On primes p for which N − p is an almost prime sum of two squares, and other class number one forms

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, so primes congruent to 3 mod 4 are sifted only up to N^(1/2 − δ). The survivors with 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. The same argument applies to the principal form of each of the nine imaginary quadratic fields of class number one, that is, the fundamental discriminants Δ = −3, −4, −7, −8, −11, −19, −43, −67, −163. For each such Δ, the number of primes p ≤ N for which N − p is represented by the principal form of discriminant Δ and has at most nine prime factors is ≫ S(N) N / (log N)^2. The bound is ten when Δ = −3 and N ≡ 1 mod 3. Restricting p to a suitable residue class forces χ_Δ(N − p) = 1. The main term and the switching bound then carry the same factor, so the certified numerical inequality is the same for every Δ. 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 statements is still being checked against the literature. Version 2 adds Theorem 1.2 (the nine class number one discriminants, Section 6) and Figures 1–2. Source code: https://github.com/Ruqing1963/almost-prime-sum-of