What is the hard part of powering a mind?
Start with the supply. Gas falling toward a black hole spirals in, rubs against itself and shines before it crosses the horizon. A petawatt, the Vigil's first power, needs only 0.11 kg of gas a second falling into Vesper. Vesper spins, and that helps: a hole that did not spin would need 0.19 kg a second for the same light. The story of that furnace is on the page about feeding a black hole.
That makes power cheap. What you do with it is compute, and computing makes heat. In space there is no air or water to carry it off: a surface can only glow it away, and a cold surface glows very faintly.
Guess first To shed one petawatt of heat at 30 K, how big does the radiating surface have to be? Compare it with the Earth.
About 43 times the whole surface of the Earth. Radiated power grows with the fourth power of temperature, so going from a 300 K surface to a 30 K one makes the area ten thousand times larger. The Vigil builds it as thin films: 1.1 × 1016 square metres of panel when both sides work, weighing 1.1 × 1014 kilograms in all.
The film mass and replication target use the assumed 0.01 kg/m² cold-film density. This is an unsourced engineering target challenge; Cooling is the economy separates that assumption from the physical area law.
Why run cold? Lower temperatures reduce the minimum energy per erasure (see Colder is faster) and let the Vigil recover work as heat flows through heat engines. The larger radiator area needed at low temperatures dominates the Vigil's construction budget.
The same fact points to a way of spotting such civilizations from far away. Anything that thinks on this scale glows in the infrared, whatever it looks like in visible light. The full page on radiators has the area calculation to play with.