You do realize we have an unlimited energy source burning down on us during the day? Energy isn't the issue here. We already overproduce during the day (though electric cars might take the load off of that in the near future).

Also you don't have to make the carbon go poof, you just have to capture it and store in in another form (that's not in the atmosphere). So yes, if you have a way to capture it and you use solar power (or any other renewable) you can reduce the carbon in the air.

Earth isn't a closed system.

The only carbon sequestration that makes any sense is small-scale, on-site or local (so you can avoid transport) biochar production via retort.

--> Biochar if you're not familiar is similar to charcoal, it's a form of "carbon black" that is elemental (isn't going to decompose or oxidize and contribute to climate change) and when added to soil helps plants (by acting as a sponge for water, nutrients, bacteria) while sequestering carbon for millennia in the soil.

For example, a landscaping company that burns it's waste to fuel a biochar retort and then using the resulting biochar to amend the soil used in the landscaping operations. (Think in cycles)

--> A biochar retort is form of furnace or fire pit that uses the flammable gasses produced by pyrolizing organic materials to fuel itself.

--> Pyrolysis is decomposition of organic material with high heat and no oxygen. It produces gases like methane which are burned in the retort producing particulates, CO2 and water (and that carbon does go back into the cycle) and leaves behind large amounts of elemental carbon black that is not going to contribute to climate change.

Sequestration by millions of backyard gardeners and little landscaping companies doing a little is better than trying to do it on a large scale because the large scale requires (as you note) resources. Hundreds of engineers and architects and workers driving to work for years so they can design and build a large device made of metal (that had to be mined, smelted, and shipped) and likely has an accompanying parking lot and office building would take years to break even sequestering as much carbon as it took to design and build it.

Sequestration as I describe here doesn't require much. For example, I make biochar using coffee cans in my fire pit .

Q: But won't burning some of the waste in the retort to heat the biochar contribute to climate change?

A: Any carbon in landscaping wastes, unless sequestered, is going to decompose into carbon dioxide (e.g. composting). Burning doesn't add any extra carbon, it's just that burning is a faster reaction than composting (but both burning and composting are part of the short term carbon cycle, biochar is not) . But because this burning is done to fuel pyrolysis it's part of an efficient process.

The real danger from burning the waste is particulate pollution, but that could be controlled with common scrubbers tech.

Imagine that you have a fire in a fireplace. You've expended j₁ joules of energy moving that firewood into your house, setting it up correctly, and igniting it.

Now, you've got another fire, using the exact same amount of wood, in the forest under a tree. The wood happens to have been piled up perfectly to ignite, and--oh, look at that, a squirrel brought an ember over, igniting that fire. You've expended j₂ joules of energy starting the outside fire, and j₂ = 0.

Ok, let both of the fires burn out. Now let's clean up the ashes, and we'll call the amount of energy required to do that c₁ and c₂. Assuming that you're moving the ashes the same distance to dispose of them, c₁ and c₂ are equal; in any case, any difference is completely unrelated to j₁ and j₂.

Carbon capture isn't trying to rebind the molecules or anything. It's just trying to vacuum up the carbon and sequester it. In fact, plants already do carbon capture really, really well, and they certainly don't require a ton of energy to do it. Amine scrubbing (an ~100 year old technology) is used to capture carbon from power plant exhaust, and it's using 0.11 mWh to scrub about one metric ton of carbon; but to create that much carbon the plant produced about ten times as much electricity.

Now, does any of this mean that carbon capture should be the first step? No! The cost is currently huge; something like $50 per ton removed, and that's even before you consider how you're going to get all the air through the amine scrubber. It's definitely more cost-effective to switch to renewables, reduce usage, etc. But the time will probably come (and honestly might already be here) where that's just not enough. When that happens, we don't need to worry about doubling our total output; increasing it by 10% or so should do it, and if we're really good at usage reduction, we can probably divert some of that saved energy toward capture.

That doesn't make sense. Turning CO2 into methane would take at least as much energy as we get from burning methane to produce CO2, but we don't have to turn it back into methane. There are probably other ways to capture it.