On Mod (Product of the First Primes), a Boolean Lattice of 2 Groups, and Goldbach's Conjecture

The product of the first primes (2..) has neighbours ± 1 with all prime divisors beyond , implying there are infinitely many primes [Euclid]. All primes between and are in the group of units in semigroup of multiplication mod . Squarefree modulus yields as disjoint union of 2 groups, with as many idempotents -one per divisor of , forming a Boolean lattice . It is shown that each complementary pair in adds to 1 mod , and each even idempotent in has successor +1 in . Hence + ≡ , the set of even residues in , so each even residue is the sum of two roots of unity, proving “Goldbach for Residues” mod (). A Euclidean prime sieve for integers is shown, which for increasing extends natural units < by ‘carry' < of weight to ′ = + . Failure of Goldbach's Conjecture () for some 2 contradicts () for some , together with Bertrand's postulate proving