Bounded-Error Goldbach Capture by Maynard Prime Clusters

Abstract

This preprint establishes a quantitative bounded-error Goldbach result in which one prime summand is required to belong to a genuine bounded prime cluster. We prove that there exist a fixed admissible tuple of even shifts {h₁,…,hₖ} and a constant H₀ such that, for every A > 0, all but O_A(X/(log X)^A) even integers N ≤ X have at least one translate N + hᵢ that can be written as p + q, where p belongs to B_{H₀}, the set of primes having another prime within distance H₀. The proof combines Maynard’s multidimensional sieve with target-dependent diagonal weights, an L¹ Goldbach estimate in arithmetic progressions, a treatment of residual prime collisions, and an explicit local quadratic-form comparison for the Goldbach singular series. The result is complementary to exact almost-all Goldbach statements: it yields an arbitrarily strong logarithmic exceptional-set bound at the cost of allowing finitely many fixed translates of the target. This is a preprint and has not yet undergone external peer review.