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

The Prime Curvature Geometry Conjecture for Goldbach (PCGC--Goldbach) introduces a geometric framework for prime pair counting in which remainder terms are controlled by an explicit bounding envelope with intrinsic exponential curvature. This paper develops and analyzes the conditional consequences of this framework, assuming either PCGC--Goldbach or its weaker variant, PCGC--Goldbach Bounds.

Under this assumption, a collection of reductions is established showing that the geometric framework provides sufficient quantitative control to imply several classical results in analytic number theory. First, PCGC--Goldbach Bounds implies Goldbach's conjecture for all even integers 2n >= 4, with explicit verification required only up to a computable finite threshold. Second, the classical Hardy--Littlewood asymptotic formula for Goldbach pair counts (Conjecture A, Goldbach form) emerges as an asymptotic consequence of the geometric bounds. Third, the framework yields relative agreement between measured and predicted Goldbach counts in the short-interval scaling regime M >= (2n)^(1/2 + epsilon), corresponding to the window sizes associated with Bombieri--Vinogradov-type phenomena.

These results show that a single geometric hypothesis suffices to organize and connect several problems that have traditionally required distinct analytic techniques. All statements in this paper are explicitly conditional. Rather than claiming unconditional progress on Goldbach's conjecture, the paper demonstrates that PCGC--Goldbach, if validated, provides a unified geometric foundation with explicit bounding structure for classical Goldbach asymptotics and short-interval behavior.