  We construct the first smooth horizonless supergravity solutions that have
two topologically-nontrivial three-cycles supported by flux, and that have the
same mass and charges as a non-extremal D1-D5-P black hole. Our configurations
are solutions to six-dimensional ungauged supergravity coupled to a tensor
multiplet, and uplift to solutions of Type IIB supergravity. The solutions
represent multi-center generalizations of the non-BPS solutions of Jejjala,
Madden, Ross, and Titchener, which have over-rotating angular momenta. By
adding an additional Gibbons-Hawking center, we succeed in lowering one of the
two angular momenta below the cosmic censorship bound, and bringing the other
very close to this bound. Our results demonstrate that it is possible to
construct multi-center horizonless solutions corresponding to non-extremal
black holes, and offer the prospect of ultimately establishing that
finite-temperature black holes have nontrivial structure at the horizon.
