Fixed-Height Weyl--Schur Sampling for Free-Tail Canonical Systems

We study the finite sampling map $H \mapsto \bigl(v_{H,Λ}(x_k + iη)\bigr)_{k=1}^M$ for trace-normed canonical systems on $[0,Λ]$ with free tail $H(s)=\frac{1}{2}I$ for $s \ge Λ$, where $v_{H,Λ}$ is the Schur transform of the Weyl coefficient. At the free Hamiltonian $H_0 \equiv \frac{1}{2}I$, we obtain an explicit first-order expansion with quadratic remainder; the linearization is a weighted Fourier--Laplace transform. This yields quantitative local identifiability and local inversion on finite-dimensional families for which the free Jacobian is injective. In the block model, the free Jacobian factors into a row factor, a Fourier sampling matrix, and exponential depth weights, giving explicit singular-value bounds and an exponential depth-conditioning barrier. By contrast, on the full free-tail class every finite sample set has nontrivial first-order invisible directions at $H_0$, so no local inverse-Lipschitz estimate can hold near $H_0$ in $L^1(0,Λ;\mathrm{op})$.