Theorem (1+1.75) on the Goldbach Conjecture

We prove unconditionally that every sufficiently large even integer \(N\) admits \(N=p+rq\), where \(p,q\) are primes, \(r=1\) or prime, and \(r\le q^{3/4}\). The proof combines a positive upper sieve with a lower double sieve incorporating paired restrictions. The argument subtracts an upper bound for the excluded semiprimes from a lower bound for the total Chen count. For the upper bound, we construct a nonnegative divisor majorant and establish its density and distribution estimates for the same coefficients. For the lower bound, we impose paired restrictions on the two-prime and four-prime terms of an eleven-term Chen weight and apply Wu’s double-sieve framework. The coefficient constructions retain the required factorization, support, and prefix conditions throughout the argument. Finite positive transitions and directed rational bounds yield explicit constants for quantitative representation counts. These also allow the factor restriction to be strengthened to r ≤ q0.74816865. The threshold for N is ineffective. The accompanying materials include: English and Chinese manuscripts. Bilingual LaTeX sources with compilation instructions. A numerical supplement containing finite rational data, source certificates, numerical witnesses, and verification scripts. README files and SHA-256 checksums. Appendix I describes the certificate correspondence and seven verification entry points, including the inputs retained by each check. Numerical verification instructions are provided in the accompanying README files.