"Proof of" is a euphemism for "waste of resources" in general.

Only in cryptocurrency, i.e. in the cargo cult of finance.

In real world finance a common stock is proof of making a useful investment at some point in the past, when the stock certificate was issued in return for capital. Same with other tradeable and non-tradeable assets.

But this isn't proof, that financial stock certificate. It's actually fiat: the clearinghouse or the national registry says who owns what and the court can change it.

The crypto version of proof is closer to a bearer bond or a bar of gold: whoever has possession has possession.

Yes, you pointed out why the analogy isn't perfect, all the while missing the point entirely.

In cargo cults it would be like complaining that the control tower made of wood has the wrong number of windows, all the while missing the fact that you don't have any aircraft.

No, the point I'm making is that there are conflicting definitions of proof here. The same word means different things.

If two crypto bros go to court, the one who refuses to follow the judges ruling gets to stay in jail. How does crypto help with that? Possession is 9/10ths, but the state is 100%.

If you're doing it seriously you'll create plots in RAM and your SSDs will be fine.

Still a waste of space, but a gentler one.

Why can't they use Folding@Home as proof of something?

For something to be used in PoW, it needs to be both hard to solve and easy to verify. Protein folding checks the first box but not the second. As an example, prime number calculations check the both boxes and there's Primecoin which bases its proof algorithm on prime chains.

This only applies if you prefer to be trustless though. If you are willing shed a bit, you can trust Folding@Home's scoreboard data (and whoever supplies the data to you) to use it as a pseudo-"PoW" and run a proper proof algorithm beneath. (which is also done by a number of coins like Banano, Curecoin, Gridcoin. You can see their people on the leaderboard with weird names like x_ALL_x, x_GRC_x or just bunch of non-sensical alphanumeric characters.)

Do you know PoW could work with backup data? It would be great if people could have a little backup space for the price of some compute, instead of cash or "free" backup.

Are you looking for Sia and Storj?

I would say Filecoin but very strange things are going on with that coin and I can't wrap my head around how you're supposed to actually store things.

Well, I used F@H as an example. Are there truly no PoW algorithms that satisfy both conditions and actually represent some useful IRL result?

It's a pretty narrow set of calculations that are very expensive but easy to verify. You have to be looking for needles in a haystack, and the haystack has to be gargantuan but not involve all that much data transfer.

So unless you feel that it's useful to find prime numbers, good luck.

The more "useful" you PoW algorithm is, the cheaper 51% attacks on your chain will be. To be maximally secure, a PoW algorithm should only generate useless data.