Diagonal Stability of Two-Block Systems with Reciprocal Single-Channel Coupling

This paper considers coupled linear subsystems $\dot{x}=-Ax-By$ and $\dot{y}=-Cx-Dy$, whose local blocks A and D may have different dimensions. For reciprocal single-channel coupling $B=\beta e_i{e}_j^T$ and $C=\gamma e_j{e}_i^T$, let ${\delta }_i\left(A\right)$ and ${\delta }_j\left(D\right)$ denote the directional diagonal-Lyapunov margins of the respective blocks: each is the supremum of a normalized coordinatewise Lyapunov reserve over admissible positive diagonal Lyapunov matrices. I prove that the full matrix is diagonally stable if and only if both local blocks are diagonally stable and $\beta \gamma <{\delta }_i\left(A\right){\delta }_j\left(D\right)$. The strict proof does not presume attainment of the local suprema. I also give a general Schur-complement test, constructive relative diagonal-Lyapunov scalings, and a dimension-free criterion for arbitrary one-way coupling. Closed planar formulas recover the classical acyclic threshold, while reproducible examples with cyclic internal graphs illustrate the broader theorem and show that Hurwitz stability need not imply diagonal stability.