@Acccumulation That's switching the computation model (including randomization), IMHO in a not helpful way. Yes, brute force always works but is prohibitive. The problem with easy instances is that given the public information, deterministic solvers can find the key quickly. That would make the best case easier than guessing, which we should try to avoid.

@User584322 Why don't you try it out? And which step do you suspect to loop?

@theprogrammer Good point. I probably intended it to be vertex count, but input size would also work for the argument.

@Stef True. The principles carry over: you can talk about average case (weighted sum over all computations) and expected case (potentially different probabilities per computation).

"All that I have to do to refute these conventional proofs" -- is that your goal? Because it's utterly unclear from your posts what you're trying to achieve there. "how this "impossible" counter-example is correctly decided" -- which you simply haven't done; all you've done is identified a class of functions that can't be $H$. (or tried to, at least).

Ah, we're getting closer. "The undecidability of the halting problem is conventionally proved with a single counter-example." -- nope! The conventional proof by contradiction constructs infinitely many counter-examples, one for each candidate for $H$ (using the terminology from Linz). Well, zero counter-examples, if you're exact, since no $H$ actually fulfills the criterion, but by assumption there would be infinitely many such $H$.

"When H recognizes what would be infinitely nested simulation it aborts its simulation of this input and halts" -- No. You don't know anything about H.

By the way: even if you managed to come up with some machine that can solve its own halting problem, that does not solve the halting problem.

There is no material here to spend more time on. I'm sorry you wasted all that time.

You can not prove or disprove the halting problem with a single example.

No: cs.stackexchange.com/questions/141080/… Hard as it is, at some point you'll have to reconsider the idea that you acually haven't understood the proof.