A Proof of the Goldbach Conjecture: Based on Three-Cycles, Three Patterns, and Finiteness of Coverings

This paper presents a complete proof of the Goldbach conjecture. We first transform the conjecture into an equivalent form and introduce the ``Move-2'' algorithm. By analyzing the period-6 structure of natural numbers, we define three-cycles and their prime candidates. Then we abstract three basic patterns that reveal the correspondence between the large and small numbers in the Move-2 algorithm. On this basis, we study the covering patterns of primes such as $5$ and $7$, and prove that there are only finitely many covering elements (bounded by √N) and that their covering points form finitely many arithmetic progressions. Using Dirichlet's theorem (there are infinitely many primes in arithmetic progressions), we show that there exists an uncovered prime candidate, hence the Move-2 algorithm must find a pair of primes. The argument relies only on elementary number theory and classical theorems, and does not depend on any unproven conjectures.