This sounds an awful lot like "God of the gaps" argumentation with "non-Turing hypercomputation" substituting for "God". If the computational approach doesn't work — that's all of human reason, to emphasize: any symbols you can scribble on paper and reason about over coffee, that's a finite algorithmic process — then there's no humanly-achievable endeavor that can solve the problem. It's forever hiding in the gaps, the shadows; it's definitionally unknowable.
So what field of science are we discussing right now? One of the more frivolous branches of philosophy: talking about something that we can't define, can't reason about, won't ever be able to reason about; something which marks no experimental evidence in our human sciences. Therefore: it doesn't exist. Apply your favorite razor. Something we can't possibly talk about, let's not talk about. Something whose existence or non-existence are indistinguishable, which leaves no shadow in the observable universe: it does not exist.
It's not a "God of the gaps" argument because I'm not claiming that "God" (that is, incomputable physics) exists. I'm simply pointing out that the fact that our entire understanding of physics is built on a computational framework doesn't actually say anything about the Universe, merely about the way we do physics. There is a priori no reason to expect that the Universe would be in any way amenable to human understanding. We have just found that in some cases, it is.
And non-computability doesn't make something philosophical or mystical. 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.
- "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.
As someone who works in a custom fabrication shop and who has been research job shop scheduling for over a decade as an attempt to get better at it, it's a computationally impossible task. There are huge numbers of "optimal" ways to schedule the shop and all of them require enormous amounts of computation. Genetic algorithms are offering the most promise right now, but the schedule has to be constantly updated because things never take as long as you think they will.
This is primarily because, unlike manufacturing, the basic unit of our system is a person, with all of their chaotic outputs. You start stacking those up and you end up with "long, fat tails" in every analysis you do. Your probabilities are spread almost flat over a broad range of outcomes.
It's fascinating and definitely falls into the category of uncomputable.
So what field of science are we discussing right now? One of the more frivolous branches of philosophy: talking about something that we can't define, can't reason about, won't ever be able to reason about; something which marks no experimental evidence in our human sciences. Therefore: it doesn't exist. Apply your favorite razor. Something we can't possibly talk about, let's not talk about. Something whose existence or non-existence are indistinguishable, which leaves no shadow in the observable universe: it does not exist.