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I would love for a proof of human work to exist, but how would you even do that? It would need to be monitoring the user activity in their email client, which isn't something that can be trusted by a server (and is pretty shady).

But that makes me think of Hashcash, that was developed to limit email spam via proof of work, but I don't think that has ever been used in practice: https://en.wikipedia.org/wiki/Hashcash (and of course wouldn't work for the proof of humanness you're talking about).

I want to give my bank my public key (preferable at a branch), so that ANY comms coming from them I can prove it came from them as.

What about automated signup confirmation emails, just as one example?