Position-Preserving Zero-Packet Optimization for the Binary Goldbach Exceptional Set

Abstract

Let E(X) denote the number of even integers n ≤ X that are not representable as a sum of two primes. We obtain a new upper bound for the binary Goldbach exceptional set by refining the zero-packet optimization in the Pintz–Zhao framework. The key observation is that the usual reduction to separate classwise caps and global mass bounds can discard information shared by fixed-class and unrestricted-union zero-density estimates. We introduce a position-preserving formulation in which these estimates are imposed on the same underlying zero measures before projection. This yields cross-class concentration constraints that are invisible in the traditional cap-and-mass geometry. We characterize the traditional projected feasible region exactly, derive non-redundant union cuts, and develop a saturation principle that forces residual zeros beyond certain defect thresholds. We then reduce the resulting zero-measure optimization to an interval-safe finite relaxation whose bounds admit exact rational certification. Using the zero-location and weighted-density inputs recorded and source-mapped in the paper, without requiring a new zero-density theorem, we establish the packet inequality at A = 263/80. Consequently, for every ε > 0, E(X) ≪ε X^(183/263+ε).