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.
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.
Madelung showed that the Schrodinger equation can be rewritten, away from the zeros of the wave function, as hydrodynamic equations for a density and a velocity potential. Wallstrom [Phys. Rev. A 49, 1613 (1994)] showed that the rewriting is not an equivalence: the hydrodynamic system admits solutions whose circulation around nodes is an arbitrary real multiple of h/m, and equivalence requires imposing circulation quantization loop by loop, by hand. We examine a single kinematic postulate, the compact-phase postulate: the phase is fundamentally circle-valued, and the circle has action-circumference h. We prove three statements. First, a continuous circle-valued phase field winds an integer number of times around every loop in the node complement, so the fractional-circulation solutions are excluded kinematically. Second, compactness alone quantizes circulation in an arbitrary unit; the postulated circumference fixes the unit to h/m, and the circumference h is displayed independently by measured ladders in superconducting rings (h/2e) and in superfluid helium (h/m), and by the Aharonov-Bohm period (h/e), all in coherent-condensate systems. Third, under an integer-free regularity condition at the nodes, the postulate together with the Madelung equations is equivalent to exact scalar quantum mechanics at strong-solution (H1_loc) grade. We state the caveat explicitly: over the node complement, given continuity, the circle-valued phase is logically equivalent to the quantization condition it subsumes. The gain is at the level of axioms: one universal kinematic postulate with a calibrated, independently measured unit in place of a per-solution family of integral constraints.
The hydrodynamic (Madelung) formulation of quantum mechanics must be supplemented by the circulation-quantization condition of Wallstrom. We ask which parts of this global datum can be fixed by conditions tied to a region, and answer exactly. For conditions on the germ of the data along a compact region K, the separating power is computed in closed form: the strongest such condition is the coset test that the period vector, taken modulo the subspace V_K = im(H¹_c(Q∖K) → H¹(Q)), lie in the image of the lattice (2πℤ)^b₁. Every direction of the circulation datum is therefore either invisible to K or functionally determined by the germ along K, with no middle ground — the Aharonov–Bohm phenomenon at the level of axiomatization (Theorem 4.1, Corollary 4.2). The endpoint of the same scale is a no-go theorem: the carrier of a jet-local law covers no cycles, so no finite conjunction of jet-local axioms — pointwise conditions of arbitrary finite differential order, with arbitrary predicates — can replace the Wallstrom condition, not even one-sidedly (Theorem 3.4). A second and orthogonal blindness is identified as well: the entire jet language — arbitrary quantifiers over points, arbitrary predicates on jets, arbitrary Boolean structure — sees only the support of the jet occupation measure, whereas the circulation integrates against that measure; the condition is therefore not definable in that language at all (Theorem 3.7). The two negative results and the dichotomy are one principle, a ladder of locality: what decides the separating power of a class of conditions is the cycle coverage of its carrier — not differential order, not the amount of information the carrier holds, and not spatial extent. The proofs are constructive, based on explicit bump realizations of jets; the constructive steps are verified symbolically in an accompanying SymPy script. Extensions cover non-compact spaces, punctured domains and nodal sets, where vortex filaments source the datum and the known node-wise conditions are boundary rather than interior-local. A role-accounting survey shows that each proposed fix of the Wallstrom problem relocates, rather than eliminates, the condition. These negative results complement the recent momentum-map resolution of Khesin and Modin: it identifies where the condition lives; we prove it cannot be traded for local laws.
This paper presents a complete derivation of the Schrodinger equation from the Onsager-Machlup stochastic variational principle, rigorously resolving the longstanding Wallstrom objection. Wallstrom proved that hydrodynamic equations derived via the Madelung transformation do not uniquely determine quantum mechanics unless phase circulation is quantized. We demonstrate that the quantization condition (the integral of the gradient of S along a closed loop equals 2pin*hbar) emerges as a necessary consequence of requiring finite action and topological stability of coherent structures. Configurations with non-integer winding numbers are shown to lead to divergent action costs or inadmissible multi-sheeted spatial representations. For nodal configurations where the probability density vanishes, we show that the smoothness of the probability current independently forces integer winding numbers, excluding non-quantized solutions recently constructed in the literature. This establishes the Schrodinger equation as the unique and inevitable description of probability fluid dynamics under the calibration D = hbar/(2m). Finally, the framework provides testable experimental predictions regarding wave packet spreading and interference visibility in the presence of environmental noise.
