@MicrmSo you are assuming it is false now?
So the logic of your proof is:
Assume twin prime conjecture is false.
Then there is one prime doing the last elimination.
If there is such a prime, then the conjecture is false.
Therefore it's true.
I think that you are not quite following what his argument boils down to.
I think what he originally intended (but did not say) was this syllogism:
(1) If there exist unique integers m and u such that for any integer n $\ge$ m > 0 u either (6n - 1)/u or (6n + 1)/u is an integer, then the conjecture is false.
(2) No such m and u exist.
(3) Therefore the conjecture is true.
The first proposition is self-evidently true because it would entail that at least one member of each 6n - 1 and 6n + 1 pair was composite. I suspect that the second proposition is also true but did not attempt a proof because even if both propositions 1 and 2 are true, proposition 3 does not follow as a consequence. The syllogism is a well known fallacy.
I may have misunderstood what he was originally trying to say. In any case, he seems now to have revised his argument.
(1a) Only if there exists unique integers m and u such that, for any integer n $\ge$ m, either (6n - 1)/u or (6n + 1)/u is an integer will the conjecture be false.
(2a) No such m and u exist.
(3a) Therefore the conjecture is true.
THAT is a valid syllogism. But proposition (1a) is no longer self-evidently true, and no proof has been forthcoming except hand-waving about "last" and "final" in the context of infinite sets. In fact, I suspect proposition 1a is false.