Beyond Perfect Numbers: The Sum of Divisors Divisibility Problem for σ(n) | n + a

The sum-of-divisors function σ(n) has been studied since antiquity, most often in connection with perfect and abundant numbers, yet its divisibility behavior under integer shifts has never been classified. The paper asks a simple question. For which integers a does σ(n) divide n + a? The result completes the trilogy on multiplicative divisibility, following the earlier analyses of φ(n) and λ(n). The proof establishes that for every fixed integer a ≥ 2, only finitely many positive integers n satisfy σ(n) | n + a. The argument combines a 2-adic wall limiting the number of odd prime factors, an overflow lemma that forces q-adic excess for large ω(n), and a finiteness schema constraining the remaining primes to a fixed modulus set. Special cases include a = 1, where σ(n) | n + 1 holds only for n = 1 and for prime n, a = 0, which yields the classical perfect numbers satisfying σ(n) = 2n, and a < 0, where no infinite families exist since σ(m) ∤ m for all m > 1. The result marks the terminus of multiplicative coherence, the point at which arithmetic structure collapses into absolute finiteness.