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The jump coefficient $G(t)=(t^2-1)/t$ has one enclosed pole at zero and no enclosed zeros, because $\pm 1$ are outside. Its winding index is therefore $\kappa =-1$. A homogeneous canonical pair isEach is analytic and nonvanishing in its required finite domain; the exterior growth $X_{-}\sim z$ encodes the negative index. Dividing the unknown by these factors gives an additive jumpThe Cauchy-integral solution for $D$ normally has an exterior $z^{-1}$ coefficient $-(2\pi i{)}^{-1}{\int }_{L}g$. Since $C_{-}=z{D}_{-}$ must decay, this coefficient must vanish. Thus the solvability condition is ${\int }_{L}g(t)dt=0$. Its only enclosed pole is zero, with residue $(1-K)/2$, soThese are the negative-index scalar Riemann-Hilbert moment conditions for index minus one. There is no arbitrary polynomial addition: an entire addition respecting $D_{-}=O(z^{-2})$ is zero by Liouville's theorem.