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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.

Initial scattering gives $s$ and boundary-time scattering gives $S$. The defocusing reduction and unit determinants give the displayed matrix forms. The initial functions $a,b$ are analytic in the upper half-plane, while the boundary functions $A,B$ are bounded in the first and third quadrants. Their finite-horizon compatibility is the half-line NLS global relation.

For the Robin boundary condition $q_x(0,t)=cq(0,t)$ with real $c$, the time Lax pair has a diagonal-conjugation symmetry under $k\mapsto -k$. It implies $A(-k)=A(k)$ and $B(-k)=-h(k)B(k)$. Together with the half-line NLS global relation, this eliminates unknown boundary scattering from an equivalent Riemann-Hilbert problem determined by the initial data and $c$.

Eliminate ${\mu }_2$ between the two connection relations and evaluate at $(0,T)$, where ${\mu }_1=I$. This yields the matrix global relationIts upper off-diagonal entry is especially useful. Since $S^{-1}=\left(\begin{array}{cc}A& -B\\ -B^{♯}& A^{♯}\end{array}\right)$, it isThe function $c_T$ is analytic in the upper half-plane and is $O(1/k)$ there by the spatial Volterra integral equation. This is the half-line NLS global relation. It expresses compatibility between initial and boundary scattering and the final state. At finite $T$ the right-hand side is generally nonzero; setting it to zero would silently impose an extra assumption.