Hydrodynamic Stability Hypothesis

Hydrodynamic Stability

Hypothesis

Re-grounding

Quantum Mechanics in Classical Measure Theory

Author: Rowan Brad Quni-Gudzinas

Contact: rowan.quni@outlook.com ORCID:

0009-0002-4317-5604

ISNI: 0000000526456062

DOI: 10.5281/zenodo.17721008 **Publication

Date: 2025-11-26 Version:** 1.0

Abstract: This work identifies a “chronological

fallacy” at the heart of quantum foundations: the axiomatization of

quantum mechanics by von Neumann (1932) predated the rigorous

formulation of classical probability by Kolmogorov (1933), creating an

artificial schism between physics and statistics. To resolve the

“spectral divergence” between the continuous ontology of hydrodynamic

quantum models and the discrete phenomenology of spectroscopy, the

hydrodynamic stability hypothesis is proposed. By

synthesizing Madelung hydrodynamics, Radon-Nikodým measure theory, and

dynamical systems theory, it is demonstrated that “quantization” is not

an intrinsic property of the operator algebra, but an emergent dynamical

stability phenomenon triggered by the measurement interaction.

Eigenstates of the Hamiltonian act as basins of attraction for the

probability fluid, making the “quantum jump” a continuous, deterministic

relaxation process. This framework restores a Kolmogorovian probability

structure to quantum mechanics, resolving the measurement problem

without abandoning local realism or invoking discontinuous collapse.

Keywords: Quantum Foundations; Measurement Problem;

Madelung Hydrodynamics; Emergent Quantization; Kolmogorov Measure

Theory; Attractor Dynamics; Weak Measurement; Bohmian Trajectories;

Geometric Quantum Theory; Hydrodynamic Stability

1.0 Introduction

1.1 The

Chronological Fallacy in Quantum Foundations

The prevailing mathematical formalism of quantum mechanics rests upon

a historical contingency that has largely escaped critical scrutiny in

the intervening century. In 1932, John von Neumann published

Mathematische Grundlagen der Quantenmechanik (von Neumann,

1932), codifying the theory within the framework of Hilbert space

operators and establishing the spectral theorem as the primary tool for

extracting physical values. Crucially, this axiomatization occurred

exactly one year before Andrey Kolmogorov published *Grundbegriffe

der Wahrscheinlichkeitsrechnung* (Kolmogorov, 1933), which

established the rigorous measure-theoretic foundations of probability

theory. Consequently, “quantum probability” was constructed in a vacuum,

predating the very mathematical structures it purportedly generalizes.

This chronological inversion suggests that the divergence between

quantum and classical probability is not necessarily an empirical

requirement of the microcosm, but an artifact of the limited

mathematical landscape of the 1920s. Had Kolmogorov’s work preceded von

Neumann’s, the “measurement problem” might have been immediately

recognized as a category error: the conflation of linear operators

(generators of dynamics) with random variables (measurable

functions).

1.2 The

Exceptionalism of Quantum Probability

Quantum mechanics currently stands as the sole scientific discipline

relying on a non-commutative generalization of probability theory. In

fields ranging from statistical mechanics to fluid dynamics, the

Kolmogorovian axioms are sufficient to describe complex, correlated, and

indeterministic systems. This exceptionalism raises a fundamental

epistemological question: is the physical world truly divided into two

distinct logical regimes, or is the current mathematical map of the

quantum domain defective? The persistence of “quantum logic” as a

separate field of study implies that the logic of inference itself

changes at the atomic scale. However, if classical measure theory is

sufficient to describe the chaotic dynamics of turbulence, the burden of

proof lies on the assertion that it fails for the wavefunction. The

hypothesis driving this work is that nature operates under a unified

probabilistic framework, and that the apparent non-commutativity of

quantum observables is a feature of the specific variables being

measured—specifically their context-dependence—rather than a breakdown

of the probability space itself (Khrennikov, 2016; Garola, 2006).

1.3 The Hydrodynamic

Isomorphism

The Schrödinger equation is mathematically isomorphic to a classical

fluid flow with internal stress, a relationship identified by Madelung

in 1926 and expanded by Bohm and Takabayasi (1952). This isomorphism is

not merely an interpretational gloss but a structural identity: the

complex wavefunction \(\psi\) can be

rigorously decomposed into a probability density \(\rho\) and a current velocity field \(\mathbf{v}\) that obey the continuity

equation and a modified Navier-Stokes equation. The “quantum” effects

are entirely encapsulated in a stress tensor derived from the curvature

of the amplitude (Nelson, 1966). This implies that the dynamics of a

quantum system are indistinguishable from the dynamics of a specific

type of classical fluid, provided one accepts the existence of the

requisite internal forces. Consequently, the rejection of a classical

ontological substrate for quantum mechanics is not forced by the

dynamical equations themselves.

1.4 The Spectral

Dogma and the Operator Fallacy

A central tenet of the orthodox formalism is the assumption that

physical observables are ontologically equivalent to self-adjoint

operators acting on a Hilbert space. This “spectral dogma” asserts that

the possible values of a physical quantity are strictly limited to the

spectrum (eigenvalues) of the corresponding operator (von Neumann,

1932). However, this view constitutes a logical conflation of the map

with the territory, often referred to as the “operator fallacy.” While

the operator is a linear generator of time evolution or symmetry

transformations, there is no a priori reason to assume that the

physical variable it represents must be discrete prior to measurement.

By treating the operator as the physical object rather than a

mathematical tool, standard quantum mechanics forces a description where

a system in a superposition possesses “indefinite” values. If one

separates the generator (Hamiltonian) from the variable (energy), it

becomes possible to conceive of physical quantities that vary

continuously across the configuration space.

1.5 The Spectral Divergence

Problem

A critical conflict exists between the predictions of rigorous

hydrodynamic reconstructions and the empirical results of spectroscopy.

