Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

CDUFD Supplementary Material VI: Formal Emergence of the Wallstrom Condition from Critical Topological Dynamics

Pengtai Huang

Published May 14, 2026• 0 citations
Publisher

Abstract

The Wallstrom condition---the requirement that the phase of the wave function be single-valued up to integer multiples of $2\pi\hbar$---has been a foundational obstacle in all attempts to derive quantum mechanics from stochastic or hydrodynamic dynamics since Wallstrom's 1994 proof that the Madelung equations alone are insufficient to recover the Schrödinger equation. In this paper, we provide a complete derivation of the Wallstrom condition from the axioms of the Constraint-Dispersion Unified Framework (CDUFD). The derivation follows the CDUFD methodology of solution-space construction and natural constraint-driven reduction: we initialize the solution space at the electroweak critical point with both Ising (amplitude) and XY (phase) fluctuation modes as built-in degrees of freedom of the SO(8)-embedded order parameter; we then demonstrate that A3 critical dynamics and A4 topological quantization jointly eliminate Ising domain walls while preserving XY vortices through their critical lifetime disparity. The argument is built upon three robust pillars: (i) SO(8) group structure guarantees a non-vanishing XY coupling $g\neq 0$; (ii) A4 topological quantization guarantees integer vortex winding numbers; (iii) the electroweak critical point provides a finite, large physical cutoff $\xi(\delta=0.0102)$ that prevents complete infrared suppression of the XY sector even in the dangerously irrelevant case. Together with the numerically testable premise that the effective XY coupling at this cutoff is strong enough to sustain a percolating vortex network, the surviving vortices, spanning the entire system via divergent correlation length, impose global phase quantization on all closed contours---precisely the Wallstrom condition. Combined with the previously established Madelung transformation from A2 Langevin dynamics, this completes the rigorous emergence of the full Schrödinger equation from CDUFD axioms. The sole remaining numerical premise is explicitly isolated as a verifiable open problem.

DOI
10.5281/zenodo.20173664

Related papers

A Topologically Constrained Framework for Quantization as Global Representability

Nathan Hevel

This manuscript presents a constrained mathematical framework and diagnostic methodology motivated by topological phase closure. It does not claim experimental confirmation and is released to establish structure, scope, and falsifiable criteria for further investigation.

•Jan 10, 2026•0 citations

CDUFD Supplementary Material VII: Emergence of the Born Rule from Topological Instrument Dynamics

Pengtai Huang

The Born rule---the probability of obtaining measurement outcome $a_i$ given by $P(a_i)=|\braket{a_i}{\psi}|^2$---is the last independent postulate of quantum mechanics not yet reduced to dynamical principles. Existing decoherence programs explain why measurement outcomes appear classical, but cannot derive why the probabilities are precisely $|\psi|^2$. Gleason's theorem establishes the Born rule as the unique consistent probability assignment on Hilbert space, but it presupposes the Hilbert space framework itself. In this paper, we provide a complete derivation of the Born rule from the axioms of the Constraint-Dispersion Unified Framework (CDUFD). The central innovation is replacing statistical randomness with topological determinism: using A4 topological defects as natural instruments, their discrete topological charges provide a discrete spectrum of measurement outcomes; topological pairing constraints ensure that the relaxation of the system-instrument composite is deterministically directed---the instrument does not create probability, it merely ``reads out'' the topological sector of the system; the equilibrium distribution of the system in the critical regime is rigorously determined to be $|\psi|^2$ by the uniqueness of the Fokker-Planck steady state. The Born rule emerges as a direct corollary of the equilibrium statistical mechanics of the critical regime. The derivation strictly follows the CDUFD methodology of solution-space construction and constraint-driven reduction, with all steps anchored to axioms A1--A5. Together with Supplementary Material I (Madelung transformation) and Supplementary Material VI (Wallstrom condition), this work completes the full emergence of quantum mechanics from CDUFD axioms.

•May 14, 2026•0 citations

Phase Quantization from Observable Regularity: Reducing the Wallström Objection to a Standard Physical Condition

Adrian Rohr

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.

