Theorem (1 + 1.8) on the Goldbach conjecture

Every sufficiently large even integer N has a representation N = p + rq. Here p, q are primes and r is 1 or a prime, with r ≤ q^(a−1) = q^(4/5) = q^0.8. This is Proposition (1 + a) of Li–Liu with a = 1.8 = 9/5. We use a weighted sieve on shifted primes. Its level of distribution exceeds one half. The proof combines thirteen row estimates with a switched upper bound. For a fixed source cutoff, the certified D-unit endpoint is D_lower = 1.83399665352036682372. Since the source mass is 4(1 − ε_A) U_C, division by four gives (1 − ε_A) D_lower = 1.833813253855014787037628 − o(1). The balanced-semiprime cost is at most 1.788209644611262112809348242339 + o(1) in those units. The remaining coefficient is 0.04560360924375267422827 − o(1).