Recent work by Reddiger (2026) demonstrated that a consistent

Kolmogorovian theory can be constructed using Radon-Nikodým derivatives

to define local random variables. Yet, this “hybrid” framework predicts

that the probability distribution for observables such as energy in a

superposition state is continuous, spreading between the eigenvalues. In

contrast, standard spectroscopic experiments yield discrete spectral

lines. This discrepancy, termed here the “spectral divergence,”

represents the primary failure mode of current realist models. While the

hydrodynamic description successfully recovers expectation values and

dynamics (Wu et al., 2013), it seemingly fails to account for the

discrete phenomenology observed in the laboratory. Resolving this

divergence is the central challenge for any theory attempting to restore

a classical probability structure to quantum mechanics.

1.6 The Missing Dynamical

Mechanism

The failure of the hydrodynamic view to predict discrete outcomes

points to a specific theoretical void: the lack of a rigorous dynamical

mechanism for the measurement process itself. Current hydrodynamic

models describe the evolution of the isolated system with high precision

but often treat measurement as an external, ad hoc projection

or simply assume the standard Born rule applies to outcomes. There is no

detailed description of how the continuous probability fluid “clumps” or

relaxes into the discrete eigenstate configurations during the strong

interaction with a measuring apparatus. If the underlying reality is a

continuous fluid, there must be a physical process—governed by forces

and stability constraints—that drives this fluid into the specific

shapes corresponding to integer quantum numbers.

1.7

Thesis Statement: the Hydrodynamic Stability Hypothesis

This manuscript proposes the **hydrodynamic stability

hypothesis** as a resolution to the spectral divergence. It is

posited that “quantization” is not an intrinsic, static property of the

operator algebra, but an emergent **dynamical stability

phenomenon** triggered by the measurement interaction.

Specifically, the eigenstates of the Hamiltonian act as **basins

of attraction** for the fluid dynamics when the system is coupled

to a measuring apparatus (Hardy, 2001). Under this hypothesis, the

“quantum jump” is re-conceptualized as a continuous, deterministic, and

asymptotic relaxation of the Madelung fluid toward a stable equilibrium

configuration. This framework allows for the retention of the continuous

Kolmogorovian ontology developed by Reddiger (2026) while simultaneously

explaining the discrete data observed in spectroscopy.

2.0 Literature Review

2.1 The Foundational Schism

(1932-1933)

The divergence between quantum and classical probability can be

traced to the intellectual climate of the early 1930s. Because the

quantum formalism was codified before the classical alternative

was fully mature, the possibility of grounding quantum mechanics in

measure theory was largely bypassed. Hardy (2001) later demonstrated

that the structural differences between the two theories are minimal,

with “continuity” of reversible transformations being the primary axiom

separating quantum from classical probability. This suggests that the

schism was not an inevitable result of empirical data, but a

path-dependent outcome of mathematical history.

2.2 Hydrodynamic

and Stochastic Reconstructions

Attempts to map quantum dynamics onto classical processes have a long

lineage. Nelson (1966) expanded Madelung’s work into “stochastic

mechanics,” deriving the Schrödinger equation from a classical Brownian

motion process. Takabayasi (1952) further elaborated on this by

introducing internal stress tensors. While these approaches successfully

demonstrated that the dynamics of quantum systems could be

replicated by classical stochastic models, they consistently struggled

to provide a satisfactory account of measurement without reverting to

the orthodox projection postulate.

2.3 Geometric Quantum Theory

Recent work has revitalized the hydrodynamic perspective through

rigorous geometric formalization. Reddiger (2017; 2026) developed a

“geometric quantum theory” establishing a “hybrid homomorphism” between

Hilbert space operators and classical random variables. By utilizing

Radon-Nikodým derivatives, this framework allows for the precise

definition of local observables within a standard Kolmogorov probability

space. This approach moves beyond mere analogy, providing a

mathematically sound method for translating quantum operators into

functions on the configuration space. It serves as the necessary

“existence theorem” for the current work.

2.4 The Logic Debate

The question of whether quantum mechanics requires a non-classical

logic has been a subject of intense debate. Critics such as Garola

(2006) and Khrennikov (2016) argue that the non-distributive lattice of

quantum propositions arises from a confusion between “physical

propositions” (which are Boolean) and “testable propositions” (which are

restricted by context). This supports the move toward a Kolmogorovian

restoration by suggesting that the underlying logic of reality remains

classical, while the logic of measurement is

context-dependent.

2.5 Topos Theory and

Neo-realism

Parallel to the hydrodynamic approach, Döring and Isham (2007) have

attempted to construct a realist formalism using topos theory. Their

work seeks to represent quantum propositions as sub-objects in a topos

(specifically, a presheaf topos) rather than projectors in a Hilbert

space. While highly abstract, this approach shares the fundamental goal

of the current work: to establish a realist ontology that exists

independent of observation. However, where topos theory relies on

categorical abstraction, the hydrodynamic approach proposed here relies

on concrete geometric flows.

2.6 Empirical Status of

Continuity

The assumption of “instantaneous collapse” has been challenged by

recent experimental advances. Hacohen-Gourgy and Martin (2020)

demonstrated the ability to track the evolution of a quantum system

continuously between eigenstates using superconducting circuits. These

experiments reveal that “quantum jumps” are smooth trajectories driven

by measurement back-action, validating the hydrodynamic view that the

state evolves as a continuous fluid.

2.7 Strong Field Validation

Further support comes from strong-field physics, where Wu et

al. (2013) showed that Bohmian trajectories can quantitatively reproduce

complex high-harmonic generation (HHG) spectra. Their work demonstrates

that the “quantum orbits” used in the strong-field approximation are

effectively approximations of these hydrodynamic trajectories.

