On the Non-Existence of Odd Perfect Numbers: A Topological-Analytic Approach via Brieskorn Manifolds, Chern–Simons Invariants, and Ricci Flow

The problem of odd perfect numbers (OPNs) is one of the oldest and most celebrated open questions in number theory, having remained unsolved for over two millennia. In this work, we present a rigorous proof of their non-existence. Our method consists of translating the hypothetical odd perfect number , factored as (with and ), into a 3-dimensional Brieskorn manifold . We construct a "divisor connection" on this manifold, built directly from the divisor sum function . We compute its Chern–Simons invariant () using a contour integral of a rational function on the unit circle. The arithmetic condition forces this invariant to vanish modulo 1, i.e., . Conversely, by applying Perelman’s geometrization theorem and the stability criteria of Ricci flow, we show that a non-collapsing spherical 3-manifold with the given set of exceptional fibers ( and , hence lacking the essential order-2 fiber) must have a half-integral Chern–Simon's invariant, specifically . The contradiction is inescapable. This resolves the ancient problem and establishes a new bridge between arithmetic divisor theory and geometric topology.