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.
Okay, but so what?
Bitcoin also didn't solve P ?= NP or any number of other problems.
And unless I'm mistaken, Satoshi did not say that it did solve the Byzantine Generals problem, especially in the specific manner that problem is formulated (with discrete General-actors, something that doesn't even exist in Bitcoin at all). At best there is a rough similarity. Correction: Satoshi did say it was a solution in
this email. But again, he formulated in a very careful manner, stating that each general has a laptop, which ends up making "majority of CPU power" equivalent to a majority of discrete General-actors.
He said exactly what it does solve. If a majority of the CPU power is not conspiring to attack the network, then it reaches consensus that is final and secure (though slowly in the case close to 50%).
It is up you as a prospective user or investor to decide whether "a majority of the CPU power" is an acceptable requirement. It seems at this point there isn't anything better, and some number of people think it is useful (most of the world does not).
Because as I explained to monsterer in the prior post, Satoshi's design is ambiguous about Byzantine faults such as censoring transactions and thus it does not solve the BGP.
And because I assert there are other ways to organize a PoW block chain design so that some of those faults can be objectively identified and reacted to (e.g. the fault of censoring transactions). The fact that in Satoshi's design these faults can't even be objectively identified (and the Sybil attack on pools is another one that destroys any objectivity), then there is no recourse other than for the system to centralize and fail.
(and the linked data doesn't even account for the Sybil attack we can't see).