Must a decision problem in $NP$ have a complement in $Co-NP$, if I can verify the solutions to in polynomial-time?

Goldbach’s Conjecture says every even integer $ >$ $ 2$ can be expressed as the sum of two primes.

Let’s say $ N$ is our input and its $ 10$ . Which is an integer > 2 and is not odd.


1.Create list of numbers from $ 1,to~N$

2.Use prime-testing algorithm for creating a second list of prime numbers

3.Use my 2_sum solver that allows you to use primes twice that sum up to $ N$

for j in range(list-of-primes)):   if N-(list-of-primes[j]) in list-of-primes:    print('yes')    break 

4.Verify solution efficently

if AKS-primality(N-(list-of-primes[j])):     if AKS-primality(list-of-primes[j]):         print('Solution is correct') 


yes 7 + 3 Solution is correct 


If the conjecture is true, then the answer will always be Yes. Does that mean it can’t be in $ Co-NP$ because the answer is always Yes?