This preprint develops an exact finite residue-theoretic framework for a symmetric complementary affine system arising from an additive diagonal encoding of primality constraints. For a finite shift set H = h₀,. . . , hₖ, the paper studies the affine family formed by the complementary pairs i + hⱼ + 1, n - i - hⱼ + 1, 0 n + 2. The paper also identifies the local root-count data entering the corresponding Hardy-Littlewood singular series and formal Bateman-Horn local factors. The results are finite and residue-theoretic; no infinitude theorem or asymptotic estimate for prime-producing indices is claimed.
Gabriel Dorel Dura (Fri,) studied this question.