Variant of Subset-sum has an $O(1)$ algorithm if $Goldbach$ is true

Question

Given $S$ of positive integers $>$ $1$ is there some combination with even $SUM$ > $2$ that is NOT the sum of two primes?

$SUM$ = 10

$S$ = $[4,6]$

$No$,

Sum of Two Primes $5 + 5 = 10$.

Combination $4+6=10$

It has an $O(1)$ algorithm if Goldbach is True (always output $NO$). Otherwise, it would seem to be NP-complete because it would require solving Subset-Sum.

Question

Will a many-one-reduction from $Subset-sum$ work for this decision problem in poly-time?

Answer 1

If there is a finite number of exceptions to the Goldbach conjecture then it is still solvable in linear time.

Note that there is a non-zero probability that the Goldbach conjecture is false. Heuristically, the probability for an infinite number of exceptions is zero.