As a context, my advisor recently lent me the book Uncle Petros & Goldbach's Conjecture to read, a story about a man obsessed trying to prove/disprove the Goldbach conjecture.
I was searching around for some information on Goldbach's conjecture, and I directly encountered Matan Cohen's proof for the conjecture here (literally the first link proposed by Google, as well as by Google Scholar if selecting "since 2016"). I assume the proof is incorrect since it is published as a preprint since 2018, but not published in a conference or journal since. Also the conjecture has not been marked as solved.
My question is : Is this proof discussed anywhere, and in particular, are the errors pointed out?
I also see the first mistake in Theorem 1.1 but there are some other issues.
One of them is that the arrangement of the natural numbers in a tree and the extra terminology is not really necessary, and with the bad notation it just makes it harder to read.
The author is just considering an even number $k$ which is between two consecutive powers of $2$, say, $2^n$ and $2^{n+1}$. The number defined as the offset of $k$ is just $A=k-2^{n}+1$ and the difference $B$, which in the tree can also be seen as the number of vertices at the left of $k$ not counting $k$, is just $B=2^{n+1}-k-1$.
This gives us a way to write formulas for $A,B$, just $$A=k-2^{\lfloor\log_2 k\rfloor}+1,B=2^{\lfloor\log_2 k\rfloor+1}-k-1,$$ so we can test computationally for each counterexample. The smallest counterexample occurs when $k=30$. In this case $A=15$ and $B=1$. They sum $2^4$ and none of them is prime. Also $A+B=2^3+2^3$. Other counterexamples are $(9,55),(15,49),(25,39),$
The counterexample $A=39,B=25$ appears when $k=102$ or when $k=88$ with $A=25,B=39$. There is no counterexample between $88$ and $102$ so that's a twin one, and they appear in some kind of symmetry, since the next counterexamples are $(49,15),(55,9)$.
Theorem 1.2 is also false. It says
Even in the case when one of the $A,B$ is prime it can fail. Take $k=18$. Then $A=3,B=13$.
Both $A$ and $B$ are prime, but $k-A=15$ is not prime. In this case, $k-B=5$ is prime, but in the case $k=22$ only $A=7$ is prime ($B=9$), and $22-7=15$ is not prime.
About Lemma 2. I completely didn't get the proof of it. It seems to attempt to prove that every even number is a sum of four primes (which is proven to be true for big enough even numbers) but then it says that since $2m=2^x+p_1+p_2$ for $p_1,p_2$ primes, is a sum of four primes then $2^x$ is the sum of two primes, which I think he does by substracting $p_1$ and $p_2$, I think, which would be absurd: If $2m=p_3+p_4+p_5+p_6$ then $$2^x=p_3+p_4+p_5+p_6-p_1-p_2$$
and that's it. You can't say more unless $\{p_1,p_2\}\subseteq \{p_3,p_4,p_5,p_6\}$ which doesn't always hold.
Lastly, proposition 1 seems like a complete overcomplication of something which would be immediate if both Lemmas were correctly proven. But the author manages to mistake even here, by saying that
while the even numbers greater than $2$ do not form a group.
So... yes, the author should at least have tested it computationally because there are lots of counterexamples. It seems really half-baked, though (More when, after searching, you find that the author claims to be a mathematician AND programmer). Like a bad attempt to confuse your advisor.