Bitcoin signed messages don't prove ownership

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Are you referring to signing a message with the key for an address that only temporarily has the correct balance?

Addresses don't have balances at all

Haha ok true. I don’t mean that in the ETH sense. I mean that UTXOs have been sent to an address that is related to a private key, which has signed a message, and not thereafter moved. That’s possible to check, right?

No, because that UTXO isn't necessarily theirs anymore

How so since in the hypothetical we’ve already checked that it hasn’t been spent? Maybe.. if they have signed a transaction and broadcasted it but it has yet to make it into a block?

Or if they transferred it off chain

Can you elaborate? Don't you have to own the private keys to sign a message?

No. Custodial wallets are supposed to let you sign messages for your addresses, and UTXOs don't necessarily need to move when you transfer the bitcoins to someone else.