Quantitative version: let f(N) be the least value of |p-q| such that

2N=p+q with p,q prime. [If no such p,q exist then let f(N)=infinity.]

Has there been computational effort devoted to determining the behavior of f(N)?

I presume/guess that

f(N)=(logN)^(3+o(1))

is a valid stronger version of Goldbach conjecture.

If all record-breaking f(N) were computed for N=1..10^12

then we'd have some evidence on this question.