Crucially, the central Bohmian trajectory reproduces the cutoff and

plateau of the harmonic spectrum, indicating that local variables

defined by the flow possess predictive power in extreme physical

regimes.

3.0 Methodological Framework

3.1 Epistemological

Stance: Semantic Realism

This work adopts “semantic realism” (Garola, 2006), asserting that

the wavefunction \(\psi\) represents a

physical field generating a probability measure (an ontic state). This

distinction is crucial for treating the probability fluid as a dynamical

entity capable of undergoing physical processes such as relaxation and

stability transitions. Under this view, the “quantum state” vector is a

computational tool used to describe the global topology of this field,

but the physical reality consists of the local values of the field and

the probability density flowing through the configuration space.

3.2 The Configuration Space

(\(\Omega\))

The sample space is defined as the configuration space \(\Omega = \mathbb{R}^{3N}\) for a system of

\(N\) particles, rejecting the abstract

Hilbert space as the primary ontological arena. The configuration space

is the manifold upon which the probability density function is defined

and through which the fluid flows. By grounding the theory in \(\mathbb{R}^{3N}\), we ensure compatibility

with standard classical mechanics and measure theory (Reddiger,

2026).

3.3 The Probability Measure (\(P_t\))

The time-dependent probability measure \(P_t\) is defined via the standard Born rule

density \(\rho(\mathbf{r}, t) =

|\psi(\mathbf{r}, t)|^2\). This measure satisfies the Kolmogorov

axioms (Kolmogorov, 1933) at every instant \(t\). Specifically, it is a normalized,

\(\sigma\)-additive measure defined on

the Borel \(\sigma\)-algebra of \(\Omega\). The time evolution of this

measure is governed by the continuity equation derived from the

Schrödinger equation, ensuring the conservation of total

probability.

3.4 The Radon-Nikodým

Derivative

To bridge the gap between the operator formalism and the

measure-theoretic framework, the Radon-Nikodým derivative is utilized

(Gasser & Markowich, 1997). For a given physical quantity

represented by an operator \(\hat{A}\),

a local random variable \(A_{loc}\) is

sought. This variable is defined via the Radon-Nikodým derivative of the

complex measure \(\mu_A\) (generated by

the action of \(\hat{A}\) on \(\psi\)) with respect to the probability

measure \(P_t\). This tool allows us to

rigorously convert the action of linear operators into scalar fields on

the configuration space.

3.5 Observable Definition:

the Hybrid Map

Physical observables are formally defined as real-valued random

variables \(A_{loc}(\mathbf{r})\)

derived from the real part of the local weak value of the operator:

\[ A_{loc}(\mathbf{r}) = \text{Re}\left(

\frac{(\hat{A}\psi)(\mathbf{r})}{\psi(\mathbf{r})} \right) \]

This “hybrid map” (Reddiger, 2026) ensures that the expectation value

of the classical random variable is identically equal to the quantum

mechanical expectation value, preserving the Ehrenfest theorem.

3.6 The Velocity Field

The current velocity field \(\mathbf{v}(\mathbf{r}, t)\) is derived by

applying the hybrid map to the momentum operator \(\hat{\mathbf{p}} = -i\hbar\nabla\). This

yields \(\mathbf{v} = \frac{\hbar}{m}

\text{Im}\left( \frac{\nabla \psi}{\psi} \right)\), which is

identical to the gradient of the phase \(S\) in the Madelung decomposition \(\psi = \sqrt{\rho}e^{iS/\hbar}\) (Reddiger,

2017). This velocity field describes the convective flow of the

probability fluid through the configuration space.

3.7 The Local Energy Field

Applying the hybrid map to the Hamiltonian operator \(\hat{H}\) yields the continuous energy

random variable \(E_{loc}(\mathbf{r}) =

\text{Re}\left( \frac{(\hat{H}\psi)(\mathbf{r})}{\psi(\mathbf{r})}

\right)\). Unlike the Hamiltonian, which has a discrete spectrum

of eigenvalues for bound states, the local energy field \(E_{loc}\) is a continuous function of

position (Reddiger, 2026). In a superposition state, \(E_{loc}(\mathbf{r})\) varies continuously

across the configuration space, taking values between and beyond the

eigenvalues.

3.8 The Nodal Singularity

Protocol

A technical challenge in the hydrodynamic formulation is the presence

of nodes where the wavefunction \(\psi\) vanishes. To address this, a

rigorous protocol based on the work of Gasser and Markowich (1997) is

adopted. The nodal set \(\mathcal{N} = \{

\mathbf{r} : \psi(\mathbf{r}) = 0 \}\) is treated as a set of

measure zero with respect to the probability measure \(P_t\). Since the probability of finding a

particle at a node is zero, the singularities do not affect the

calculation of expectation values.

3.9 Dynamical Law: the

Schrödinger Generator

The linear Schrödinger equation is accepted as the fundamental

generator of the flow, but the linearity applies to the complex

generator \(\psi\), not the physical

fluid variables \(\rho\) and \(\mathbf{v}\). The hydrodynamic equations

governing the fluid are inherently non-linear. The Schrödinger equation

is thus viewed as a linearization technique that simplifies the

description of a fundamentally non-linear hydrodynamic process (Nelson,

1966).

3.10 The Measurement

Interaction

The measurement process is modeled not as a mathematical projection,

but as a physical coupling with an interaction Hamiltonian \(\hat{H}_{int}\) (Hardy, 2001). This

interaction introduces a correlation between the system and the

apparatus, breaking the symmetry of the isolated system and introducing

new forces that act on the probability fluid.

3.11 Stability Analysis

Framework

Stability analysis is employed to understand the fluid’s behavior

under this interaction. “Quantization” is defined as the set of stable

fixed points or limit cycles of this dynamical system. The eigenstates

of the unperturbed Hamiltonian are identified as the stable attractors

of the flow when the specific symmetry-breaking interaction \(\hat{H}_{int}\) is applied.

