A Decomposition of the Wallstrom Condition

The Madelung equations reproduce the Schrödinger equation only when supplemented by the Wallstrom condition of integer circulation, \( ∮_γ p ∈ 2πℏℤ \) for the phase one-form p, which the hydrodynamic equations do not supply: on the positive-density region the circle- and line-valued phase geometries are locally isometric, and explicit finite-energy stationary solutions with arbitrary real circulation exist (Wallstrom; Reddiger–Poirier). We show the condition decomposes into two independent parts, one of which can be derived. The Madelung gradient energy endows the local amplitude–phase value space with the flat polar metric g = dr² + (r²/κ²) dS², r = √ρ, κ = √(8mα), with α the Fisher coefficient. Within a minimal two-dimensional phase-equivariant category we prove that the line cover and circle quotients exhaust the globalizations of this geometry; each has a canonical metric completion adding a single phase-blind point at zero amplitude, and exactly one completion is a smooth Riemannian surface there. Smoothness fixes the phase period uniquely to 2πκ — for the quantum Fisher coefficient α = ℏ²/8m, to 2πℏ — and identifies the completed target with the Euclidean plane ℂ, the wavefunction \( ψ = √ρ e^{iS/ℏ} \) arising as the pullback of its regular Cartesian coordinate. Madelung data that descend to a globally defined scalar field then obey integer circulation by degree theory; fractional-circulation solutions survive locally but fail descent. Target regularity thus fixes the phase period, while integer winding irreducibly requires global scalar descent. Gauge coupling yields the flux-shifted (fluxoid) circulation law, and multiply connected and twisted sectors are delimited explicitly.