Vectorial Asymmetry in the Cartesian Grid: A geometric and Analytical to Goldbach's conjecture

This paper proposes a geometric transposition of Goldbach's Conjecture (p_1 + p_2 = 2n) into a two-dimensional Cartesian model. The problem is redefined as the search for unobstructed intersections between the transversal line described by the equation Y = -X + 2n and the orthogonal grid generated by composite numbers. Through probabilistic modeling based on the Prime Number Theorem, we apply an Odd Parity Filter and analyze the mathematical expectation E(n). It is demonstrated, both algebraically and asymptotically, that the linear growth of the search space strictly dominates the logarithmic resistance of the composite number distribution. Finally, employing proof by contradiction (reductio ad absurdum), prime gap analysis, and the Hardy-Littlewood Circle Method, we formalize the structural inevitability of the prime intersection, corroborated by mirror symmetry across the diagonal axis.