Lamport signatures and other CAT tricks

The concern arises when a peer moves channel coins to a new address, and the state recognized by the other party does not match the final state. To address this, one needs access to all the non-deterministic components of the script to reconstruct the transaction and spend the outputs based on the latest state maintained.

The query also delves into two distinct situations: first, when a peer possesses a later state than what is currently known, necessitating the placement of the settlement transaction onto the blockchain to claim one's outputs; secondly, when there is a requirement to adapt one's later state to a peer's already published on-chain state. This process, referred to as rebinding, aligns the private state with the public state that has been committed to the chain.

When dealing with lost state during settlement transactions, it is typically expected that the remote peer will execute the transaction on the blockchain, allowing identification and claiming of one's outputs. The Asynchronous Proof of Ownership (APO) mechanism comes into play specifically for the rebinding process. APO facilitates the verification and alignment of one's private state with the counterpart's publicly committed state, ensuring the ability to spend outputs accordingly.