Prime Curvature Geometry and the Structure of Additive Prime Deviations

This paper develops a geometric framework for analyzing remainder terms in additive prime and prime-like problems. The approach is based on a multiplicative decomposition of the deviation into an explicit small-prime residue structure and a medium-to-large-prime tail exhibiting intrinsic exponential curvature. This decomposition leads to the formulation of the PRIME CURVATURE GEOMETRY HYPOTHESIS (PCGH), together with an associated curvature constant \(\Omega\), intended to govern pointwise remainder envelopes beyond the scope of the classical Fundamental Lemma of sieve theory. [1] Specializing the framework to the Goldbach problem yields the PRIME CURVATURE GEOMETRY CONJECTURE FOR GOLDBACH (PCGC–GOLDBACH) and an explicit curvature constant \(\Omega_{prime}\). The resulting conjectural bounds are compatible with Hardy–Littlewood A predictions in the asymptotic limit, while providing a geometric mechanism that organizes deviation behaviour across finite analytic windows and cutoff scales. An explicit bounding envelope for the Goldbach remainder is derived and shown to control the exact remainder uniformly. This envelope is proved not to be an asymptotic proxy: the exact remainder exhibits non-vanishing exponential curvature within cutoff cells, preventing pointwise asymptotic equivalence. This distinction clarifies the limits of scale-based approximations and isolates the geometric source of deviation. The present paper is restricted to the derivation and formal statement of the geometric framework, associated hypotheses, and bounding envelopes. Numerical validation, certified bounds over large finite ranges, and reduction theorems relating PCC–Goldbach to Hardy–Littlewood–A and Bombieri–Vinogradov–type hypotheses [2][3] are developed in subsequent work.