Reductions to Prime Curvature Geometry: Conditional Theorems for Goldbach, Hardy–Littlewood A, and Short–Interval Problems

The Prime Curvature Conjecture for Goldbach (PCC–Goldbach) proposes a geometric framework for controlling remainder terms in Goldbach pair counting through an explicit bounding envelope with intrinsic exponential curvature. In this paper we investigate the conditional consequences of this framework, assuming either PCC–Goldbach or its weaker bounding variant, PCC–Goldbach Bounds. We establish a series of reductions showing that PCC–Goldbach Bounds implies several classical results in analytic number theory. First, we prove that PCC–Goldbach Bounds implies Goldbach’s conjecture for all even integers 2n≥4, with explicit verification required only up to a computable finite threshold. Second, we recover the classical Hardy–Littlewood asymptotic formula (Conjecture A, Goldbach form) as a direct consequence of the geometric bounds. Third, we show that PCC–Goldbach Bounds yields relative agreement between measured and predicted Goldbach counts in the short–interval scaling regime M ≥(2n)^(1/2+ε), matching the window sizes typically associated with Bombieri–Vinogradov–type hypotheses. These reductions demonstrate that a single geometric hypothesis suffices to organize and imply a range of results that have historically required distinct analytic methods. All statements in this paper are explicitly conditional. Rather than claiming unconditional progress on Goldbach’s conjecture, we show that PCC–Goldbach, if validated, provides a unified geometric foundation for classical Goldbach asymptotics and short–interval behavior.