Ĥ.q0 wM ⊢* Ĥ.qx wM wM ⊢* Ĥ.qy ∞
Ĥ.q0 wM ⊢* Ĥ.qx wM wM ⊢* Ĥ.qn

The above is adapted from (Linz:1990:319).
It shows that Turing machine Ĥ copies its input at (q0) and begins
executing an embedded copy of the original halt decider with this input
at (qx).

The (qy) state indicates that the halt decider decides that its input
would halt. The ((qn)) state indicates the input would not halt. The
appended (qa) and (qb) states cause Ĥ to infinitely loop if the halt
decider decides that its input would halt.

Linz, Peter 1990. An Introduction to Formal Languages and Automata.
Lexington/Toronto: D. C. Heath and Company.

http://www.liarparadox.org/Peter_Linz_HP(Pages_315-320).pdf

*Three irrefutable truisms lead to one conclusion*

Truism(1a)
The simulation of [Ĥ] by its simulating halt decider Ĥ would never halt
unless this simulation is aborted.

Truism(1b)
(1a) is another way of saying that when the copy of H at Ĥ.qx only
simulates its input [Ĥ] its input never halts.

Any input running under a simulating halt decider that never aborts its
input will have the same behavior as an input run under a simulator.

This [simulating halt decider] is not decider in the technical sense,
because it never halts. The fact that it never halts proves that its
input is not halting.

Truism(2)
Every simulation of input P that would never halt unless simulating halt
decider H stopped simulating it <is> a non-halting computation. This
remains true even after H stops simulating P.

I am merely generalizing the the halt deciding principle that overcomes
the pathological self-reference error of the halting theorem.

Truism(3)
Ĥ([Ĥ]) Halts because it aborted the simulation of its input.

Conclusion:
Ĥ is a halting computation and its input is not a halting computation
therefore Ĥ decides its input [Ĥ] as non-halting correctly.

http://www.liarparadox.org/Halting_problem_undecidability_and_infinitely_nested_simulation.pdf

--
Copyright 2021 Pete Olcott

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

["Followup-To:" header set to comp.theory.]
On 2021-05-20, olcott <NoOne@NoWhere.com> wrote:
> Ĥ.q0 wM ⊢* Ĥ.qx wM wM ⊢* Ĥ.qy ∞
> Ĥ.q0 wM ⊢* Ĥ.qx wM wM ⊢* Ĥ.qn
>
> The above is adapted from (Linz:1990:319).
> It shows that Turing machine Ĥ copies its input at (q0) and begins
> executing an embedded copy of the original halt decider with this input
> at (qx).
>
> The (qy) state indicates that the halt decider decides that its input
> would halt. The ((qn)) state indicates the input would not halt. The
> appended (qa) and (qb) states cause Ĥ to infinitely loop if the halt
> decider decides that its input would halt.
>
> Linz, Peter 1990. An Introduction to Formal Languages and Automata.
> Lexington/Toronto: D. C. Heath and Company.
>
> http://www.liarparadox.org/Peter_Linz_HP(Pages_315-320).pdf
>
>
>
> *Three irrefutable truisms lead to one conclusion*
>
> Truism(1a)
> The simulation of [Ĥ] by its simulating halt decider Ĥ would never halt
> unless this simulation is aborted.

Corrections.

Firstly, you made a typo: the halt decider is H, not H-hat.

Secondly, the truth is more general:

"The simulation of [Ĥ] by *ANY* simulating halt decider would never
halt unless this simulation is aborted."

When the simulating decider is H, then ... the simulation is NOT
aborted. So this truism specifically does not apply to H in any useful
way.

When we apply the truism specifically to H, we get a VACUOUS TRUTH.

> Truism(1b)
> (1a) is another way of saying that when the copy of H at Ĥ.qx only
> simulates its input [Ĥ] its input never halts.

That "when" is always. H only simulates its input.

If you want to talk about some other thing that simulates its input
with abort checks ... you must not call it H.

> This [simulating halt decider] is not decider in the technical sense,
> because it never halts. The fact that it never halts proves that its
> input is not halting.

(Not really; a really broken decider could fail to halt on a halting
input.)

> Truism(3)
> Ĥ([Ĥ]) Halts because it aborted the simulation of its input.

False; Ĥ([Ĥ]) does not halt. Ĥ embeds H, and H is a non-aborting,
simulate-only, run-to-completion machine.