If aliens existed we could readily see copious surface engineering in galaxies: "The germs of existence [on] earth if they could freely develop themselves would fill millions of worlds in the course of a few thousand years"(Malthus) https://waitbutwhy.com/2014/05/fermi-paradox.html … https://www.centauri-dreams.org/2012/01/23/rethinking-setis-targets/comment-page-1/ …
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Oops I misread, the structure doesn't cross the event horizon, it just points at it (it does have to be close enough so that the event horizon cross-section dominates the sky -- don't know if that is possible).
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It would be in orbit around the black hole, so as long as it's not colliding with other objects in orbit it won't cost any energy
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