorname{rank}((X_0-X_0^*)/(2i))=m$. Singular $C$ or $D$, unit-circle roots at $\pm1$ and arbitrary Jordan structure inside the disk are allowed. A homotopy and the argument principle place exactly $n$ eigenvalues of the regularized pencil inside the disk; Rouché's theorem passes a factorization to the limit, so $X_0^{-1}C$ has exactly $m$ unit-circle eigenvalues, all simple. A Stein identity for the imaginary part, with the derivative at each simple root, gives rank at least $m$; the stable generalized eigenspace lies in its kernel, giving rank at most $m$.