The Goldbach Conjecture Proven Using Exponential Phase Contradiction

We give direct and elementary proof of the Goldbach conjecture, which asserts that every even integer greater than 2 can be expressed as the sum of two prime numbers. The method employs an exponential phase representation of integers, mapping primes to points on the complex unit circle and analyzing their additive combinations through trigonometric identities. By assuming the existence of a counterexample, we derive a contradiction from the resulting phase alignment conditions, which cannot be satisfied simultaneously for all cases. The proof treats both congruence classes of even integers, R ≡ 0 (mod 4) and R ≡ 2 (mod 4), and covers all decompositions N = 2k with k even or odd. This approach avoids the use of analytic number theory, the Riemann zeta function, or deep asymptotic estimates, relying solely on elementary number-theoretic properties, parity arguments, and basic trigonometric identities.