A LETHE address is registered nowhere. It is the CREATE2 address of a contract that has not been built yet, and the commitment to the key sits inside that contract's init code, so the address is a hash of the key, and of nothing else. Before it spends, it holds funds while carrying no code, no storage and no entry in any list. Nothing on chain says it is a LETHE address.
Authority is a Winternitz one time signature: 67 chains of keccak256, each walked forward by the number its digit of the message names. To verify, you walk the rest of the way and see whether you land on the commitment. A signature therefore shows how far each chain was walked, which is exactly why the key may sign once and never again.
Normally that is the awkward part of hash based signatures, and people build Merkle trees to work around it. Here it costs nothing: the address was only ever going to be used once anyway. The one time key and the one time address are the same shape, so they fit without a tree, without a setup and without a bound.
A spend is a signed blob, not a transaction. Anyone can carry it to the chain and take the tip it offers, and the digest does not name them, so a stranger who steals the job gains the gas bill and nothing else. No address you control ever pays for this account, and that gas payment is the link that usually gives a fresh address away.
When the account finally runs, it is created, it acts, and it returns empty code in the same transaction. Nothing is deployed. Its nonce is burned, so CREATE2 can never land there again. The address is closed in the same breath as the key that opened it.