Is it correct to refer to hashes found through proof of work as "proof of work numbers"?
I wouldn't say it is incorrect but it isn't exactly correct either. I'd say The phrase "Proof of Work" (PoW) applies to the overall method and not to any specific data item.
I think you could sensibly say that the sufficiently low hash of a block is a kind of proof of the work that is likely to have been done in obtaining it. This only really applies over a statistically significant sample size I guess.
Calling the hash the proof of work number doesn't seem to me to be a particularly useful description. I guess that if you are going to hang that label on a single data item, that is probably the obvious candidate.
Since no one is in charge of bitcoin, there is no one authority who decides what terminology is correct.
The hash function takes three inputs: (1) a pointer to the previous block, (2) identifier of the node doing the computation, and (3) a random (32-bit) one-time whole number.
The data that is hashed does not necessarily include any reference to the node doing the computation. Some mining pools may insert a pool identifier into their coinbase transaction but that is not required at all.
Your list omits a lot of other data that is in the data hashed. It includes other items in the block header, such as a hash of data in the transactions in that block.
Proof of Work requires computing a hash of three components, hash(B, A, W)
I don't recognise that from Satoshi Nakamoto's whitepaper on Bitcoin nor from Adam Black's 2002 whitepaper on his 1997 HashCash proposal.
I suspect that W, the proof of work number, corresponds roughly to the nonce in Bitcoin's block header. But as Pieter Wuille pointed out, the Bitcoin nonce alone is insufficient for that role.
Adam black wrote
The hashcash CPU cost-function computes a token which can be used as a proof-of-work.
Black defines a non-interactive token as T = MINT(s, w) where I believe s is a service name and and w is the amount of work the user will have to expend on average to compute a token.
If you look at Wikipedia's article and others, I think it is fairly common to see reference to a "proof of work algorithm" or a "proof of work system".
I suspect looking for a "proof of work number" in Bitcoin is not especially helpful to people new to Bitcoin who are trying to understand how it works.
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