The Divisor Mirror: A Geometric Invariant for Integer Structure

Description: Every positive integer N carries a natural reflection symmetry whose deviations reveal its prime structure. This paper introduces the divisor-mirror imbalance invariant W(N), defined as the total absolute distance of all 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 cannot see. Core Mathematical Results: 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. This identifies these as the unique "maximally polarized" states of integer structure. The Semiprime Gap Theorem: Demonstrates that for any semiprime N = pq, the excess value E(pq) = W(pq) - (pq-1) is exactly equal to the prime gap q - p. This allows for the geometric detection of factor separation without direct factorization. The Prime Filter Lemma: Establishes that for primorials and squarefree integers, the differences between complementary divisor pairs act as a natural prime filter. Any prime factor of these differences must either already divide the divisor or be larger than the largest prime factor of N. Second-Moment Stratification: Introduces the scale-free shape ratio R(N) = W(N)^2 / V(N), proving that integers stratify into distinct asymptotic classes (primes, semiprimes, and highly composite numbers) based on their internal divisor geometry. Significance: This work positions the square-root axis as a fundamental inversion mirror in arithmetic. It provides a rigorous foundation for studying integer structure through the lens of symmetry, balance, and positional invariants.