The stated problem bounds do not include being able to tell whether someone controls >50% of the hash rate. That isn't in the paper at all. The wording of the paper is "As long as a majority of CPU power is controlled by nodes that are not cooperating to attack the network". It doesn't matter whether they cooperate via pools or otherwise, either way it is outside the bounds.
Without considering the Sybil attack, then one isn't solving the Byzantine fault issue, i.e. isn't solving the Byzantine Generals problem (which is the correct title of this thread). Just because Satoshi failed to mention that he hadn't solved what he was implying to have solved, doesn't make that just having a majority of the hashrate is the only consideration in a PoW solution to the Byzantine Generals problem.
There is no Sybil attack possible on the problem
as stated. "A majority of CPU power" is a physical quantity which can't be Sybil attacked. Period.
The Byzantine Generals problem does not state "A majority of CPU power" as the problem. I already stated that is Satoshi's requirement but as the correct title of this thread points out, Satoshi's stated requirement is not a solution to the Byzantine Generals problem. Period.
One of the attack vectors in solving the Byzantine Generals is the Sybil attack. The Byzantine Generals problem is all about the need to trust that 2/3 of the generals are loyal without centralization where all generals are the same person, i.e. that there is no Sybil attack.
Anyone who has studied all the variants of consensus algorithms (as I have) will know clearly that Sybil attacks are always resolved via centralization of the protocol.
This is why as I looked for an improvement over all of what has already been tried, I was cognizant of that I would need to accept centralization in some aspect and so I began to look for the possibility of controlling centralization with decentralization, i.e. a separation of orthogonal concerns which is often how paradigm shifts arise to solve intractable design challenges.
Every consensus design creates centralization. This will always be unavoidable due to the CAP theorem. The key in my mind is to select carefully where that centralization should be.
- Satoshi's PoW consensus design centralizes because a) SHA256 has orders-of-magnitude lower electrical cost on ASICs, b) full nodes must centralize (maximize pooled hashrate) to win the battle over who will have the most profitable verification costs (which can be accomplished with a Sybil attack), and c) variance of block rewards require maximizing pooled hashrate (at least up to double-digit percentages and Sybil attack incentives kick in from there).
- Stellar's SCP consensus design centralizes because although it can't diverge, it requires that slices are not Sybil attacked to avoid eternal preemption (being jammed stuck forever).
- Ripple's consensus algorithm diverges unless it is centralized trust, as confirmed by Stellar's divergence before it switched to the SCP algorithm.
- Iota's (any DAG's) consensus diverges unless centralization can force the mathematical model that payers and recipients encode in their interaction with the system.
- Ethereum never solved the issue that verification of long running scripts can't be decentralized. They are now off another deadend tangent (consensus-by-betting, Casper, shards) trying to deny the CAP theorem.
- PoS is centralization.
Extracting the generative essence of an issue is what I do. That is where I have made my career in the past and will do so again.