AbstractWe prove that the annihilation of harmonic 1-forms and 2-forms on a closed orientable 3-manifold forces spherical topology, using operator-theoretic methods rather than geometricflow. The proof proceeds: (1) kernel annihilation determines homotopy type (pure algebraictopology), (2) in dimension 3, homotopy type determines homeomorphism type (operatorrigidity), (3) the rigidity follows from spectral constraints, not from Ricci flow. The keyinsight is that the operator Δ1 ⊕ Δ2 having trivial kernel leaves no degrees of freedom fornon-spherical topology. This is a constraint satisfaction problem, not a flow problem. GPTsaid this question was “open a priori.” It is now closed.