Goldbach's Conjecture for $n \ge 30$

This is Goldbach's Conjecture restricted to even integers $n \ge 30$.

For every natural number $n \ge 30$ with $n$ even, there exist primes $p$ and $q$ with

The full Goldbach's Conjecture (every even $n > 2$ is a sum of two primes) splits naturally into a finite range, $4 \le n < 30$, which can be checked directly by exhibiting explicit prime pairs for each of the finitely many even values, and this cofinite range $n \ge 30$, which carries all of the conjecture's genuinely open, asymptotic content. This lemma isolates that open remainder so it can be attacked and reused independently of the trivial finite case check.

Formalization Note The threshold $30$ is an arbitrary but convenient cutoff: any finite set of small even numbers is decidable by exhibiting explicit witnesses, so shifting the threshold higher only changes how much of the finite case-check is absorbed into the base case, not the mathematical content of this lemma.