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

Abstract 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 kₚ = ( (p-1) !+1) /p and kq = ( (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.