On primes p for which N − p is an almost prime sum of two squares

Let N be a sufficiently large even integer. We show that the number of primes p ≤ N such that N − p is a sum of two squares having at most nine prime factors, counted with multiplicity, is at least (1/10) S(N) N / (log N)^2, where S(N) is the binary Goldbach singular series. The proof follows the vector sieve framework of Nath and Xie and combines two semi-linear sieves at the Bombieri–Vinogradov level of distribution with Richert's logarithmic weights. At this level the semi-linear lower bound function vanishes for s ≤ 1. Because of this, primes congruent to 3 mod 4 are sifted only up to N^(1/2 − δ). The survivors that have two large prime factors congruent to 3 mod 4 are then removed by a switching argument. The final numerical inequality is verified with outward-rounded interval arithmetic. This record contains the paper (PDF and AMS-LaTeX source), the interval-arithmetic verification script certify.py, the exploratory floating-point scans used to choose the parameters, their outputs, and the figures. This is a preprint. The analytic sieve lemmas have not yet been independently refereed, and the novelty of the statement is still being checked against the literature. Source code: https://github.com/Ruqing1963/almost-prime-sum-of-two-squares