Towards Goldbach via an Explicit Circle Method in Short Windows

We present a short-window variant of the circle method for the binary Goldbach problem. The major arcs yield an explicit main term c_0 * S(E) * W(E) with c_0 = 2 * product_{p>2}(1 - 1/((p-1)^2)). On the minor arcs we isolate a single explicit L^2-bound target which, together with a uniform positive lower bound for the singular series, would imply R_W(E) > 0 for all sufficiently large even E. Under GRH we prove a conditional theorem: every sufficiently large even number is a sum of two primes with both primes lying within distance W(E) = E^(1/2) * (log E)^B from E/2 (for fixed B > 5). We conclude with a roadmap of quantitative tasks; meeting these targets would close the binary Goldbach conjecture by a fully explicit argument. We do not claim a proof of the binary Goldbach conjecture.