The Divisor Mirror: A Geometric Invariant for Integer Structure

Abstract

Every positive integer N carries a natural reflection symmetry whose deviations reveal its arithmetic structure. This paper introduces the divisor-mirror imbalance invariant W(N), defined through the total absolute distance of the divisors of N from the square-root axis. By measuring how asymmetrically divisors distribute across this inversion axis, W(N) detects structural features that classical multiplicative functions do not directly capture. The paper establishes several exact results concerning this invariant. The Classification Theorem proves that W(N) reaches its minimum value of N-1 if and only if N is a prime or the square of a prime, identifying these as the unique maximally polarized states of integer structure. The Semiprime Gap Theorem shows that for N = pq, the excess E(N) = W(N) - (N-1) equals the prime gap q-p, providing a geometric measure of factor separation. The Prime Filter Lemma establishes structural constraints on prime factors arising from complementary divisor differences in squarefree integers and primorials. The paper also introduces a scale-free second-moment ratio for examining how divisor geometry varies across different arithmetic classes. More broadly, the work develops the square-root axis as a natural inversion mirror in arithmetic and treats divisor structure as a geometric object rather than solely as a multiplicative one. The results provide a mathematical foundation for studying integer structure through symmetry, balance, reflection, and positional invariants.