The Arithmetic Structure of the Divisor Mirror

This paper develops several arithmetic consequences of the divisor mirror invariant introduced in The Divisor Mirror. 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 fixed anchor m and growing prime p. Perfect numbers are shown to generate infinite constant-offset families, and several exclusion theorems restrict the search space for the fixed-point equation E(N) = N, whose only known solution is N = 392. Established results are stated as theorems with proofs; the uniqueness of N = 392 and several structural interpretations of the invariant remain open and are treated separately as future directions.