This note consolidates seven theorem-grade Wallstrom closure results for the SGOC programme's density-phase framework, and precisely identifies the two remaining open problems. The Wallstrom problem is the failure of the Madelung hydrodynamic reformulation of quantum mechanics to automatically enforce the integer quantisation of circulation of the quantum phase. The present note does not prove unrestricted Wallstrom closure. It maps the problem branch by branch. Seven branches are closed: (i) Primary single-valued complex spectral branches: Wallstrom quantisation is automatic on every connected nodal-complement component. (ii) Flat solid-torus collar branch: the condition reduces to one period on the core generator, with a gauge-covariant integer winding-number refinement. (iii) Contractible carrier windows: local face-curvature data determine local abelian edge-phase gauge classes. (iv) Closed surface carriers with global carrier data: the downstream abelian complement obstruction collapses. (v) Boundary torus of the flat toral collar branch: strong-global carrier transport is automatic and the branch data rigidify the carrier architecture up to gauge. (vi) Genus-zero carriers with local data and proper-support localisation: the ambient period torsor disappears and the discrete carrier branch closes canonically up to gauge. (vii) Radial discovery-direction spectral branch: the recovered nodal-free ground state closes immediately. The two remaining open problems are: the state-space selection theorem for the primary complex branch, and the phase-to-carrier transport/localisation theorem for the discrete carrier branch. Keywords: Wallstrom problem, Madelung hydrodynamics, quantum phase quantisation, carrier transport, holonomy, SGOC, circulation quantisation Related Programme: Spectral Geometry of Coherence (SGOC)
The completed Chronoflux Atlas recovers the quantum regime as the coherent phase branch of conserved temporal transport. Continuity supplies the current, temporal order and positive kinetic response recovered in DPL-00 through DPL-04. Coherence adds a non-vanishing complex state with a physical phase and action scale. Once that descendant is admitted, variation returnsphase transport, topology quantises circulation, the Fisher-gradient term returns the Bohm quantum potential, and the density-phase equations recombine exactly into the Schrödinger equation. Quantum behaviour is therefore not attached to continuity by analogy. It is recovered as the lawful dynamics of a coherent continuity branch. The coherent action gives a conserved density current and a quantum Hamilton-Jacobi equation. On the domain where the amplitude does not vanish, the phase one-form is locally exact. Its global periods are integer winding numbers, giving circulation in units of ℎ/𝑚. Those integers persist under every continuous admissible evolution and can change only when a defect crosses the return path or the coherent amplitude vanishes. The apparent discontinuity of a quantum phase slip is thus the topological consequence of continuous evolution through a configuration where the phase ceases to be defined. The same structure returns the exact Madelung representation of a selected Schrödinger branch. On configuration space it yields the Born current, current velocity and quantum potential. Adding an actual configuration following that velocity is the Bohmian ontological completion, not an unstated consequence of continuity. Equivariance then proves persistence of quantum equilibrium. At field level, a shift-symmetric action closes a classical current, its quadratic perturbations recover a characteristic metric, and stationary canonical data produce a recoverable bosonic quantum state. Gauge selection, spin, fermionic statistics, outcome selection and Standard Model promotion remain separately indexed descendants. The paper also closes the non-ideal boundary. Multiplying a density by an arbitrary positive weight generates an exact continuity source and an exact transformed quantum potential; it cannot be presented as a free admissibility correction. Any departure from the closed branch requires a declared self-adjoint operator or trace-preserving open-system law. Admissibility, Recoverability and Persistence then distinguish a genuine quantum branch from incomplete phase data, singular nodal flow or an unreturned measurement model. The result closes coherence, quantised circulation and quantum response within the recovered tree of nature as such. Keywords: coherence; phase; circulation; quantum potential; Madelung transform; Schrödinger equation; Bohmian guidance; equivariance; quantum field reconstruction; Chronoflux Atlas
We pose the converse Madelung question: not whether Fisher information can reproduce quantum mechanics, but whether it is necessary. We work with minimal, physically motivated axioms on density and phase: locality, probability conservation, Euclidean invariance with a global phase symmetry, reversibility, and convex regularity. Within the resulting class of first order local Hamiltonian field theories, these axioms single out the canonical Poisson bracket on density and phase under the Dubrovin and Novikov assumptions for local hydrodynamic brackets. Using a pointwise, gauge covariant complex change of variables that maps density and phase to a single complex field, we show that the only convex, rotationally invariant, first derivative local functional of the density whose Euler Lagrange term yields a reversible completion that is exactly projectively linear is the Fisher functional. When its coefficient equals Planck constant squared divided by twice the mass, the dynamics reduce to the linear Schrodinger equation. For many body systems, a single local complex structure across sectors enforces the same relation species by species, fixing a single Planck constant. Galilean covariance appears through the Bargmann central extension, with the usual superselection consequences. Comparison with the Doebner and Goldin family identifies the reversible zero diffusion corner with linear Schrodinger dynamics. We provide operational falsifiers via residual diagnostics for the continuity and Hamilton Jacobi equations and report numerical minima at the Fisher scale that are invariant under Galilean boosts. In this setting, quantum mechanics emerges as a reversible fixed point of Fisher regularised information hydrodynamics. A code archive enables direct numerical checks, including a superposition stress test that preserves exact projective linearity within our axioms.