Is the following the correct way to translate Argument To Proof of Work
Argumentum Ad Probationem Operis
The intention is to translate it in the same way as Argument to the Person
Argumentum Ad Hominem
The basic structure you want is argumentum ad… with an accusative noun: that is, an argument aimed at something.
Amusingly, the most common word for "proof" is also argumentum, in the sense of proving something in court, or finishing a mathematical theorem. Instead, I'd use indicium, which means evidence you can present before a court.
For "work", since this is specifically work designed to be difficult without actually accomplishing anything useful, I'd say labor: literally "toil" or "drudgery".
All together, that would be argumentum ad indicium laboris.
However, this sounds somewhat unwieldy, and is kind of an obvious calque: it's a literal translation from English, not an idiomatic one. For an idiomatic translation, I would use onus. Literally it means "burden", but in the sense of "something you have to suffer through" or "a duty that has to be done" (as in "burden of proof", a literal translation of Latin onus probandi). So an argumentum ad onus is proving your point by demonstrating what you've been through, which in this case is the one-way computation.