= Solution
The common axial <Killing vector field> fixes the same sign convention for both spins. Initial rest and alignment of the separation with the symmetry axis give no orbital <angular momentum>. At large separation, the total axial charge is therefore $J_1+J_2$. Since <axisymmetric radiation carries no axial angular momentum>, all of this charge remains in the remnant after the radiation has escaped and the exterior has settled:
$$
\boxed{J_f=J_1+J_2.}
$$
Here the spins are signed quantities about the same axis, not their magnitudes. The conclusion assumes the exact <axisymmetry> stipulated throughout the merger and no remaining exterior matter carrying <angular momentum>; the final stationary vacuum remnant is a <Kerr black hole> under the usual uniqueness hypotheses.
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