I have seen the following in my observation. I am so sorry, if I am wrong. I do not know how far I am right. But, I feel that, it might be correct. If correct, Please let me know that way to proceed further to prepare a proof.
every number can be expressible either, prime + prime or prime + non prime or non prime + non prime or non prime + prime as shown below. To support GC, I would like to take an example to show these combination. For example: number 10 (>2) and consider that non prime as NP and prime is P.
10 = 1 + 9 or 2 + 8 or 3 + 7 or 4 + 6 or 5 + 5
I am sure there is prime + prime in every even integer.
We can notice that every integer > 2 can be expressible any one of the above form. If P + P is not taken place in the above example, then there is no question of discussion of GC. Now, find the number of primes in fraction. If the fraction is > ¼, then we need not to discuss further proof of GC. No matter what fraction we have taken, we find the same. If the fraction of primes is 1/8, then the probability of finding N + P are 6/16, but we require 8/16 to avoid any P + P sums. If the fraction of primes is 1/10, then the probability of finding N + P are 8/25, but we require 10/25 to avoid any P + P sums. The probability is always strongly in favor of a P + P. So, we can conclude the GC.
Here: n is non prime and p is prime