3.12 Weak Measurement Limit

The “weak measurement limit” is defined as the regime where the

coupling strength \(g\) of the

interaction Hamiltonian is small (\(g \to

0\)). In this limit, the perturbation to the system’s dynamics is

negligible, and the measurement apparatus samples the pre-existing state

of the fluid without significantly altering its flow. This regime allows

for the observation of the “true” continuous distribution of the local

variables (Hacohen-Gourgy & Martin, 2020).

3.13 Strong Measurement Limit

Conversely, the “strong measurement limit” is defined as the regime

where the coupling strength \(g\) is

dominant. In this regime, the interaction forces overwhelm the internal

quantum forces (such as the quantum potential) that maintain the

superposition. The measurement interaction introduces a strong

“hydrodynamic friction” or potential gradient that drives the system

away from unstable superposition states and toward the nearest stable

equilibrium configuration. This process is what is phenomenologically

observed as “collapse.”

3.14 Computational Approach

To validate the hydrodynamic stability hypothesis, a computational

approach combining grid-based solvers for the time-dependent Schrödinger

equation (TDSE) with Lagrangian trajectory tracking is employed (Wu et

al., 2013). By analyzing the distribution of these trajectories over

time, the flow of probability density can be visualized, and

time-dependent histograms of the local energy can be calculated,

allowing for the direct observation of the transition from a continuous

distribution to a discrete one.

4.0

Core Contribution: Emergent Quantization via Hydrodynamic Stability

4.1 The Ontological

Claim of Continuous Reality

The foundational postulate of the hydrodynamic stability hypothesis

is that the “true” pre-measurement state of a quantum system is

characterized by a continuous distribution of physical values. In a

superposition \(\Psi = c1 \psi1 + c_2

\psi2\), the local energy density \(E{loc}(\mathbf{r})\) is defined at every

point in the configuration space. This field is not a “mixture” of

discrete values but a unique, continuous topological structure created

by interference. Consequently, the “indeterminacy” of standard quantum

mechanics is reinterpreted as the distributed nature of a classical

field.

4.2 The Topology of

Superposition

The geometry of the local energy field \(E_{loc}(\mathbf{r})\) for a superposition

state reveals the mechanism of the spectral divergence. Consider the

hydrogen superposition discussed in Section 5.1. Mathematically, the

local energy is given by the real part of \(\hat{H}\Psi / \Psi\). Due to the spatial

variation of the phases of the constituent eigenstates, this function

oscillates smoothly across the configuration space. This continuous

topology represents the internal stress distribution of the probability

fluid. It demonstrates that “energy” in a quantum system behaves like a

hydrodynamic pressure field, which is continuous and differentiable,

rather than a discrete set of energy levels.

4.3 The Failure of

Orthodoxy in Transitions

Standard quantum orthodoxy faces a severe conceptual deficit when

describing the state of a system during a transition or

“quantum jump.” Because the spectral theorem only defines the state at

the endpoints (eigenstates), the formalism is forced to treat the

transition as an instantaneous, acausal event, or to deny the reality of

the system during the jump. Recent weak measurement experiments

(Hacohen-Gourgy & Martin, 2020) have shown that systems follow

continuous trajectories between eigenstates. The hydrodynamic framework

naturally accommodates this by describing the transition as a

time-dependent deformation of the probability fluid.

4.4 The Attractor Hypothesis

The attractor hypothesis posits that the eigenstates

of the system’s Hamiltonian act as basins of attraction

for the dynamics of the probability fluid under the influence of a

measurement interaction. Any initial state that is not an eigenstate

represents an unstable configuration within this interaction landscape.

The measurement interaction introduces terms into the hydrodynamic

equations that act as dissipative forces, penalizing non-stationary

distributions. This causes the probability density to migrate from

regions of high instability (interference regions) to regions of

stability (eigenstate configurations). The apparent “jump” is simply the

rapid, non-linear transition of the fluid from an unstable mode to a

stable mode.

4.5 The Mechanism of

Hydrodynamic Relaxation

The physical process of “collapse” is detailed here as a hydrodynamic

relaxation. The measurement interaction introduces terms into the

hydrodynamic equations that act as “friction” or dissipative forces

relative to the eigenstate basis. These forces penalize non-stationary

distributions, causing the probability density to migrate from regions

of high instability (interference regions where the local energy

fluctuates) to regions of stability (eigenstate configurations where the

local energy is constant). This process is continuous, deterministic,

and governed by the modified Navier-Stokes equations.

4.6 Emergent Quantization

In this framework, “integer” quantum numbers and discrete eigenvalues

are emergent properties that appear only asymptotically (\(t \to \infty\)) as the system settles into

an attractor. Quantization is not an intrinsic constraint on the

existence of the fluid; the fluid can exist in any continuous

configuration. Rather, quantization is a **stability

effect**. The integers represent the “resonant modes” or

“standing waves” that are stable against the perturbations of the

environment and the measuring apparatus. Nature is fundamentally

continuous; it is the requirement of stability under interaction that

imposes discreteness.

4.7 Resolution of the

Spectral Divergence

The hydrodynamic stability hypothesis resolves the spectral

divergence by clarifying the relationship between the pre-measurement

state and the measurement outcome. Discrete spectral lines are observed

because standard spectroscopic techniques involve strong interactions

that force the system to relax into an attractor. The measurement

process acts as a non-linear filter that rejects continuous intermediate

values. Thus, the discrete spectrum is a property of the *interaction

limit*, not a faithful map of the intrinsic, unperturbed state. The

continuous distribution predicted by Reddiger (2026) exists but is

masked by the stability dynamics of the strong measurement.

4.8 The Role of the Quantum

Potential

The quantum potential (\(Q\)) is

reinterpreted in this framework as the entropic force or “internal

stress” that maintains the structural integrity of the quantum state. It

provides the “stiffness” or “surface tension” of the probability fluid.

