We establish a quantitative structural obstruction for linear operators acting diagonally on the Fourier basis in the context of the binary Goldbach problem. Any uniformly bounded linear operator TN whose multipliers satisfy the high-frequency decay condition MN = o ( (log N) ^−2) annihilates the main term of the weighted Goldbach representation function. The result extends to general bounded operators via orthogonal projections, providing a rigorous explanation for the failure of purely linear filtering methods and the structural indispensability of bilinear techniques. Conditional refinements under the Bombieri–Vinogradov theorem and the Generalized Riemann Hypothesis are discussed.
Ozorio Olea Arnaldo Adrian (Thu,) studied this question.