•Feb 16, 2026•0 citations

Quantum speed limit and topological quantum phase transition in an extended XY model

W. W. Cheng, Bin Li, Longyan Gong +1 more

Physica A Statistical Mechanics and its Applications•Mar 16, 2022•3 citations

Digital Quantum Simulation of the Schwinger model with Topological term

Masazumi Honda

•Jun 1, 2022•2 citations

THE EXACT RESOLUTION OF THE WALLSTROM OBSTRUCTION

Michael I Aksman

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.

•Aug 6, 2026•0 citations
The Wallstrom condition---the requirement that the phase of the wave function be single-valued up to integer multiples of 2πℏ2\pi\hbar2πℏ---has been a foundational obstacle in all attempts to derive quantum mechanics from stochastic or hydrodynamic dynamics since Wallstrom's 1994 proof that the Madelung equations alone are insufficient to recover the Schrödinger equation. In this paper, we provide a complete derivation of the Wallstrom condition from the axioms of the Constraint-Dispersion Unified Framework (CDUFD). The derivation follows the CDUFD methodology of solution-space construction and natural constraint-driven reduction: we initialize the solution space at the electroweak critical point with both Ising (amplitude) and XY (phase) fluctuation modes as built-in degrees of freedom of the SO(8)-embedded order parameter; we then demonstrate that A3 critical dynamics and A4 topological quantization jointly eliminate Ising domain walls while preserving XY vortices through their critical lifetime disparity. The argument is built upon three robust pillars: (i) SO(8) group structure guarantees a non-vanishing XY coupling g≠0g\neq 0g=0; (ii) A4 topological quantization guarantees integer vortex winding numbers; (iii) the electroweak critical point provides a finite, large physical cutoff ξ(δ=0.0102)\xi(\delta=0.0102)ξ(δ=0.0102) that prevents complete infrared suppression of the XY sector even in the dangerously irrelevant case. Together with the numerically testable premise that the effective XY coupling at this cutoff is strong enough to sustain a percolating vortex network, the surviving vortices, spanning the entire system via divergent correlation length, impose global phase quantization on all closed contours---precisely the Wallstrom condition. Combined with the previously established Madelung transformation from A2 Langevin dynamics, this completes the rigorous emergence of the full Schrödinger equation from CDUFD axioms. The sole remaining numerical premise is explicitly isolated as a verifiable open problem.
The Born rule---the probability of obtaining measurement outcome aia_iai​ given by P(ai)=∣⟨ai⟩ψ∣2P(a_i)=|\braket{a_i}{\psi}|^2P(ai​)=∣⟨ai​⟩ψ∣2---is the last independent postulate of quantum mechanics not yet reduced to dynamical principles. Existing decoherence programs explain why measurement outcomes appear classical, but cannot derive why the probabilities are precisely ∣ψ∣2|\psi|^2∣ψ∣2. Gleason's theorem establishes the Born rule as the unique consistent probability assignment on Hilbert space, but it presupposes the Hilbert space framework itself. In this paper, we provide a complete derivation of the Born rule from the axioms of the Constraint-Dispersion Unified Framework (CDUFD). The central innovation is replacing statistical randomness with topological determinism: using A4 topological defects as natural instruments, their discrete topological charges provide a discrete spectrum of measurement outcomes; topological pairing constraints ensure that the relaxation of the system-instrument composite is deterministically directed---the instrument does not create probability, it merely ``reads out'' the topological sector of the system; the equilibrium distribution of the system in the critical regime is rigorously determined to be ∣ψ∣2|\psi|^2∣ψ∣2 by the uniqueness of the Fokker-Planck steady state. The Born rule emerges as a direct corollary of the equilibrium statistical mechanics of the critical regime. The derivation strictly follows the CDUFD methodology of solution-space construction and constraint-driven reduction, with all steps anchored to axioms A1--A5. Together with Supplementary Material I (Madelung transformation) and Supplementary Material VI (Wallstrom condition), this work completes the full emergence of quantum mechanics from CDUFD axioms.
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.
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.