> Sure, you may notice, but you can't prove my new chain is forged. I'll say the backup chain you're using as proof is the forgery...

If the items in the chain have digital signatures, you can prove that the people they're signed by contradicted themselves (or lost control of their signing keys). In Bitcoin people have felt that this wasn't good enough by itself, perhaps because there's not necessarily recourse against a counterparty who double-spends money, but you can still show that the double-spend attempt happened and who (in the sense of a signing key/address) was responsible for it.