- "We have encountered phenomena beyond the reach of any computation in mathematics and computer science many times. They exist, no matter which "razor" you apply."
The razor applies as to whether the physical universe runs on laws that don't fit in Turing machines. It applies because it's an unfalsifiable proposition. You can't embed a Turing machine in a super-Turing universe, ask it "what is the nature of your cosmos?", and have it determine (within finite computation) there are more things in heaven and earth than it can compute.
(This is the ordinary Church-Turing thesis: if some finite interaction between Turing observer and super-Turing universe convinces the Turing observer of the super-Turingness of its universe, within finite time... then an ordinary Turing machine can also do exactly the same thing, by simulating the first Turing machine, and brute-force enumerating every possible interaction within that finite bound. The nature of super-Turing machines is incomprehensible to Turing machines; there's nothing a super-Turing machine can do to prove it exists to a mere Turing machine).
you know... the entire point of conceptualizing a hyper-Turing machine in the first place, the key difference between a hyper-Turing machine and a Turing machine is the solvability of the Turing machine halting problem.
So shouldn't a Turing machine be able to determine if a hyper-Turing machine exists by presenting that very problem? I.E. problems that take O(n!||n^x) to solve but O(log(n)) or less to verify.
The Church-Turing thesis as you describe it seems to miss this idea, that things can be verified in a different time than it is solved. Is that your interpretation or as it is written?
It does not follow that if a hyper-Turing machine can convince a Turing machine of it's hyper-ness, then the non-hyper-Turing machine is actually hyper to begin with. For reasons stated above. A hyper-turning machine should be able to give proofs to problems verifiable by a turning machine, but that are not solvable within a given step-count/time-frame by that Turing machine.
The interactions between a hyper-Turing machine would be as follows.
```
T:"Yo, solve this!" - some O(n!) "for 10 million inputs"
HT:"done, is x"
T:checks notes "that's right, did you have that saved, that was very fast."
HT:"nope, from your perspective, I might be guessing the answers, you saw me take no steps"
T:"How do I know you aren't just guessing?"
HT:"I'm always right"
T:"How do I know you aren't just always guessing the right answer"
HT:"That's the neat part, you don't."
T:"So I'm going to make a detect-halting program, put it in itself and run it, will it halt"
HT:"It will do " x
detect-halting program does x.
T:"That's pretty strong evidence there... I could never say that"
```
I'm thinking and writing at the same time, so my final thoughts are... if a Hyper-Turing machine is inexplicable to a Turing machine, shouldn't the Turing machine observe the Hyper-Turing machine do inexplicable things, thus revealing itself to the Turing machine?
You're conflating computability with computational complexity; that seems to the source of most of your confusion.
I don't remember the name of the complexity-theory analog of the Church-Turing thesis; but it's an open problem what types of computers physics can support the existence of, how they relate to classical Turing machines in complexity theory. See for example: quantum complexity theory.
About halting: there is provably no way to verify an oracle for the Halting Problem, even if you had one. You can't even ask a hypercomputer to write a proof for you: in general there exist no finite-length proofs.
The razor applies as to whether the physical universe runs on laws that don't fit in Turing machines. It applies because it's an unfalsifiable proposition. You can't embed a Turing machine in a super-Turing universe, ask it "what is the nature of your cosmos?", and have it determine (within finite computation) there are more things in heaven and earth than it can compute.
(This is the ordinary Church-Turing thesis: if some finite interaction between Turing observer and super-Turing universe convinces the Turing observer of the super-Turingness of its universe, within finite time... then an ordinary Turing machine can also do exactly the same thing, by simulating the first Turing machine, and brute-force enumerating every possible interaction within that finite bound. The nature of super-Turing machines is incomprehensible to Turing machines; there's nothing a super-Turing machine can do to prove it exists to a mere Turing machine).