Prime Factor Marking and Signed Correlations in Binary Goldbach

We examine the progress of the PIHR Goldbach programme after Gold 47. The principal advances are an explicit accounting of a squareclass reserve, a quantitative obstruction to subtracting semiprimes from a small fixed sieve lower bound, a block application of Chen’s double sieve in Wu’s formulation, and an exact prime detecting weight on a rough sequence. The double sieve calculation gives a positive lower bound for complements with no prime factor at most a fixed power slightly larger than one quarter. On this support every complement has at most three prime factors, counted with multiplicity. Marking the factors below a power slightly smaller than one half removes the three factor contribution exactly when combined with a Liouville projector. We prove an asymptotic formula for the corresponding unsifted marked sum using Bombieri Vinogradov. The simultaneous roughness and Liouville conditions introduce additional moduli and a signed correlation outside that calculation. We give their exact finite expansion and retain the full prime complement weight.