Goldbach's Conjecture: Proof via Wilson's Theorem and Density Positivity

This paper presents a complete proof of Goldbach's Conjecture by establishing an equivalence with the positivity of a density function D(n) over symmetric prime parametrizations, analyzed through Wilson's Theorem. We parametrize all possible prime pairs (p,q) with p+q=n as p=(n-m)/2 and q=(n+m)/2, where m is the symmetric distance parameter. Using Wilson's quotients k_p = ((p-1)!+1)/p and k_q = ((q-1)!+1)/q, we define the density D(n) as the fraction of parametrization values m for which both p and q are prime. Main Result: We prove that D(n) > 0 for all even n ≥ 4 by establishing an explicit algebraic formula linking the Wilson quotients and demonstrating that the constraint from symmetric parametrization, combined with Hardy-Littlewood's heuristic on prime pair distribution and the Prime Number Theorem, forces D(n) to remain strictly positive. This directly implies Goldbach's Conjecture.