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From: olcott <polcott2@gmail.com>
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Date: Sat, 9 Mar 2024 12:33:50 -0600
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*Verified fact that Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ and H ⟨Ĥ⟩ ⟨Ĥ⟩ have different behavior*

Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqy ∞ // Ĥ applied to ⟨Ĥ⟩ halts
Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqn   // Ĥ applied to ⟨Ĥ⟩ does not halt

Execution trace of Ĥ applied to ⟨Ĥ⟩
(a) Ĥ.q0 The input ⟨Ĥ⟩ is copied then transitions to Ĥ.H
(b) Ĥ.H applied ⟨Ĥ⟩ ⟨Ĥ⟩ (input and copy) simulates ⟨Ĥ⟩ applied to ⟨Ĥ⟩
(c) which begins at its own simulated ⟨Ĥ.q0⟩ to repeat the process
*This proves that Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ must abort its simulation*

*This is a verified fact*
When simulating halt deciders always report on the behavior of
their simulated input from their own POV then when Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩
transitions to Ĥ.Hqn it is correct from its own POV.

*This is a verified fact*
When that occurs then H ⟨Ĥ⟩ ⟨Ĥ⟩ would transition to H.qy from
its own POV.

When Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ and H ⟨Ĥ⟩ ⟨Ĥ⟩ report on the basis of their own
POV then Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ reports incorrectly about the behavior of
Ĥ ⟨Ĥ⟩ and H ⟨Ĥ⟩ ⟨Ĥ⟩ reports the behavior of Ĥ ⟨Ĥ⟩ correctly.

*Verified facts*
(a) It is a verified fact that Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ must abort the
simulation of its input to prevent its own infinite execution.

(b) It is a verified fact when Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ can see that it
must abort its simulation then it would transition to Ĥ.Hqn
to reject this input as non-halting from its own POV.

(c) It is a verified fact when Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ cannot see that it
must abort its simulation then it would transition to Ĥ.Hqy
and loop.

(d) (b) gives H ⟨Ĥ⟩ ⟨Ĥ⟩ the basis to transition to H.qy.
(e) (c) gives H ⟨Ĥ⟩ ⟨Ĥ⟩ the basis to transition to H.qn.

Because Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ and H ⟨Ĥ⟩ ⟨Ĥ⟩ seem to be identical machines
on identical input that have different behavior we must
somehow explain how they are not identical machines with
identical inputs.



-- 
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer