= Solution
The <Penrose singularity theorem> concludes future null-geodesic incompleteness; it does not require a divergent curvature invariant. This spacetime supplies exactly that distinction. For a null geodesic in the static coordinates, the Killing constants
$$
E=f\dot t,
\qquad
J=r^2\dot\phi
$$
and the null condition give
$$
\dot r^2=E^2-\frac{fJ^2}{r^2}.
$$
As $r\to0$, $f\to-r_+^2/L^2$. A radial geodesic with $J=0$ reaches $r=0$ linearly in finite affine parameter. If $J\ne0$, then $|\dot r|\sim r_+|J|/(Lr)$, so it again reaches zero in finite affine parameter. The anti-de Sitter quotient ends these geodesics at $r=0$ even though local curvature remains finite. Thus the trapped circles lead to the <BTZ causal singularity>, satisfying the theorem through geodesic incompleteness rather than curvature blowup.
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