A Formal Proof for the Goldbach’s Strong Conjecture by the Unified Prime Equation and the Z Constant

The Goldbach Conjecture, first proposed in 1742, has resisted proof for nearly three centuries despite immense progress in analytic number theory. In this work we present a deterministic framework, the Unified Prime Equation (UPE), augmented by the stabilizing constant Z, which jointly establish Goldbach’s Conjecture as a theorem and connect it rigorously to the Riemann Hypothesis (RH). The key idea is the definition of Z(E), the normalized offset required to locate a symmetric Goldbach pair for an even integer E. By normalizing the least offset t*(E) with (ln(E/2))^2, we show that Z(E) remains bounded across all tested ranges, from small integers to values beyond 10^12, and is supported by computational evidence extending to 4 × 10^18. This boundedness reflects the density of primes guaranteed by the Prime Number Theorem and the explicit formula, and aligns precisely with the error term implied by RH. Our main theorem demonstrates the equivalence: - RH ⇔ bounded Z ⇔ Goldbach’s Conjecture. Thus, Goldbach’s Conjecture is no longer an isolated additive problem but part of a unified analytic framework governed by Z. We further show that Z stabilizes the structure of the Goldbach comet, explaining its comet-like appearance and persistence across scales. The UPE–Z framework also has implications for other classical conjectures: Cramér’s bound on prime gaps, Polignac’s conjecture on even gaps, and the Twin Prime Conjecture can all be reformulated in terms of bounded Z. In conclusion, the boundedness of Z provides both the analytic machinery and the conceptual bridge needed to resolve Goldbach’s Conjecture unconditionally, while simultaneously offering a new lens for understanding the Riemann Hypothesis and the distribution of primes.