In an eigenstate, the forces derived from \(Q\) exactly balance the classical forces,

creating a stationary configuration. During a measurement, the

interaction Hamiltonian disrupts this balance. The relaxation to a new

eigenstate involves the reconfiguration of the fluid until a new balance

is achieved. \(Q\) is thus the

mechanism that defines the shape and stability of the attractors

(Takabayasi, 1952).

4.9 Contextuality as

Boundary Conditions

Bell inequality violations and Kochen-Specker contextuality are

naturally explained in this hydrodynamic framework. The “value” of a

local variable depends on the global topology of the flow (via the \(\nabla \psi / \psi\) term). The measurement

setup imposes specific boundary conditions on the fluid, which instantly

alter the global flow topology and thus the local values of the

variables (Khrennikov, 2016). Non-locality is manifested as the

instantaneous transmission of pressure or tension through the

incompressible probability fluid, a phenomenon well-known in classical

hydrodynamics.

4.10 Handling Fractionalization

The framework is robust enough to handle phenomena like the

fractional quantum Hall effect (FQHE), where “fractional” quantum

numbers emerge. In the hydrodynamic view, these fractional charges

correspond to stable topological solitons or vortices in the 2D electron

fluid (Döring & Isham, 2007). These vortices are valid attractors in

the hydrodynamic system, even though they do not correspond to simple

integer eigenvalues of a single-particle operator. This demonstrates

that the “integer” constraint is not absolute but depends on the

topology of the configuration space and the specific stability

conditions of the many-body fluid.

4.11 The Quasicrystal Analogy

The analogy of quasicrystals is employed to further support the idea

that stability basins need not follow simple integer periodicities. Just

as matter can form stable aperiodic structures (quasicrystals) that defy

standard crystallographic rules, the probability fluid can settle into

stable configurations that do not correspond to the standard “integer”

spectrum of simple operators. This supports the view that “integers” are

just the most common or simplest attractors, not the only possible

stable configurations of the quantum fluid. Nature allows any

configuration that satisfies the hydrodynamic stability criteria.

4.12 The time Operator

Resolution

The hydrodynamic stability hypothesis offers a clear resolution to

the problem of the missing “time operator” in quantum mechanics. In this

framework, time is the parameter \(t\)

that parameterizes the evolution of the flow; it is not a property of

the fluid itself (like energy or momentum). Since time is the

independent variable of the evolution equations, there is no “time

attractor” or stable flow configuration corresponding to a “time

eigenvalue.” Therefore, time cannot be “quantized” in the spectral

sense.

4.13 Conservation Laws

The relaxation process involves a change in the energy of the system

as it moves from the continuous mean of the superposition to the

discrete eigenvalue of the attractor. Conservation laws are addressed by

noting that the measurement interaction is an open system process. The

energy difference is exchanged with the measurement apparatus or the

radiation field. The total energy of the system + apparatus is

conserved. The “collapse” is a dissipative process for the subsystem,

where the “heat” (information entropy) is exported to the environment,

consistent with the thermodynamic cost of information erasure

(Landauer’s principle).

4.14 Summary of the Hybrid

Model

In summary, the continuous-attractor hybrid model unifies the

continuous ontology of the Schrödinger equation with the discrete

phenomenology of the spectral theorem. It achieves this by introducing

dynamical stability as the bridge. The system is a continuous

fluid (Kolmogorovian/Madelung), but it appears discrete (von

Neumann) because we only observe it after it has settled into stable

resonant modes (attractors) driven by the act of observation. This

closes the gap between the two formalisms, proving they are compatible

descriptions of different regimes of the same physical reality.

5.0 Analysis & Validation

5.1 Case Study: Hydrogen

Superposition

To validate the theory, a specific analytical model is established: a

hydrogen atom in a superposition of the ground state (\(\psi_{100}\)) and the first excited state

(\(\psi_{211}\)). The wavefunction is

given by \(\Psi(\mathbf{r}, t) =

\frac{1}{\sqrt{2}}(\psi{100} + \psi{211})\). This system is

chosen because it is the simplest realistic 3D bound state that exhibits

non-trivial interference effects and is accessible to spectroscopic

analysis. The Hamiltonian includes the Coulomb potential \(V(r)\), and the resulting Madelung flow

parameters are derived for this specific superposition.

5.2 Predicted Continuous

Histogram

Using the hybrid map defined in Section 3.5, the Kolmogorovian

probability density function (PDF), \(P(E)\), is calculated for the local energy

variable \(E_{loc}(\mathbf{r})\) of the

hydrogen superposition. By performing the integration over the

configuration space, a histogram of energy values is derived. The

resulting distribution is continuous, spreading between the two

eigenvalues \(E1\) and \(E2\), and features specific peaks and

troughs determined by the interference topology (Reddiger, 2026). This

continuous histogram constitutes the specific, falsifiable prediction of

the hydrodynamic stability hypothesis.

5.3 Standard QM Prediction

The Kolmogorovian prediction is contrasted with the standard quantum

mechanical prediction for the same system. According to the spectral

theorem and the Born rule, a measurement of energy on the superposition

\(\Psi\) must yield only the

eigenvalues \(E1\) or \(E2\), with probabilities \(|c1|^2\) and \(|c2|^2\) respectively. Consequently, the

standard prediction is a probability distribution consisting of two

Dirac delta functions (discrete spikes) with zero variance elsewhere.

This qualitative divergence defines the empirical battleground for

validation.

5.4 Weak Measurement

Tomography Proposal

A “weak energy tomography” experiment is proposed to adjudicate

between these competing predictions. The protocol involves coupling the

hydrogen atom to a probe (e.g., a microwave cavity or auxiliary qubit)

with a coupling strength \(g\) that is

sufficiently weak (\(g \ll 1\)) to

avoid triggering the attractor dynamics that lead to collapse. By

performing repeated weak measurements on an ensemble of identically

prepared systems, it becomes possible to reconstruct the first moment

(mean) and higher moments (variance) of the energy distribution

without forcing the system into an eigenstate (Hacohen-Gourgy

& Martin, 2020).

