I was thinking more along the lines that solar (or whatever) energy were used to heat limestone to release (and capture) the CO2 from that limestone and produce the lime.
The captured CO2 could then be pumped into the ground along the lines of
which suggest that captured CO2 could be pumped underground and effectively soaked up by the peridotite.
The surface level Lime (after being mixed with water to generate heat/electricity on demand) then absorbs CO2 out of the atmosphere and turns to limestone which .... full circle.
The process results in a net air to ground transition of CO2 based upon solar/wind/wave energy to drive the process (not a perpetual motion machine) whilst providing an energy store (Lime) that services energy on demand production.
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