>No, because there is no plausible computational principle giving the answer to the Busy Beaver problem embodied in that system. Notably, it's a turing machine.

Yes, there is no plausible computational principle giving the answer to the Busy Beaver problem in any system, because it is not computable. The point was that you can't trust a machine simply because it produces the known answers correctly and keeps going the same way.

>Doing so would not necessarily involve infinite dimensions - you are deriving those from the behavior of Turing machines.

You need an infinite resource available to get anywhere past finite automata, you just don't have to actually use infinite resources until you get past TMs. Non-automatic models of computation aren't relevant to measurable behavior of dynamical systems in the real world.

>That's a great argument if the universe is Turing-equivalent. That may be the case, but how to prove it?

No, it isn't. It's an observation that 'effective' has a standard definition which precludes hypercomputation. Any effective computation is simulable by a Turing machine; that's not the physical Church-Turing thesis, it's the vanilla version. (edit: and the reason I put the word in there originally is that any AGI implemented with computers would be in that boat, while many models AGI theorists prefer would not be, but are intended to represent real-world systems that would. Thus, they are often claiming to have effective means to non-effective ends.)

>An example non-Turing universe is one where Real numbers are physical, I.e. it is fundamentally non-discretizable. I have no idea if that would be sufficient to allow hypercomputation, but it breaks the TM isomorphism.

This is an example of falling back on infinite information in finite space. If space is continuous, it contains all the uncomputable reals. If you doubt this requires infinite information, consider that these include the incompressible strings of infinite length. A system moving through such a space would adopt states that require infinite information to describe infinitely often. It still wouldn't allow us to show any hypercomputation, though; our ability to observe and communicate remains finite, and all finite observations are explicable by finite machines, well within computability.