Half-line mKdV spectral global relation ($S^{-1}s=e^{4i{k}^{3}T{\hat{\sigma }}_3}{\mu }_3(0,T,k)$)

For the three corner-normalized Lax pair eigenfunctions, let $s={\mu }_3(0,0,k)$ and $S=e^{4i{k}^{3}T{\hat{\sigma }}_3}{\mu }_2(0,T,k{)}^{-1}$. Their constant connection matrices give the displayed global relation. If their second columns are $(b,a{)}^T$ and $(B,A{)}^T$, its upper-right entry is $Ab-Ba=e^{8i{k}^{3}T}{\mu }_{3,12}(0,T,k)$. The right side is analytic in the lower half-plane. In the linear limit this is the backward-sign Airy half-line global relation at Fourier parameter $2k$, coupling all three boundary traces rather than allowing three independent boundary conditions.