Well, I don't know much about proof of resource (unless you're talking about proof of work?). I see where you're coming from, and it seems logical that prime controllers to be spread throughout the network--I'm assuming you're thinking "after all, they're just nodes, right? And bitcoin's nodes are spread all over the place!".
No not POW i distinctly remember reading something about Proof Of Resource.
There's a big "but" here. If you spread out prime controllers, that means that anyone (including malicious actors) can get their hands on one. What happens when someone who operates one of those prime controllers (or hypothetically, a person holding this operator at gunpoint) decides to commit a double spend (they broadcast two different transactions with the same inputs to the network)? It would be impossible for the prime controllers to come to a consenseus on which transaction is legitimate--they would have to rely on proof-of-work to decide. The reasons why bitcoin has a confirmation time at all is exactly this--proof of work (and proof of stake, to a degree) is the only way to get consenseus without trust. In short, any person operating a prime controller would be able to double-spend before the first confirmation, and it would be impossible to trust zero-confirmation transactions. Because of this, GAW needs to trust prime controllers to not double spend... which means they can't distribute them to the public.
Ok Sounds logical but what if it needs a number of Prime controllers to agree on the transaction.
My biggest problem with the idea of one centralized Prime controller is bandwidth and latency. I don't have numbers ready but i suspect the bitcoin network has a tremendous number of transactions per second that are broadcast . If all those transactions must be transmitted to one central node it would not be instantaneous because some of those networks have high latencies.
It would be impossible for the prime controllers to come to a consenseus on which transaction is legitimate--they would have to rely on proof-of-work to decide.
Now this could be the fallback position if a number of Prime controllers could not verify the transaction they fall back to POW to verify it.
So for my feeling the centralized prime controller could never work.
Sorry for the huge quotes beforehand, but I have to type quick to avoid posts from getting buried.
On your comment about proof of resource: Ahhhh, proof of resource! Just googled that, I vaguely remembered it now that I think back. It's very similar to proof of work, except instead of using CPU power to mine, you use your hard drive space. This doesn't actually make confirmation time faster though, there still will be a set blocktime. (The end result is the same as proof of work)
On your comment about the number of prime controllers to confirm the transaction: Actually, using the total number of prime controllers is great thinking! Satoshi actually mentions this in the original bitcoin whitepaper how using "one node gets one vote" could be used as a solution for getting consenseus--but there's a key problem with it. If anyone can host a prime node, then people can host multiple prime nodes. In fact, people would be able to SPAM prime nodes! (They could host hundreds of nodes on VPS servers for example), or even just host hundreds of nodes on one computer. In short, an attacker would be able to attack the network by simply spamming nodes (or DDOSing existing nodes, or a combination of the two), in order to "swing" the vote in their direction.
GAW could instead only authorize specific IP addresses (or something like that in order to regulate the number of nodes), but then you get another problem--GAW will get exclusive control over which prime controllers are authorized or not--which then means that GAW can create new controllers at will--which then means that GAW can spam nodes--which means that GAW has final control over the system.
On your comment about the proof of work fallback: Yes, proof of work could be used as a fallback--but its sort of confusing, because it looks like GAW will have the only copy of the blockchain. If GAW servers go down, how do POW miners know which block to build on? And if proof of work is used as a fallback, then it sort of defeats the purpose of the system--one guy could just constantly attack the system, forcing the network to fall back on proof of work every block, which negates the "killer app" of hashcoin.
(This post has been edited, but only for spelling and formatting)