A Quadratic Root-Difference Approach to Goldbach Partitions

This paper reformulates the even-sum condition in Goldbach's conjecture as a quadratic equation, and introduces a compensation framework based on the ratio \(S\) between the roots and the even integer \(E\). This structure naturally selects the working point \(S = e^{\pi}\), corresponding to a stable and non-degenerate geometric corridor. The prime density outside this corridor is sufficient to support Goldbach pairings in the supercritical range \(E > 94\), ensuring that the original formulation is unaffected by classical obstructions and remains consistent with the Hardy--Littlewood main-term scale.