A New Sieve Framework Circumventing the Parity Problem

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.