How does this system handle problems with the lower chains after they have been "locked-in"?

The rule is that if a block in the child chain is pointed to by its parent, then it effectively has infinite POW?

The point of the system is that a node monitoring the parent chain only has to watch the header chain for its 2 children.

A parent block header could point to an invalid block in one of the child chains. That parent block could end up built on top of before the problem was discovered.

This would mean that a child chain problem could cause a roll-back of a parent chain. This violates the principle that parents are dominant over child chains.

Alternatively, the child chain could discard the infinite POW blocks, since they are illegal.

P1 -> C1

P2 -> ---

P3 -> C3

P4 -> C5

It turns out C4 (or C5) was an invalid block

P5 -> C4'

P6 -> ---

P7 -> C8'

This is a valid sequence. Once P7 points at C8, the alternative chain displaces C5.

This displacement could require a compact fraud proof to show that C4 was an illegal block and that C5 was built on it.

This shouldn't happen if the miner was actually watching the log(N) chains, but can't be guaranteed against.

I wonder if the proof of stake "nothing is at stake" principle applies here. Miners aren't putting anything at stake by merge mining the lower chains.

At minimum, they should get tx-fees for the lower chains that they merge mine. The rule could require that the minting reward is divided over the merge mined chains.