Integer Invariants from Stokes and de Rham Period Integrality on Multiply Connected Domains

Fifth paper in a series on holonomic closure. In the four preceding papers we exhibited integer invariants arising from covering-space constraints in quantum mechanics, rotor transport, DNA linking topology, and coset partitions of free groups. Here we identify the single mathematical mechanism underlying all four: Stokes' theorem applied to closed 1-forms on domains with topological obstructions. A closed 1-form on a simply connected domain has vanishing circulation around every loop. On a multiply connected domain, de Rham duality implies the circulation around each non-contractible loop equals the pairing with the homology class—an integer determined by the cohomology class. This is the common mechanism underlying all four cases. The Madelung circulation condition, rotor holonomy, the Gauss linking integral, and permutation cycle closure are each shown to be Stokes' theorem instantiated on a specific domain with a specific obstruction. What is usually imposed as single-valuedness of the phase factor follows from integrality of de Rham periods once the nodal obstruction is recognized.