Abstract

This paper introduces the divisor-mirror imbalance invariant, W(N), defined as the total absolute distance of all divisors of an integer N from its square-root axis. We provide a rigorous classification of W(N) across the natural numbers, establishing that W(N) = N - 1 if and only if N is a prime or the square of a prime. By introducing the shape ratio R(N), we reveal a predictable geometric architecture underlying composite numbers, offering a novel framework for analyzing integer structure and prime distribution.