A (1 + 1.7482) Theorem for the Goldbach Problem

This research manuscript studies representations of sufficiently large even integers in the form N = p + rq, where p and q are primes and r is either 1 or a prime satisfying r ≤ q^b. Writing H_N = C_N N / (log N)^2 for the Chen singular-factor normalization, the manuscript presents the lower bound liminf D_b(N) / H_N ≥ 1.96176252 − 6.7617368 log(1/b), for 1/2 < b < 1, where N tends to infinity through even integers. In particular, the stated consequences include D_{3/4}(N) ≥ 0.0165 H_N and D_{0.74816964}(N) ≥ 10^−5 H_N for sufficiently large even N. The argument combines a positive upper sieve for excluded ordered semiprimes, an eleven-term Chen weight, paired restrictions of sieve terms, restricted distribution estimates, and finite rational certificates. The manuscript is undergoing further mathematical verification and formalization. Supporting computational and formal verification materials are being prepared. OpenAI ChatGPT/Codex assisted with mathematical exploration, proof development, computational work, and manuscript preparation. The author remains responsible for the mathematical claims and final manuscript. This record preserves the English and Chinese v3 manuscripts dated 28 October 2026. Numerical materials cited in reference [2] are available in the v2026.10.10 release. The ongoing Lean formalization is maintained separately. See README.txt for the scope and status of the accompanying materials.