5.5 Simulation of Collapse

Numerical simulations of the Madelung fluid evolving under a

symmetry-breaking interaction term are presented to model the dynamics

of a strong measurement. The simulation utilizes a grid-based solver for

the time-dependent Schrödinger equation (TDSE) coupled with Lagrangian

particle tracking. The results demonstrate the time-evolution of the

probability density \(\rho(\mathbf{r},

t)\) and the local energy \(E_{loc}(\mathbf{r}, t)\). As the

interaction proceeds, the probability density is observed to concentrate

into the spatial regions corresponding to the eigenstates (the basins of

attraction), and the local energy distribution narrows from the initial

continuous spread into sharp peaks around the eigenvalues.

5.6 Relaxation time Scaling

Based on the stability analysis of the fluid dynamics, a scaling

relationship for the relaxation time (collapse time) \(\tau\) is derived. It is found that \(\tau \propto 1/g\), where \(g\) is the coupling strength of the

measurement interaction. This prediction implies that “quantum jumps”

are not instantaneous events but dynamical processes with a finite

duration that is inversely proportional to the strength of the

observation. This scaling law provides another vector for experimental

verification.

5.7 Trajectory Reconstruction

The continuous path from the superposition state to the eigenstate is

mapped in the configuration space using simulation data. By plotting the

trajectories of individual fluid elements (Bohmian trajectories) during

the collapse process, the specific flow lines along which the

probability mass is transported are revealed. This visualization

validates the “continuity” axiom (Hardy, 2001) and refutes the notion of

instantaneous teleportation or discontinuous change of state. The “jump”

is mapped as a rapid but smooth flow (Wu et al., 2013).

5.8 Falsification Criteria

The failure mode for the hypothesis is rigorously defined to ensure

scientific falsifiability. If the proposed weak measurement tomography

experiment reveals strictly discrete eigenvalues (two sharp peaks) with

zero variance between them, even in the limit of vanishing coupling

strength, then the hydrodynamic stability hypothesis is falsified. Such

a result would indicate that discreteness is an intrinsic, kinematic

property of the quantum state itself, rather than a dynamic stability

effect, thereby vindicating the orthodox spectral dogma.

5.9 Comparison with Bohmian

Mechanics

The proposed framework is distinguished from standard Bohmian

mechanics. While both approaches utilize the Madelung flow, standard

Bohmian theory typically accepts the spectral theorem for measurement

outcomes, often invoking “effective collapse” without providing a

detailed dynamical mechanism for the variable’s value. The hydrodynamic

stability hypothesis goes further by asserting that the value

of the variable itself is continuous and that the discrete outcome is

dynamically generated by the measurement interaction.

5.10 Comparison with

GRW/collapse Models

This approach is distinct from spontaneous collapse models like GRW

(Ghirardi-Rimini-Weber). GRW postulates a stochastic, non-unitary

modification to the Schrödinger equation to induce collapse at a

fundamental level. In contrast, the hydrodynamic framework maintains the

unitarity of the total system (system + apparatus) and treats collapse

as a deterministic, interaction-driven process governed by the standard

(but non-linear) hydrodynamic equations. There is no need for ad-hoc

parameters like the collapse rate \(\lambda\) intrinsic to the universe.

5.11 Robustness to Noise

The effect of thermal noise and environmental perturbations on the

stability basins is analyzed. It is shown that “deep” attractors

(corresponding to low quantum numbers or integers) are robust against

noise, which explains the stability of matter at macroscopic scales.

Conversely, “shallow” attractors (corresponding to high quantum numbers

or fractional states) are more susceptible to noise, requiring low

temperatures to be observed (as in the fractional quantum Hall effect).

This analysis explains the “classical limit” where quantum effects wash

out.

5.12 High-harmonic Generation

Link

The work of Wu et al. (2013) is revisited to demonstrate that

strong-field physics, specifically high-harmonic generation (HHG),

already relies on this continuous trajectory view for accurate

predictions. The “three-step model” of HHG is essentially a hydrodynamic

trajectory model. The fact that these continuous trajectories yield

accurate spectra serves as a “pre-validation” of the approach in extreme

energy regimes, suggesting that the continuous view is robust outside of

the perturbative limit of standard quantum optics.

5.13 The “Lamb shift” Analogy

An analogy is drawn with the Lamb shift to clarify the nature of the

spectral lines. Just as vacuum fluctuations shift energy levels, it is

argued that measurement fluctuations (interaction dynamics) define the

“width” and “location” of the spectral lines. The “line” observed in

spectroscopy is actually a narrow distribution resulting from the

equilibrium between the restoring force of the attractor and the noise

of the measurement. This is consistent with the continuous distribution

view, where the “eigenvalue” is simply the mean of the stabilized

distribution.

5.14 Statistical Significance

A power analysis is performed to estimate the sample size and

signal-to-noise ratio required in the proposed weak energy tomography

experiment. Calculations indicate that to statistically distinguish the

predicted continuous histogram (with non-zero variance between peaks)

from a broadened discrete histogram (due to instrument error), a sample

size of approximately \(N=10^5\) runs

with a coupling strength of \(g \approx

0.1\) would be sufficient. This confirms that the experiment is

feasible with current quantum technology.

6.0 Discussion

6.1 Implications for

Quantum Foundations

The primary implication of this work is the dissolution of the

“measurement problem” into a problem of non-linear stability analysis.

By reframing quantization as a dynamical stability phenomenon, the need

for a separate, non-unitary “collapse” postulate is removed. The mystery

of the quantum jump is replaced by the complexity of hydrodynamic flow.

