Additive–Multiplicative Nodes, Harmonic Weights, and an Effective Threshold Toward Goldbach

We study a weighted Goldbach setup with a Fejér triangular window around E/2. Using the circle method with explicit L2 control and a fourth-moment bound for S(alpha) = sum_{n<=E} Lambda(n) e(alpha n), we get an effective threshold: for W(E) = E^(2/3)(log E)^4, if the fourth moment <= C4E^2*(log E)^6, then every even E >= E0(C4) has a prime–prime representation within the window. This implies the multiplicative ratio b/a is close to 1. A small-prime pre-sieve (“Monty”) speeds up verification of the finite initial range. For C4=350 we computationally verify all even E up to E0 ~ 12,756,358. An appendix gives an explicit bound C4 <= 384, moving E0 to ~ 27,724,525.