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> Not allowed by thermodynamics. Otherwise you could create free energy by continuously burning and recapturing the same carbon.

That doesn't have to be the case; the carbon you capture may not be in a burnable form. E.g. if carbon dioxide were captured, additional energy would be required to split off the oxygen before the carbon could be 'burned' again.

Just capturing might be possible - it doesn't have to be in form that can be easily used to generate energy.

It would suffice to use less non-carbon-based energy to capture the CO2.