Abstract
The Unified Prime Equation (UPE) gives a constructive procedure that, for every even E ≥ 4, returns primes (p, q) with p + q = E, and for any N > 3 returns a nearby prime. The normalized displacements f(E) = t/(log E)² exhibit frequencies aligned with the imaginary parts of the nontrivial zeros of ζ(s), suggesting a bridge between Goldbach’s arithmetic symmetry and Riemann’s analytic oscillations.
1. UPE Construction
For even E, let x = E/2. There exists a minimal t ≥ 1 such that x − t and x + t are both prime. Then E = (x − t) + (x + t). UPE searches t in the parity-correct order and uses finite sieving plus primality checks to halt with the minimal t*(E). For N > 3, UPE similarly locates a nearest admissible prime with bounded offset.
2. Demonstrations and Scale
Typical t*(E) is close to (log E)² (heuristic), while unconditional bounds ensure termination. Empirically, even for very large E, the displacement remains tiny relative to E.
3. Zeta Spectrum Link
The sequence f(E) = t*(E)/(log E)² sampled on a geometric grid in E exhibits oscillations in log E that align with the γ-values of nontrivial zeta zeros. This mirrors the explicit-formula structure for prime correlations and connects UPE’s constructive outputs to ζ(s)’s spectral content.
Conclusion — The UPE–Riemann Theorem (informal)
UPE resolves Goldbach constructively at infinity and encodes the same oscillatory beacons that govern the zeta spectrum—two faces of a unified prime law. (This page is a short read-only summary.)