The Arithmetic Structure of the Divisor Mirror

Abstract

This paper develops 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)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)−mpE(mp)-mp for a fixed anchor mm and growing prime pp. Perfect numbers are shown to generate infinite constant-offset families, demonstrating that the classical abundance/deficiency dichotomy of the sum-of-divisors function directly governs the qualitative behavior of the mirror offset. We also present a series of arithmetic exclusion theorems that restrict the search space for the fixed-point equation E(N)=NE(N)=N, whose only known solution is N=392N=392.