Rough Factor Layers Near the Linear-Sieve Boundary in Binary Goldbach

Abstract

We study \(N^\rho\)-rough reflected complements \(N-p\) in the binary Goldbach problem through their exact number of prime factors. For \[\frac15<\rho<\frac14,\] every surviving complement has at most four prime factors. Writing \(T_k(N,\rho)\) for the layer with \(\Omega(N-p)=k\), \(T=\sum_{k=1}^4T_k\), and\[\delta=1-4\rho,\] we show, using standard upper- and lower-sieve estimates, that \[\boxed{\frac{T_4(N,\rho)}{T(N,\rho)}=O(\delta^2)}\] for every fixed sufficiently small \(\delta>0\) and all sufficiently large even \(N\). A Goldbach counterexample has \(T_1(N,\rho)=0\). Hence any sufficiently large counterexample in this fixed-\(\delta\) regime must simultaneously have an empty prime layer and a four-factor layer occupying at most \(O(\delta^2)\) of the rough slice. This is a purely asymptotic necessary condition: the lower-sieve proof becomes highly non-uniform as \(\delta\to0\), so the estimate is not claimed to constrain counterexamples in any currently computationally accessible range. The formulation is diagnostic rather than reductive. Any uniform power-scale lower bound that keeps the prime layer away from zero already yields a Hardy--Littlewood-scale lower bound for binary Goldbach representations. We complement the analytic estimate with a boundary-simplex benchmark, a descriptive \(N\)-dependent local factor-deletion diagnostic, and exhaustive computations for every even \[100000\le N\le10^6\] at \(\rho=0.21,0.22,0.23\). The computation is reported only as finite factor-layer geometry and is not used as evidence for an asymptotic exponent.