Can we use the proof of the weak Goldbach conjecture to also prove the strong Goldbach conjecture

Question

Why doesn't proof of the weak Goldbach conjecture also prove the strong Goldbach conjecture?

Actually I am referring to this link. My question is why the logic used in this question cannot be used to constitute a formal proof of the strong Goldbach conjecture? The comments made on that question were not very clear for understanding so I am asking this again and expecting a detailed and better explanation. I am willing to refine this question even more, if required.

EDIT

Thanks a lot @lulu for pointing that out. I concur that it's very vague and not a formal proof according to mathematical standards. But I would counter that as follows:

Here "O" represents any odd number greater than 5 and we are considering "a" to be an odd prime number (such "a" always exists in pair of three primes whose sum is an odd number).

Lowest possible value of "a" is 3 because below that we have the only even prime number i.e. 2 Then no matter what value the odd number "O" takes greater than 5 if we subtract 3 from it we get an even number.

Now the next odd number after 5 is 7 and subtracting 3 we get 7 – 3 = 4 (This is the smallest supporting example of the proof).

Then no matter what values "O" and "a" take as long as "O" is odd greater than 5 and "a" is odd prime greater than 2 their subtraction i.e. "O – a" will always result in an even number greater than or equal to 4.

And this is exactly what the strong Goldbach conjecture states – every even natural number "greater than 2" is the sum of two prime numbers.

Answer 1

Consider this list of numbers (known as Lourrran's numbers) : $O, 3, 5, 7, 11, 15, 23, 25,$ ... and all odd numbers greater than $25$.

Weak conjecture : Every odd number $n$ greater than $3$ can be written as a sum of 3 Lourrran's numbers.

Easy to check that this conjecture is correct.

Strong conjecture : Every even number greater than $6$ can be written as a sum of 2 Lourrran's numbers

Easy to check that this conjecture is false, there is no couple $(a,b)$ of Lourrran's numbers such that $a+b=24$.

Edit :

Of course, I have created those L-numbers just for this message.

You say :

when we have a suite of odd numbers (Primes numbers without 2 in your case),

knowing that any odd number can be written as a sum of 3 numbers from this suite

should imply that :

any even number can be written as a sum of 2 numbers from this suite.

I show that :

Based on a suite of odd numbers (see L-numbers),

This suite matches the first predicate (any odd number can be written as a sum of 3 L-numbers)

and this suite does not match the 2nd predicate.

So what you say is wrong.