This work introduces GC-60, a conceptual model for prime number exploration based on the systematic elimination of internal products within a structured arithmetic domain. Unlike classical sieves derived from Eratosthenes, where primality is an operative notion used to eliminate multiples, GC-60 never identifies or uses prime numbers during the elimination process. Instead, eliminations arise exclusively from generating products of elements belonging to a reduced domain defined by modular constraints. In this framework, primality emerges as a residual property rather than a guiding principle. The method is not proposed as a performance-oriented alternative to segmented or wheel-based sieves, but as a conceptual dual formulation of prime detection, aimed at exploring logical structures related to multiplication, compositeness, and semiprimes. A computational implementation using a modulo 60 structure and JIT compilation is presented solely to demonstrate operational consistency, not to claim algorithmic superiority. GC-60 is intended as a tool for conceptual investigation in number theory, in continuity with prior exploratory work on semiprimes (GC57).