the "recoil" ebnds up being received by the centrifuge, already turning, it sends that energy with the spin. so there is a recoil, but it wont be felt
That's just spreading the recoil over mass (through the centrifuge and the carrying body) and time; given the velocities they're talking about, I don't buy into the "you won't feel it" argument.
Consider throwing a baseball: you're essentially doing exactly what this is doing, and you definitely feel the recoil (try throwing a pitch while standing on warm ice sometime). Granted, you feel the recoil
prior to releasing the ball, but you haven't gotten rid of it. For the rate of fire this thing is supposed to have, the ball is going to be picked up, accelerated (generating the recoil), and thrown so fast that the recoil might as well be upon firing.
Now, if you pre-load the disk with all the rounds you're going to fire, then spin it up to speed, you'll be generating recoil along all vectors at once, with a net force of zero, which is what I'm assuming you're saying. In that case, you're right, there'd be no effective recoil. But if they're talking about thousands of consecutive rounds, there's no way you can achieve that. There has to be a feed system putting balls on the disk, at which point they're transitioning from stationary (relative to the weapon) to high velocity (again, relative to the weapon) in a short time frame, generating effective recoil.
Although, as I think about it, I suppose you could solve this problem by loading balls in pairs at diametrically opposed points on the disk. Hrm. Of course, you'd have to also fire them in pairs, or you're going to end up with recoil (albeit on a weird vector compared to conventional arms) when the centripetal force of one round goes away and the centripetal force of the other doesn't, which should lead to a net acceleration of the whole.
In any event, the system is generating net acceleration along a given vector when it fires, you simply cannot avoid the fact that there's a matching compensatory net acceleration. This doesn't change just because it's spinning, or you've invented a jetless drive.