Common fairness standards make the Untax be preferable to the C-swap.
The gist of the proof is this:
1. The Untax is fair.
2. Society prefers not to adopt a Poll-swap.
3. C-Swap = Untax + Poll-swap.
4. Society prefers an Untax to a (Untax + Poll-swap).
5. Society prefers an Untax to a C-swap.
The main potential loophole is that implementing the untax could somehow change society's view of the Poll-swap. This might negate step 4. But the Untax has no effect on society that would warrant such a change in view unless the Untax itself were unfair and this could be remedied by re-directing the Untax payments with the Poll-swap.
If the Untax were based on taxing, say, fruit and refunding the revenues per person, this would be a legitimate objection. By social norms, a fruit-untax is an unfair redistribution of wealth, and it could be corrected with a Poll-swap that eliminated the fruit tax. This suggests that a different Poll-swap might help correct the unfairness of a fruit-untax since it would use fruit-tax revenues for the public good instead of simply redistributing them. Hence step 4 would not hold for a fruit untax and depends on the fairness of a carbon untax.
Because, Chapter 18 shows that the Untax is fair, the poll-swap cannot be justified as a correction of its unfairness. Neither can it be justified on other grounds because nothing of significance has changed to make society reverse its position on poll-swaps. For this reason step 4 does hold.
Finally, let's review the reasons behind the two main foundations of this argument, (1) the untax is fair, and (2) the poll-swap is rejected by society.
The untax is fair. Everyone has an equal right to climate stability and energy security. So if we could privatize these rights efficiently by giving them out equally but with the total limited to the socially optimal level, that would be fair. These rights should be tradeable. Chapter 18 shows that the resulting individual costs and revenues are the same as from the untax. Hence the untax is also fair.
*The poll-swap is rejected. For over a hundred years now, our society has rejected taxes that charges everyone an equal dollar amount—say $1000 per person. Economist are well aware that collecting a poll, or capitation, tax to pay down another tax would increase economic efficiency. But this is no different from implementing a poll tax from the start, so the approach is considered to unfair and is never proposed.
It makes no more sense to propose a poll tax to pay down one of our other taxes than to propose to do that with a carbon tax, when we could just as easily implement a carbon untax. The carbon swap proposal only makes sense if the untax option is foreclosed, which it is not.