// Simplified Linz Ĥ (Linz:1990:319)
void P(u32 x)
{ u32 Input_Halts = H(x, x);
if (Input_Halts)
HERE: goto HERE;
}

int main()
{ u32 Input_Halts = H((u32)P, (u32)P);
Output("Input_Halts = ", Input_Halts);
}

When the simulation of the Turing machine description ⟨P⟩ of a Turing
machine P on input I never halts we know that P(I) never halts.

Simulating halt deciders must abort their simulation of all inputs where
the pure simulation of this input would never halt.

Simulating halt deciders act as a pure simulators of their input until
this input demonstrates non-halting behavior.

This allows simulating halt deciders to totally ignore their own
behavior in making their halt status decision.

It is this feature of the adapted halt deciding criteria that eliminates
the pathological self-reference (Olcott 2004) from the halting problem
counter-example templates.

Simulating halt deciders never have behavior that effects their halt
status decision because they only act as pure simulators until after
they have made this decision.

Using the above reasoning we can know that when the execution trace of P
on input P shows that there is no code in P that escapes the infinitely
nested simulation of P on input P, then we know that P on input P meets
the definition of a computation that never halts: (a pure simulation
that never halts).

https://www.researchgate.net/publication/351947980_Halting_problem_undecidability_and_infinitely_nested_simulation

--
Copyright 2021 Pete Olcott

"Great spirits have always encountered violent opposition from mediocre
minds." Einstein