From One-Sided Prime Distribution to Two-Sided Symmetry: Reconstructing Goldbach’s Conjecture to Find a Complete Proof

Abstract

I previously presented a formal proof approach to Goldbach via the Unified Prime Equation (UPE) [Bahbouhi Bouchaiba-b, preprints.org 2025] [1,2]. The principal criticism that remains is precise: how to pass from one–sided prime existence (primes in short intervals on each side of E/2) to the simultaneous two–sided coincidence required for a symmetric pair at the same offset t. This paper focuses on demonstrations only. We formalize the Z–scale, isolate the single analytic obstacle as a covariance term over symmetric offsets, prove unconditional lemmas for admissible offsets and one– sided prime mass, and state a sharp Conditional Theorem that converts the one–sided mass into a guaranteed two–sided hit within H = κ (log E)^2. We also give an empirical theorem and a reduction: under standard distributional hypotheses (Elliott–Halberstam–type covariance control), UPE terminates for every even E ≥ E0 within the Z–corridor. All known theorems are explicitly cited.