Quadratic Compensation and Single-Prime Convergence in Goldbach Pair Generation

\begin{abstract} We develop a curvature-sensitive framework for the Goldbach Conjecture that anchors analysis at the symmetry apex $x=E/2$ and quantifies prime-pair generation via a logarithmic mass function. Central to our approach is a quadratic compensation mechanism, through which we establish the governing inequality \[ \delta_{\log}^{\mathrm{prime}} \le \delta_{\log}^{\mathrm{root}}. \] This constraint links additive and multiplicative energies on the line $x+y=E$, identifies the apex as the locus of maximal curvature and probabilistic density, and encodes out-of-range behavior through a discriminant that measures analytic curvature. Within this compensated manifold, prime pairs are not treated as probabilistic accidents but as structurally guided configurations. We further explore how primes “live” along the even-energy axis $E$, showing that the logarithmic window around $E/2$ functions as a generative corridor in which compensated prime pairs persist.