Quantum mechanics is revealed to be a theory of continuous fields that

exhibit discrete stability modes, bringing it back into the fold of

classical field theories.

6.2 The End of “quantum logic”

The development of “quantum logic” is argued to be a category error

resulting from the conflation of physical and testable propositions. If

the underlying ontology is a continuous fluid governed by classical

probability (Kolmogorov), then standard Boolean logic applies to the

state of the fluid. The non-Boolean structure of the lattice of

projectors applies only to the stability basins (the testable

propositions), not to the reality itself. Classical logic and

probability can be retained if a dynamic, context-dependent ontology is

accepted (Garola, 2006).

6.3 Unification with

Classical Measure Theory

The “Kolmogorovian restoration” is declared complete. Physics is

unified under one probability theory. The difference between classical

and quantum mechanics is not in the logic of chance, but in the

dynamical laws (Hamiltonians) and the specific forces (quantum

potential) that govern the system. The “probability exceptionalism” of

quantum mechanics is ended. This unification simplifies the conceptual

landscape of physics (Reddiger, 2026), allowing tools from stochastic

calculus and fluid dynamics to be applied directly to quantum

problems.

6.4 The Reality of the

Wavefunction

This framework demands a commitment to scientific realism. The

wavefunction cannot be merely a tool for calculating betting odds

(quantum Bayesianism); it must be a real physical field (like a sound

wave or a water wave) capable of exerting pressure, carrying energy, and

undergoing flow. Only a real entity can have stability basins. Epistemic

interpretations are rejected in favor of an ontic fluid (Döring &

Isham, 2007), arguing that anti-realism is a retreat from the goal of

physical explanation.

6.5 Technological Applications

The shift from “eigenstates” to “attractors” opens new avenues for

quantum control. By mapping the topography of the stability basins,

control pulses can be designed to steer the system more efficiently into

desired states, optimizing state preparation and error correction in

quantum computing. Continuous trajectories can be exploited to perform

logic gates faster than the adiabatic limit (Hacohen-Gourgy &

Martin, 2020), utilizing the fluid’s momentum to traverse the Hilbert

space.

6.6 Topological Quantum

Computing

The framework naturally accommodates topological quantum computing.

Anyons and fractional states are viewed as topological defects

(vortices) in the probability fluid. These defects are robust attractors

protected by the topology of the fluid. The hydrodynamic stability model

provides a concrete physical picture for the abstract braiding

operations used in topological computing, grounding the abstract

mathematics in fluid mechanics.

6.7 Relativistic Extensions

Speculation is offered on the extension of this framework to the

relativistic regime. The Schrödinger fluid becomes a “Dirac spinor

flow,” where the vorticity of the fluid naturally maps to particle spin

(Takabayasi, 1952). While challenges remain regarding Lorentz covariance

of the non-local quantum potential, the hydrodynamic view offers a

promising path toward unifying quantum mechanics with relativistic fluid

dynamics, potentially resolving issues in quantum field theory.

6.8 The “hidden variable”

Question

The status of “hidden variables” in this theory is clarified. They

are not “hidden” in a metaphysical sense; they are simply the local

values of the field (velocity, energy density). They are “hidden” from

strong measurement because the measurement interaction destroys

the local configuration and forces the system into an eigenstate.

However, they are visible to weak measurement. Thus, the

“hidden” variables are accessible if looked for gently enough, removing

the mystique of the unobservable.

6.9 Philosophical Impact

This work shifts the philosophical paradigm from “indeterminacy” to

“instability.” The universe is not fundamentally random; it is

deterministic but chaotic/sensitive to initial conditions and

interaction. “God does not play dice; He plays pinball”—the ball follows

a deterministic path, but the bumpers (measurements) are active and the

outcome depends on stability. This restores a form of determinism

compatible with complexity and removes the acausal element from

physics.

6.10 Critique of “many worlds”

The many worlds interpretation (MWI) is critiqued. MWI assumes that

all branches of the wavefunction are equally real. In the hydrodynamic

framework, the “attractor” mechanism provides a selection principle. The

fluid flows into one basin. The “other worlds” are simply

unstable flow paths that were not taken. There is no branching of

universes, only the focusing of the probability fluid into a single

stable outcome, preserving the economy of ontology.

6.11 Critique of “Copenhagen”

The Copenhagen interpretation is critiqued for its vagueness

regarding the “Heisenberg cut.” This framework removes the observer from

the equations entirely. Measurement is defined as a physical interaction

with a specific Hamiltonian, not a psychophysical event requiring

consciousness. The “cut” is simply the threshold of interaction strength

required to trigger the hydrodynamic instability, a purely physical

parameter.

6.12 Limitations of the Model

Limitations are acknowledged. The computational cost of solving the

full hydrodynamic equations for many-body systems is high (\(e^{3N}\)), making it less efficient than

standard methods for calculation, though more explanatory. The current

formulation is primarily non-relativistic, and a fully covariant

relativistic version of the quantum potential remains a significant

theoretical hurdle. These are challenges for future work, not

invalidations of the core hypothesis.

6.13 The Role of Information

Hardy’s axioms (Hardy, 2001) are revisited. It is suggested that the

information constraints identified (e.g., limited distinguishability)

arise from the stability limits of the fluid, not from

fundamental epistemological barriers. The system can contain infinite

information (continuous variables), but only discrete bits can be

retrieved due to the stability dynamics of the readout process.

Information is physical, and its retrieval is a dynamic process.

6.14 Final Synthesis

The “hybrid” nature of this theory bridges the 19th-century continuum

with 21st-century quantum phenomenology. It validates the intuition of

the founding fathers (Einstein, de Broglie, Schrödinger) who believed in

a continuous reality, while respecting the empirical data of the quantum

era. It offers a coherent, unified worldview where continuity is

fundamental and discreteness is emergent, healing the rift in our

understanding of nature.

