General Algebraic Classification Framework for Odd Integers and Its Potential Applications to Prime-Related Conjectures

Prime screening and structural distribution of primes are core subjects in additive and analytic number theory. Classical sieves (Eratosthenes, Brun, Chen’s weighted sieve) adopt passive enumeration elimination, lacking a unified algebraic structural basis and universal applicability. This paper constructs a three-layer closed algebraic classification system for all odd integers. All odd positive integers take the uniform form 2t+1\ (t) ; every odd composite greater than 1 can be uniquely written as (2n+1) (2k+1) with n, k^*, separating into square composite (n=k) and cross-product composite (n k). Excluding unit element 1, the remaining odd integers exactly equal all odd primes. We supply complete proof for the core discrimination theorem, numerical validation examples, algebraic compatibility derivation with classical sieves, quantitative counting demonstration and full application analysis for Goldbach conjecture. The framework unifies the logical foundation of mainstream sieve methods and supports subsequent quantitative research on prime density and infinitude theorems.