Two Commuting Circle Actions: A Shared Minimal Condition for Torus Topology Across Spectral, Dynamical, and Physical Systems

Two Commuting Circle Actions: A Shared Minimal Condition for Torus Topology Across Spectral, Dynamical, and Physical Systems In spectral geometry this condition appears through the ℤ × ℤ indexing of Laplacian eigenmodes on the torus. In Hamiltonian dynamics it appears through the Arnold–Liouville theorem, where commuting integrals generate invariant tori in phase space. In toric symplectic geometry the same structure arises through Hamiltonian torus actions and moment maps. Toroidal physical confinement systems naturally admit commuting toroidal and poloidal coordinate flows on nested magnetic surfaces, realising the same algebraic structure. The paper assembles these independent instantiations into a unified…