Source of breathable air to fill Dyson sphere shell
So let's say you have a Dyson sphere that is 5 kilometers thick . Now, it isn't solid. It's meant to be inhabited. There's the inner shell which is completely covered with solar panels and any other manner of power generation facing the star. It's 500 meters thick. There's the outer shell which again is 500 meters thick, but is covered in armored plating, defensive weaponry, and the occasional hanger door. Between these two shells is an empty 4 kilometer area that is FILLED with breathable air. People living in the sphere would inhabit centrifugally spinning towers that act as support columns between the shells, but I digress. Where would one go about getting all that air? This is assuming that the creators of the sphere are humanesque (more or less), and require the same air we do.
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So the good news is that if you have the capability to construct a Dyson sphere, you have the capability of supplying it with resources.
The sheer amount of matter needed is huge, and you'd need to steal the planets and other debris from many other stars to get enough. And since you're stealing all that stuff you can get their gasses too.
If you don't like the idea of moving dozens of solar systems worth of planets across light years of space, the other option is to make an energy to matter converter. Then you just need a whole lot of energy, which can be provided by the sun.
Best course would be to collect the asteroid belt, kupiter belt, and any other stray rocks that are floating around, turning them into a Dyson swarm with as many energy collectors as you can put up, and start feeding it all to an e->m engine. Use that to expand your collectors until they form a Dyson ring, and then enclose the globe.
This would be an exponential process, since the more you enclose the more energy you have to work with.
At this point you've got your inner shell, and you should have all the energy you need to finish the outer shell and then start making atmosphere to fill it.
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