The Arithmetic Structure of the Divisor Mirror

This paper developments several arithmetic consequences of the divisor mirror invariant, building on the geometric framework of "The Divisor Mirror" (2026). Working from the excess function E(N), we collect and organize the explicit formulas the invariant admits: the zero-classification theorem, the semiprime gap formula, closed forms for prime powers, and the affine family theorem governing E(mp) - mp for a fixed anchor m and growing prime p. Perfect numbers are shown to generate infinite constant-offset families, showing that the classical abundance/deficiency dichotomy of the sum-of-divisors function governs the qualitative behavior of the mirror offset directly. We also present a series of arithmetic exclusion theorems that restrict the search space for the fixed-point equation E(N) = N, whose only known solution is N = 392.