I don't understand the busy beaver stuff, but is this argument similar:

On a 32-bit computer, there are 2^32 bytes of memory or 256^(2^32)=2^34359738368 possible states. A program is something that takes you from one memory state to another and you want to figure out if the state transitions form a cycle?

Similar idea, but with Turing machines rather than physical computers.

Simplified a bit, the n-th busy beaver number is the number of 1s which can be written out by an n-state Turing machine.

For 2-symbols and n-states, the sequence goes: 6, 21, 107, >= 47176870, > 7.4×10^36534, > 10^10^10^10^18705353 — and those inequalities are because the numbers themselves cannot be computed directly, you have to run the program to see what it does.

https://en.wikipedia.org/wiki/Busy_beaver

I think this approach quickly enters the realm of transcomputational problems [1], which, given the limits of this universe is equivalent to "undecidable" for all practical considerations.

[1] https://en.wikipedia.org/wiki/Transcomputational_problem

Busy-beaver only grows fast given unlimited memory. On a finite machine it’s limited by the total number of states the device has. Granted that can be 10^1,000,000,000,000+ these days, but it’s still limited so BB(10^10) on actual hardware = BB(10^100).

If you somehow know BB numbers, you know the size of finite memory, which is sufficient to run corresponding busy-beaver. Limiting mathematics to what is physically possible is not useful. We don't know what is ultimately physically possible.

We have some upper bounds on what is physically possible: https://arxiv.org/abs/quant-ph/9908043

Wait a minute, does this paper include various hypercomputers or is it only about conventional computation?

Although I personally believe hypercomputers are not physically possible, I'm not aware of any general proof that they are physically impossible. I've seen papers that suggest physical models for hypercomputers. Notice that there is a hierarchy of hypercomputers of varying powers and some of them can solve the Halting Problem.

The paper includes quantum computers, it does not include computers that exploit CTCs or similar things whose existence we can only speculate upon. Note however that CTCs don't give you hypercomputation.

Do you have a link discussing physical models for hypercomputers?

Most papers criticize the idea and describe physical limits. I don't remember which one on the physical possibility of hypercomputation I read (a long time ago), but there are some papers such as the following:

https://oronshagrir.huji.ac.il/sites/default/files/oronshagr...

https://link.springer.com/article/10.1007/s11047-009-9114-3

https://sci-hub.do/10.1142/S0129626412400105

An overview is in A. Syropoulos: Hypercomputation: Computing Beyond the Church-Turing Barrier. Springer 2008, chapters 8-9.

Gotta love the subsection headline in the third paper mentioned above: "A CTC Based Relativistic Hypercomputer That Actually Works Without Problems." ;-)

Busy beavers have an infinite tape to work with, so that's a pretty big difference.