Nothing about this research violates conservation of energy. The article as written is advocating using excess solar or wind energy as input to this CO2->CH4 conversion process (which is electrolysis based) so that some of that energy can be reused later by burning methane. Later, as in when the wind isn’t blowing or the sun is t shining.
The title is "Using copper to convert CO₂ to methane could be game changer in mitigating climate change" which is wrong for the reason i told and which you did not address.
"Excess solar power" is the hypothetical clean power grid. I see that its supposed to be the "missing" power source, but its not helping saving any carbon as long as there are fossil plants in the grid. Temporarily replacing a fossil plant saves more carbon (that can be left in the ground for longer) than powering the conversion to methane (by a factor of at least 3, inverse of eff of plant). And that is not enough, the carbon needs to stay in the ground forever, not just some days longer.
So its not a game changer, its barely even relevant to the CO2 levels. Might have some applications but its not the solution it is presented as (Like the previous "solutions" that cropped up every other week on HN and are now long forgotten).
