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

We study the finite sampling map $H\mapsto (v_{H,\mathrm{\Lambda }}(x_k+i\eta ){)}_{k=1}^M$ for trace-normed canonical systems on $[0,\mathrm{\Lambda }]$ with free tail $H(s)=\frac{1}{2}I$ for $s\ge \mathrm{\Lambda }$, where $v_{H,\mathrm{\Lambda }}$ 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 $L^1(0,\mathrm{\Lambda };\mathrm{o}\mathrm{p})$.