> This is a simple restatement of the original claim, and it is false. The program being discussed is subject to the Turing Halting Problem. The program being tested, the same. Lean, the prover and the final authority, the same. All are subject to this fundamental limitation.

I disagree, which is why I added all of the additional commentary you appear to have ignored to restate your original claim instead. I'm not sure why I should restate my response to these points when they're still available and awaiting response above.

> How the original author chose to express himself is not my problem, it is his. I have the simple responsibility to take him at his word. Anything else would be disrespectful.

When someone's words on a project as complicated as this seems to violate the basic foundations of computer science it's both your problem & disrespectful to claim you know for certain the problem is because it's a common beginners mistake. This may be something we cannot come to an agreement on morally, but I suppose it won't really matter for the rest of the mathematical conversation which continues below.

> I didn't define it, Alan Turing did, in 1936. It's not a debating point, it's a fundamental limitation. All Turing-complete code sources have this limitation. Read more here: https://en.wikipedia.org/wiki/Halting_problem .

Wikipedia is a poor source to cite, but when I follow it I see no claim or definition by Turing for what non-trivial is. I see claim of what Rice meant by non-trivial in their eponymous theorem in 1951, but that's neither from 1936 nor a relevant definition for the current discussion so I must assume you mean somewhere else in Turing's actual paper I'd need to check.

Which takes us to the actual 1936 paper rather than Wikipedia's summary https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf. I see 3 mentions of triviality, none of which appear to give a definition of what a non-trivial provably haltable example is:

1. Discussion of the remainder of the theorem itself being trivial [on page 31 of his paper, page 260 of the journal]

2. Since CC_0 is already been shown provable the conditional proof of the A(M)->CC_0 is trivial by the rules of implication [p32, 261 of the journal]

3. A trivial replacement of the variable naming scheme allows translation between the two notations without changing the calculus of them. [p34, 263 of the journal]

None of these seem to define what a non-trivially provable program (Turing Machine/Algorithm) is in context of the halting problem, so I again ask can you tell me where and what actual definition in Turing's actual 1936 paper you are using to define what a non-trivial program is so that I may apply this definition to the current conversation?

As another aside, one of my favorite "simplest" examples of a non-trivially provable program we know never halts despite the general case result of the halting problem:

  for every group of positive integers (a,b,c,n) with n > 2:
    if a^n + b^n = c^n:
      halt

To prove this never halts you have to prove Fermat's last theorem. Which we have, and so we know this must never halt as you iterate infinitely over the positive integers, but it took one of the most complicated mathematical proofs known to show it.

There are certainly definitions of non-trivial where this is still considered trivial, but I'm at a loss to what part of Turing's paper gives such a definition.