I'm a bit confused about the topic. It made me think that someone had managed to extend the Incompleteness Theorems to a lower level.
And the introductory paragraph is unacceptable. It is confusing the fundamental concept of proof with the human-based process of presenting a proof. It seems to have been written solely for shock value.
If there is a proof of twin primes, then it exists whether or not a person makes a mistake in presenting it. When the proof of Fermat's Greater Theorem was first presented, it also had a flaw. But they found it, retracted the proof, fixed it, and represented it. The particular proof of twin primes may not be as amenable to adjustment, but that is no reason to make it seem as if the situation is hopeless.
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Rrhain
WWJD? JWRTFM!