Outside the box?
No one is saying breaking down CO2 into C + O2 is impossible. Green plants do it all the time. The point is that doing so requires at least as much energy as the oxidation of C yields in the first place. No 'thinking outside the box' can suspend the laws of Conservation of Energy.
This is perpetual motion. It ain't gonna happen. Its not a question of "how do we go about doing this"? Its more like "once we ignore the laws of physics, the problem is easy."
However, if you still think this is just an issue of unleashing the creative power of capitalism, I know a young scientist/entrepreneur who will gladly accept your investment!