Arturo´s criterion for additive prime blocks: A necessary condition based on local density for finite Goldbach closure

A necessary condition is presented for the set of odd primes p ≤ n to generate all even numbers up to 2n via sums, without using primes greater than n. We prove that the additive block "closes" only when n is the largest of a prime quadruple, i.e., n ≡ 9 (mod 10). As an application, the largest known prime M = 2^136279841 - 1 does not satisfy the closure property. This avoids over-induction in finite versions of the Goldbach Conjecture.