Why did you wrongly claim that the fundamental principle used to determine equilibrium temperatures is "irrelevant"? If you actually understand how conservation of energy at equilibrium works, then you must be able to recognize that enclosing a heated plate warms it. So why do you keep insisting otherwise? Do you need physics lessons, or have you betrayed humanity by deliberately spreading civilization-paralyzing misinformation?
I have done nothing of the sort.
Are you saying that you have changed the nature of the experiment, such that it is no longer in vacuum?
The original experiment does not involve "enclosing a heated plate", except to the extent that it was already enclosed. In the experiment that has (always, as far as I am concerned) been under discussion, there is a heat source S, a passive plate P that is heated by that source, and an enclosure (which I have called W for "wall") that is actively cooled. Everything inside the enclosure is in vacuum, so that ALL heat transfer is by radiation only. No convection, no conduction.
Are you referring to the same experiment? If so, then I will repeat what I have already stated several times. And I will also repeat that if you have an argument with it -- other than your straw-man argument above, that is -- you go argue it with the proper parties, not with me. But I am indulging you to this extent.
1) Even if the passive plate completely surrounds the source, then in any real-world situation it is impossible for it to ever quite reach the same temperature as that source, even if only because the surface area is (however slightly) greater than that of the source. We have discussed this before. Therefore at equilibrium temperature Ts will always be warmer -- even if only a little -- than the passive plate Tp.
2) By the Stefan-Boltzmann equation, NET heat transfer will always be from hotter to colder. And since Ts - Tp is a positive number, net heat transfer is from the source to the plate. The plate cannot cause the heat source to be hotter because that would require NET heat transfer in the other direction. But that is a violation of the Stefan-Boltzmann law. (There is no need to re-derive how we apply the S-B law here. Again, that would be re-hashing old news.)
By asserting that at equilibrium the passive plate can cause the source to be hotter, you are contradicting the S-B law. You can make all the other arguments you like to try to sidestep this, but eventually you're just going to step in it again. Pun very much intended.
I have stated this all before. I repeat that you are making a mistake. But in order to find out what it is, you are going to have to address your argument to the person you are attempting to refute. Your argument is not with me and trying to make it with me is childish. Given that, and the abusive nature of your past behavior, I refuse to help you further. No more hints.