The parity problem is a fundamental limitation in classical sieve theory: any sieve based on a self-dual quadratic form cannot distinguish integers with an odd number of prime factors from those with an even number. In this paper, we construct a new sieve framework that circumvents this barrier. The framework rests on three pillars: (1) a linear (non-quadratic) counting function, (2) a parameter space with CRT product structure enabling mean-square estimation, and (3) a Product Decomposition Theorem that provides strong algebraic cancellation of off-diagonal terms. We prove rigorously that the classical parity problem theorem does not apply to this framework, both formally (the counting function lacks the self-dual quadratic form) and substantively (the estimation chain never degenerates into a quadratic form). The framework demonstrates that the parity barrier, long considered intrinsic to sieve theory, is actually an artifact of the quadratic algebraic structure employed by classical sieves.
Haizhu Wu (Wed,) studied this question.