THEOREM 1: There are numbers k and s and a program A(n,m) satisfying the
following conditions.

1. If A(n,m) halts, then C_n(m) diverges.
2. For all n, C_k(n) = A(n,n) and C_s(n) = Ck(s).
3. A(k,s) halts and for all n, A(s,n) diverges.

Here C_n(*) is a program with index n in some exhaustive enumeration of
all possible programs.

The proof is here.
https://xnewberry.tripod.com/Theorem.pdf

I think I asked this question before, but now the proof is taking
totally different approach.

On Tuesday, July 6, 2021 at 10:23:39 PM UTC-4, Newberry wrote:
> THEOREM 1: There are numbers k and s and a program A(n,m) satisfying the
> following conditions.
>
> 1. If A(n,m) halts, then C_n(m) diverges.
> 2. For all n, C_k(n) = A(n,n) and C_s(n) = Ck(s).
> 3. A(k,s) halts and for all n, A(s,n) diverges.
>
> Here C_n(*) is a program with index n in some exhaustive enumeration of
> all possible programs.
>
> The proof is here.
> https://xnewberry.tripod.com/Theorem.pdf
>
> I think I asked this question before, but now the proof is taking
> totally different approach.

could you perhaps rewrite the proof to make use of the "other diagonal"?
that is, instead of depending on A(n,n), the infinite cantor diagonal,
try A(0,n) A(1, n-1), ... A(n-1, 1), A(n, 0)?

the "other diagonal" has neat finite properties.
for a sample, i use the following array of thirty two 32 bit numbers
to remove the - symbol from an inner loop equation:
num diagovreason [] = { // zero curses upon the idea of aleph null of the other diagonal
(num)4294967294, (num)4294967293, (num)4294967291, (num)4294967287,
(num)4294967279, (num)4294967263, (num)4294967231, (num)4294967167,
(num)4294967039, (num)4294966783, (num)4294966271, (num)4294965247,
(num)4294963199, (num)4294959103, (num)4294950911, (num)4294934527,
(num)4294901759, (num)4294836223, (num)4294705151, (num)4294443007,
(num)4293918719, (num)4292870143, (num)4290772991, (num)4286578687,
(num)4278190079, (num)4261412863, (num)4227858431, (num)4160749567,
(num)4026531839, (num)3758096383, (num)3221225471, (num)2147483647 };