"Can you point to some of the evidence you used to arrive at the conclusion that the high-tech civ is centered in the core, or that they're gating civilizations entering it?"
A high-tech civilization in the core is IIRC spelled out in A Deepness in the Sky because the planet there comes from the core. On its own that means nothing, but combined with A Fire upon the Deep it means there's more down there than the Upper Beyond realizes.
Gating civilizations entering it is a bit of my own interpolation. One alternative is that powers simply die out after 10 years because there is some reason they need to advance to the next level, but they're locked out by the Zones.
"BOTH the speed of computation AND the laws of physics (whether FTL travel and communication are physically possible)"
My read is that that is cause and effect, though, not two effects. The amount of computation you are allowed is limited; super high tech becomes impossible, then FTL, then our tech, then even any system that performs simple and meaningful mechanical computations. Even a car engine can be viewed as performing computations to stay running.
This is a particularly sci-fi element. I don't think it's immediately obvious how to separate the universe into "meaningful" computation and the sort of computation a stationary rock is "performing" as its atoms jiggle around, such that one can distinguish at the level of the laws of physics themselves between a rock and a car engine or other simple mechanical device. Then again, I'm not a superintelligent civilization with nearly unbounded resources and deeper access to the laws of physics than I have. (Arguably, under the circumstances, it could simply be an enormous computation of its own, in which they aren't doing it via clever rewrites of the universe, but basically by executive fiat, or, to put it another way, the standard may literally be "I know it when I see it" for some sufficiently intelligent "judge", rather than any sort of physical law.)