= Solution
With $K=1$, the additive jump is $g(t)=1/[2(t^2-1)]$, analytic inside the contour. Its <Cauchy integral> has $D_+(z)=g(z)$ and $D_-(z)=0$. Therefore $C_+=(z^2-1)g=1/2$, $C_-=0$, and
$$
\boxed{\phi(t)=\tfrac12.}
$$
The moment is indeed $(\pi i)^{-1}\int_L(\tau+1/\tau)(1/2)d\tau=1$, as required. The principal-value Cauchy sum is $1/2$, and the original left side becomes $[(t^2+t-1)-(t^2-t-1)]/2-1=t-1$. Thus the recovered function satisfies both the equation and the self-consistency condition, and the homogeneous negative-index argument proves uniqueness.
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