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This research explores an algebraic identity related to the Goldbach Conjecture within the sphere helix framework.

June 4, 20260 citationsOpen Access

A Geometric Construction for the Goldbach Conjecture: Sphere Helix Identity and Gauss Partition Method

Key Points

This research explores an algebraic identity related to the Goldbach Conjecture within the sphere helix framework.
Utilized the sphere helix dynamics with the Riemann ζ function for the analysis.
Combined the Gaussian spiral concept with Gauss partition methods to establish relationships between primes.
Examined the implications of symmetric prime pairs for even numbers using Dirichlet's theorem.
Demonstrated that symmetric prime pairs can be constructed for every even number.
Found that primes are guaranteed on the Gaussian spiral subset.
Confirmed the role of the Riemann ζ function signature in the algebraic identity.

Abstract

Within the sphere helix framework of the Riemann ζ function (V12, DOI: 10. 5281/zenodo. 20504171), this paper establishes the exact algebraic identity cos (tₚ) + cos (tq) = 1/R, translating the Goldbach condition p + q = 2R into the dynamical language of the sphere helix. The 1/R on the right side comes entirely from V = 1/2 — the signature of the ζ function in sphere helix dynamics. The argument combines the Gaussian spiral (prime intersection subset of two orthogonal curve families in the yz-projection), the non-vanishing ζ·β ≠ 0 (guaranteeing primes on the spiral), and the Gauss partition method (Dirichlet's theorem on primes in coprime classes + sphere symmetry preserving coprime classes under symmetric pairing), concluding that symmetric prime pairs are unavoidable for every even number. This paper, together with V12 (Riemann Hypothesis) and the Collatz paper (DOI: 10. 5281/zenodo. 20504449), shares the common foundation e^ (iπ) = −1. Bilingual: Chinese and English PDFs included.

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