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.
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.
The Madelung equations reproduce the Schrödinger equation only when supplemented by the Wallstrom condition of integer circulation, ∮γp∈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=√ρeiS/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.
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)
This paper studies the fixed-background many-body extension and the Wallstrom obstruction within the scalar density-phase framework of capacity geometry. Within the same local first-order homogeneous density-phase action class as the general scalar forcing theorem, the forcing argument extends immediately to declared configuration-space backgrounds and yields the standard spinless many-body Schrödinger equation without any factorisation ansatz. The bare density-phase equations do not force Wallstrom quantisation: a punctured-disc family gives explicit finite-action counterexamples, and the sharpened locallocking relation reduces that family only continuously, not integrally. The strongest continuum closure theorem is a domain-selection statement: on the intrinsic nonvanishing complex-field domain the forced action is the Schrödinger action, Wallstrom counterexamples are excluded, and circulation quantisation is automatic by degree theory. What remains open is whether the SGOC construction selects that single-valued complex domain rather than only the larger density-phase representation. A second contribution is a precise theorem map for the discrete carrier-transport approach. Native phase-sensitive SGOC data live on the Fourier/triad skeleton, while the discrete Wallstrom problem needs carrier data on a spatial triangulation of a nodal-complement component. The missing theorem is a genuine native-to-carrier transport theorem. On closed orientable surface carriers the remaining cases split into three transport regimes. Once a global carrier edge-phase field exists, the total-curvature scalar obstruction vanishes automatically by simplicial Stokes; once non-flat support is confined to a proper carrier,homology-killing enlargement and relative H^1 complement globalisation handle the rest of the downstream topology. Selector-side reductions are also consolidated. Before localisation, the remaining selector-side existence content is one universal H^2 transgression class. After localisation to a proper carrier, existence of a strong completion is automatic, and the only surviving noncanonical part is the ambient surface period torsor — an affine A^{2g}-torsor on genus g — with complement residues canonically separated off. Exchange sectors and spin are additional topological or multicomponent input, not outputs of the present scalar variational route. Keywords: - Schrödinger equation - Capacity geometry - Wallstrom obstruction - Many-body quantum mechanics - Madelung equations - Circulation quantisation - Domain selection - Carrier transport - SGOC programme Related identifiers (references cited in this paper): - 10.5281/zenodo.19156067 (cites) — Capacity Geometry: REG, Dirichlet Tension, and Quasi-Local Mass (SGOC Volume 2, Part 1) - 10.5281/zenodo.19156316 (cites) — The Canonical Capacity Coordinate and the Unique Quadratic Tension in the Capacity Gauge (Part 1A) - 10.5281/zenodo.19189901 (cites) — The Schrödinger Equation from Capacity Geometry: General Forced-Action Theorem on a Fixed Capacity Background - 10.5281/zenodo.19259771 (cites) — Radial Surcharge Inversion and Free-Boundary Host Selection - 10.5281/zenodo.19267944 (cites) — Free-Boundary Surface Selection for a First-Order Nonlinear Profile Equation in Three Dimensions Related identifiers (external references): - 10.1103/PhysRevA.49.1613 (cites) — Wallstrom, T. C., "Inequivalence between the Schrödinger equation and the Madelung hydrodynamic equations," Physical Review A 49(3) (1994), 1613–1617
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.
Quantum theory carries several discrete or numerical inputs that are routinely described as "not derived" from an underlying substrate. This paper asks a sharper question: for which of these inputs can "not derived" be upgraded to a proof of undefinability, and for which is this structurally impossible? Padoa's classical method of primitive notions answers the first kind and, as shown here, cleanly refuses the second. The inputs are classified by their logical position relative to a similarity type: an input is amenable to Padoa's method exactly when it is a predicate on a fixed type, and is not amenable when it is a parameter of the type itself or a weight on an ensemble. Both poles are exhibited. At one pole, circulation quantization: reformulating a companion result, the predicate is shown Padoa-undefinable over the structure of realized jets, the nontrivial part being the existence of a model pair; here the method also computes the exact reduct threshold at which definitions begin to exist, direction by direction. At the other pole, the spinor sign: the model pair is nearly definitional and the entire mathematical weight sits in an explicit locality hypothesis about the reduct, stated rather than assumed. Finally, the amplitude parameter (the reduced Planck constant) and Born-type typicality weights are shown to fall outside the method. For the amplitude parameter a dichotomy is proved for the single-theory formulation; for typicality weights a semantic obstruction is documented with a counterexample. In both cases the two available formulations give, respectively, an answer without blindness and blindness without content — the reason these inputs are excluded structurally. The boundary is offered as a result, not an apology: the classification tells one in advance which foundational "inputs" can be argued about by this method at all. Companion to "Locality, cycles, and the Wallstrom condition: an exact dichotomy and an undefinability theorem" (Zenodo, DOI 10.5281/zenodo.21727985). A self-contained SymPy script verifying the paper's symbolic claims is provided as supplementary material.
The Madelung transformation rewrites the Schrödinger equation as compressible hydrodynamics. Although locally equivalent, the converse implication has historically been considered incomplete because classical continuum hydrodynamics admits multi-valued velocity potentials and arbitrary circulation, which are strictly excluded from quantum mechanics. This discrepancy, widely known as the Wallstrom objection, has remained the principal obstacle to purely hydrodynamic interpretations of quantum theory.We resolve this problem definitively by establishing that physical states are represented on a finite, band-limited spectral lattice. Utilizing the exact Whittaker–Shannon reconstruction theorem and globally single-valued lattice phase variables, we derive circulation quantization as an inescapable topological consequence of the lattice structure, rather than as an independent or ad hoc axiom. By proving that band-limited interpolation strictly preserves topological homotopy classes, we close all previous mathematical gaps. The resulting theory completely replaces arbitrary continuum phase postulates with rigid discrete topological constraints, preserving the standard Schrödinger dynamics while completely dismantling the Wallstrom obstruction.