A Symmetric Sieve Framework: Complete Proofs of the Goldbach Conjecture, the Legendre Conjecture, and the Composite Factor Prime Representation Theorem

We present a unified sieve framework that simultaneously resolves the Goldbach conjecture, the Legendre conjecture, and establishes the Composite Factor Prime Representation Theorem. The core method constructs a double arithmetic progression via the Chinese Remainder Theorem, combined with a “resonance breaking” technique to estimate exponential sums and a “physical space periodic cancellation” method to control error terms. We partition moduli from inclusion–exclusion into three ranges, each handled by appropriate tools. The total error is bounded by O(p 1+o(1)), while the main term is ∼ Cp2/(log p) 2 , ensuring existence of prime pairs. By varying forbidden residue classes, we prove: • Goldbach conjecture: every even M ≥ 4 is the sum of two primes. • Legendre conjecture: for n ≥ 1, there exists a prime between n 2 and (n + 1)2 . • Composite Factor Prime Representation Theorem: any composite N = m · n is expressible as exactly m primes (and also n primes). The proof uses only classical tools (Chinese Remainder Theorem, Fourier analysis, Mertens’ theorem, symmetric polynomial inequalities). No unproven hypotheses are assumed. Comments and feedback are welcome. Please feel free to contact me via wuhaizhu0512@163.com