Finite-horizon Robin spectral elimination ($\Gamma -\tilde{\Gamma }=\frac{h(k)e^{-4i{k}^{2}T}{c}_T^{♯}(-k)}{d(k)\Delta (k)}$)

Here $\Gamma =B^{♯}/(ad)$ and $\Delta =\mathcal{D}/(2k+ic)$. The difference follows algebraically from the half-line NLS global relation and the Robin symmetry. Multiplication by the spatial-temporal jump factor gives $e^{2ikx-4i{k}^2(T-t)}$, decaying in the second quadrant for $t<T$. A triangular analytic change of the Riemann-Hilbert problem removes this finite-horizon defect while retaining the reconstruction and transferring all genuine pole information to the known determinant. No unproved large-time decay of the boundary values is needed.