Even integers as sums of two sums of two squares with few prime factors

Let N ≡ 2 mod 4 be a sufficiently large integer. We show that the number of n ≤ N such that n and N − n are both sums of two squares and Ω(n(N − n)) ≤ 11 is at least (3/2) S(N) N / (log N)^2, where S(N) is the binary Goldbach singular series. With other parameters the method shows that at least (3/5) S(N) N / (log N)^2 such n satisfy max(Ω(n), Ω(N − n)) ≤ 8. The result extends to the principal form of each of the nine imaginary quadratic fields of class number one (discriminants −3, −4, −7, −8, −11, −19, −43, −67, −163). For even N, the bound is 11 when N is not divisible by the ramified prime (for −4: N ≡ 2 mod 4; for −8: v_2(N) ≤ 2; for −3: N ≡ 2 mod 3). In the remaining low-order cases the bound is 12 or 13, because there every representation must contain extra factors of the ramified prime. The sequence n(N − n) has level of distribution 1, but this level is shared by all sieve components. Sifting the inert primes up to N^(1/2) at level N lands exactly on the sieving limit s = 2 of the linear sieve, so they are sifted only up to N^(1/2 − δ). A survivor that is not a pair of represented integers has two large inert prime factors on the same side. These survivors are removed by a switching upper bound at the Bombieri–Vinogradov level. The main term and the switching term have the same local factors for every modulus, so the singular series cancels and the same certified numerical inequality applies to all nine discriminants. Richert's logarithmic weights control the number of prime factors jointly and on each side. The final numerical inequalities are verified with outward-rounded interval arithmetic. This record contains the paper (PDF and AMS-LaTeX source, 8 pages, 2 figures, 2 tables), the interval-arithmetic verification script certify.py, the local-condition analysis for the nine discriminants, the floating-point parameter scans, the figure script, and their outputs. This is a preprint. The analytic sieve lemmas have not yet been independently refereed. A web literature search found no overlapping result; a database search has not been done. Companion paper: R. Chen, On primes p for which N − p is an almost prime sum of two squares, and other class number one forms, preprint (2026), doi:10.5281/zenodo.23136112 Source code: https://github.com/Ruqing1963/bilateral-two-squares-sieve