> extracting carbon from the air always takes more energy that was gained by burning the fossil fuel to make that carbon.

This sounds true at first glance, but is it always? It seems like it's obviously true for simple reactions like burning methane and converting it back to methane since the energy you get from it is equal to or less than the energy required to turn it back into methane, but might there be some other chemical reactions where you can turn it into something less energetic than the molecule it came from?