The Divisor Mirror: A Geometric Invariant for Integer Structure

Every positive integer N carries a natural reflection symmetry whose deviations reveal its prime structure. The divisors of N pair as (d, N/d) around the square-root axis √N; we introduce the divisor-mirror imbalance invariant W(N) = Σ |d − √N|, measuring how asymmetrically the divisors of N distribute across this inversion axis. A key algebraic reduction eliminates √N entirely, expressing W(N) as a purely rational sum over divisor pairs. Our main result (Theorem 3.1) characterises exactly when W(N) achieves its minimum value N − 1: precisely when N is prime or the square of a prime. For semiprimes pq, the excess E(pq) = W(pq) − (pq − 1) equals the prime gap q − p (Theorem 4.1): the invariant detects factor separation geometrically, without direct access to the factors. For primorial numbers, which serve as natural calibration instruments for the divisor mirror, we prove a Prime Filter Lemma (Theorem 5.1), stated here in its fully general form for arbitrary integers, with the primorial case as a clean specialisation. These results position W(N) as a genuinely new arithmetic invariant, distinct from classical divisor sums, whose study connects to a companion paper developing its deeper arithmetic consequences.