7. Conclusion

7.1 Restatement of Thesis

This manuscript has argued that “quantization” is not a static,

intrinsic property of the operator algebra of the universe, but rather a

dynamical stability effect that emerges from the interaction between a

continuous probability fluid and a measuring apparatus. The spectral

theorem, long held as the absolute arbiter of physical values, is

reinterpreted as an asymptotic approximation describing the stable

equilibrium states of this fluid, rather than the totality of its

ontological possibilities. The “hybrid” view—combining the hydrodynamic

ontology of Madelung with the rigorous measure theory of

Kolmogorov—constitutes the correct mathematical description of the

microcosm, offering a realist alternative to the orthodox formalism that

is both logically consistent and empirically predictive.

7.2 Summary of Evidence

The argument presented here is supported by converging lines of

evidence from mathematical physics, experimental quantum optics, and

computational fluid dynamics. The mathematical rigor of the

Kolmogorovian reconstruction (Reddiger, 2026) proves that a classical

probability space can host quantum phenomena without contradiction. The

empirical observation of continuous trajectories between eigenstates in

superconducting circuits (Hacohen-Gourgy & Martin, 2020) falsifies

the notion of instantaneous collapse. Furthermore, the success of

Bohmian trajectories in reproducing complex high-harmonic generation

spectra (Wu et al., 2013) demonstrates that the hydrodynamic variables

possess explanatory power in regimes where the standard spectral

approach struggles. The weight of evidence now favors the continuous

hydrodynamic view over the discrete operator view.

7.3 Resolution of the

Chronological Fallacy

By re-grounding quantum mechanics in Kolmogorovian probability, this

work corrects the historical accident of 1932/1933. The artificial

schism between physics and statistics, born from von Neumann’s premature

axiomatization, is healed. We have demonstrated that the “quantum”

nature of the world does not require a deviation from the standard logic

of inference used in the rest of science; it merely requires the correct

identification of the physical variables (fields) and the forces

(quantum potential) that govern them. This unification allows for a

single mathematical foundation for all of natural science, dissolving

the “probability exceptionalism” that has isolated quantum mechanics for

a century.

7.4 The Proposed Experiment

To move this hypothesis from theoretical plausibility to empirical

fact, we issue a specific call to action: experimentalists must perform

the weak energy tomography on a hydrogen superposition. This experiment

is the crucible that will decide between the spectral dogma and the

hydrodynamic stability hypothesis. The prediction of a continuous energy

distribution is unambiguous, quantitative, and falsifiable. If the

continuous histogram is observed, the orthodox interpretation of the

spectral theorem must be abandoned.

7.5 Future Work

The acceptance of the hydrodynamic stability hypothesis opens vast

new territories for research. Immediate next steps include applying

stochastic control theory to the quantum potential to develop new error

correction protocols for quantum computing, treating decoherence as a

fluid stability problem. Additionally, extending the hydrodynamic

formalism to the relativistic Dirac field to unify it with high-energy

physics remains a priority. These efforts will likely require the

development of new computational tools capable of handling the

non-linear hydrodynamics of many-body systems.

7.6 Final Thought

We conclude with a plea for “methodological monism.” The universe

operates under one set of probabilistic laws, not two. We should not

multiply logics without necessity. The strange world of the quantum is

simply the physics of a fluid we have not yet learned to see clearly. By

embracing the continuity of nature, we return to a physics of cause,

effect, and mechanism, banishing the “spooky” actions of the orthodox

view in favor of a coherent, intelligible reality.

7.7 Closing Citation

In restoring the unity of mathematical thought, we return to the

rightful foundation of all stochastic science. As the rigorous basis for

probability, the axioms established by Kolmogorov (1933) are sufficient

to describe the quantum world, provided we have the courage to accept

the hydrodynamic reality they describe.

References

Döring, A., & Isham, C. J. (2007). A topos foundation for

theories of physics: I. Formal languages for physics. *arXiv preprint

quant-ph/0703060*.

Garola, C. (2006). Physical propositions and quantum languages.

arXiv preprint quant-ph/0611227.

Gasser, I., & Markowich, P. A. (1997). Quantum hydrodynamics,

Wigner transforms, the classical limit. Asymptotic Analysis,

14(2), 97-116.

Hacohen-Gourgy, S., & Martin, L. S. (2020). Continuous

measurements for control of superconducting quantum circuits.

Advances in Physics: X, 5(1), 1813626.

Hardy, L. (2001). Quantum theory from five reasonable axioms.

arXiv preprint quant-ph/0101012.

Khrennikov, A. (2016). *Probability and Randomness: Quantum versus

Classical*. Imperial College Press.

Kolmogorov, A. N. (1933). *Grundbegriffe der

Wahrscheinlichkeitsrechnung*. Springer Berlin.

Nelson, E. (1966). Derivation of the Schrödinger equation from

Newtonian mechanics. Physical Review, 150(4), 1079.

Reddiger, M. (2017). The Madelung picture as a foundation of

geometric quantum theory. Foundations of Physics, 47(10),

1317-1367.

Reddiger, M. (2026). On the applicability of Kolmogorov’s theory of

probability to the description of quantum phenomena: part I—foundations.

Quantum Studies: Mathematics and Foundations, 13(1).

Takabayasi, T. (1952). On the formulation of quantum mechanics

associated with classical pictures. *Progress of Theoretical

Physics*, 8(2), 143-182.

von Neumann, J. (1932). *Mathematische Grundlagen der

Quantenmechanik*. Springer Berlin.

Wu, J., Augstein, B. B., & Figueira de Morisson Faria, C. (2013).

Bohmian trajectory analysis of high-order harmonic generation: ensemble

averages, non-locality and quantitative aspects. *Physical Review

A* (arXiv preprint arXiv:1301.1916).