= Area bound for an axisymmetric Kerr merger
For initially well-separated <Kerr black holes> with no orbital <angular momentum>, exact <axisymmetry> gives $J_f=J_1+J_2$. Put $X=(A_1+A_2)/(8\pi)$. The <black-hole area theorem> and the <irreducible mass> identity imply
$$
M_f^2\geq\frac12\left(X+\frac{(J_1+J_2)^2}{X}\right).
$$
Indeed, $X\geq|J_1|+|J_2|\geq|J_f|$, and $\tfrac12(Y+J_f^2/Y)$ is nondecreasing for $Y\geq X$. Opposite spin signs lower $J_f^2$ at fixed initial areas and hence raise the allowed radiated-energy bound; this need not be achievable in an actual merger.
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