Is there any feasible way to create a long-term sustainable habitat on the sun?
My target region for habitation is the temperature minimum at the bottom of the solar atmosphere. Metals would have to be flown in through the outer atmosphere, but at a mere 4100 K, it's at least cool enough for basic molecules like CO2 and water to form. I'm hoping to condense these out to serve as the basis of some farming agriculture, from which I'll derive oxygen and solid carbon for my colonists.
The main problem is whether I can get the AC up and running. Without it, all of my colonists are going to rapidly do some combination of boiling and burning. However, I'm not sure what technologies could be used to cool a habitat down from a temperature of 4100 K. I'm also not sure if there are any materials which could remain solid at 4100 K if I need some sort of shell on the outside of my habitat that I can't cool down on its own.
Is there any possible way to do this with sufficiently advanced technology that could theoretically exist without a massive paradigm shift occurring in our knowledge of physics?