Symmetric Pairs on Primorial Wheels

Abstract

Placing a mirror at the center of a sieve reveals more structure than the usual picture of crossed-out multiples. At a center built from a primorial, the first sieve stages leave an exactly countable wheel of symmetric pairs. Every later prime meets that wheel along a small family of arithmetic progressions, turning the question of persistence into a finite covering problem. This elementary viewpoint connects reduced residue systems, the Chinese remainder theorem, and sieve ideas in a form accessible to undergraduate number theory. It also explains precisely why a tempting symmetry claim fails at the prime 2, while the primorial-wheel and later-prime conclusions remain intact.