We've been saying on and on, that's not true. Radiation re-emitted by the atmosphere only slows the rate of heat transfer. Therefore, heat still flows from hot to cold. It just does so more slowly. This is not in violation of second law, it's perfectly allowed by it.
This process isn't lossless, and the only reason warming results is because there is still continued input of energy from the sun. The earth isn't a closed system. Therefore, there is also no violation of conservation of energy and no one is claiming this is anything like a perpetuum mobile. Heat isn't "trapped" indefinitely, it just takes a longer time to leave.
I've given sources that state why that is the case, and all you've done to say differently is to 1) give a source that talked about something else entirely (electron energy levels rather than heat) and didn't actually back up your claim (energy differential vs. absolute energy) and 2) say that somehow the fact that the earth isn't a perfect black body completely removes the surrounding energy from the Stefan-Boltzmann equation (according to here, it just adds another term to the function, it doesn't remove parts of it), which you also haven't backed up with a source.
Saying "it doesn't apply because the earth is not a black body" isn't enough. Why doesn't it apply? What changes about the physics when you're not dealing with black bodies? Why does radiation from the environment become irrelevant according to you in that situation?
So give me something that proves that the radiation from (colder) surroundings cannot reduce the rate of heat loss. Give me something that proves the surrounding temperature term from the rate of heat loss equation I linked doesn't apply at all to objects that are not perfect black bodies. Until you can do so, you have not shown any cause to reject my argument thus far.
And since we're talking about fundamental physics that doesn't just apply to climate science, I'd prefer it if you can use sources that aren't related to climate change to avoid bias. I've also deliberately been doing that.