An eventual linear condition-number bound

For a fixed real $\rho>0$, let $H_n(\rho)$ be the $2n\times2n$ block Toeplitz matrix with blocks $iB_k-\rho C_k$, $-1\le k\le2$, arising from cubic $C^1$ splines in time in a space-time discretisation of the Schrödinger equation. The matrices $H_n(\rho)$ are nonsingular for every sufficiently large $n$, and $\kappa_2(H_n(\rho))\le K_\rho n$ with $K_\rho$ independent of $n$. The linear system is written as a four-state transfer recurrence with the exact boundary conditions. A scalar generating function is shown to have numerator and denominator without common roots for every $\rho>0$. An explicit classification of the four nonzero characteristic roots, two on the unit circle, one inside and one outside, covers the exceptional parameter $\rho=\sqrt{10}$. The resulting finite Green matrix has uniformly bounded blocks, so the inverse is bounded by a multiple of $n$, while the norm of $H_n(\rho)$ is bounded uniformly in $n$.