You may have a point with the random bytes. That is as close to worthless as makes no difference.
But regarding your "Bitcoin Prime": your system isn't backed by proof of work. My point is that bitcoins are not just a set of random bytes, or even a chain of signed transactions. They are also secured by a proof of work chain, and without this they are not bitcoins, by definition.
So even though you can come up with a set of bitcoin primes, one for each bitcoin, there's no proof of work backing your system. You can't reuse the proof of work that backs bitcoins without knowing all of the private keys, so you'd have to start from scratch.
But regarding your "Bitcoin Prime": your system isn't backed by proof of work. My point is that bitcoins are not just a set of random bytes, or even a chain of signed transactions. They are also secured by a proof of work chain, and without this they are not bitcoins, by definition.
So even though you can come up with a set of bitcoin primes, one for each bitcoin, there's no proof of work backing your system. You can't reuse the proof of work that backs bitcoins without knowing all of the private keys, so you'd have to start from scratch.