Quantization as Hydrodynamic Relaxation: Eigenstates as Attractor Basins in a Dissipative Madelung Fluid

#1. Introduction

Quantum processors must repeatedly measure qubit states to extract computational results, and the fidelity of those measurements sets a hard ceiling on usable circuit depth. The Tianyan‑287 superconducting platform reports single‑qubit gate fidelity $99.90\%$, two‑qubit gate fidelity $99.56\%$, and readout fidelity $98.7\%$ [2]. These numbers translate into error budgets that limit circuit depth and increase the overhead of error‑correction protocols, directly affecting the cost and scalability of quantum‑computing deployments. The conventional collapse postulate treats measurement as a fundamentally stochastic, discontinuous event that cannot be modeled within deterministic Schrödinger evolution. If measurement could instead be described as a continuous, deterministic relaxation of a quantum fluid, hardware designers could incorporate engineered dissipation to steer the system toward desired outcomes, potentially reducing readout error and simplifying error‑mitigation strategies.

This paper investigates precisely that possibility. By augmenting the Madelung hydrodynamic equations with a linear drag term, we obtain a dissipative fluid whose stationary eigenstate densities remain exact fixed points. Linear stability analysis shows that all perturbation modes decay at a universal rate $\gamma/2$, with an overdamped crossover at $\gamma=\hbar k^{2}/m$. Closed‑form level‑to‑level relaxation times are derived, revealing a characteristic acceleration of relaxation for higher excitations. The Born rule is introduced as an explicit, testable ensemble assumption rather than an axiom.

Contributions.

• C1. Derivation of the damped Madelung equations and demonstration that eigenstate densities are exact fixed points of the flow (Section 4.1).
• C2. Linearized analysis showing universal decay rate $\gamma/2$ for all wavenumbers, with overdamped crossover at $\gamma=\hbar k^{2}/m$ (Section 4.2).
• C3. Closed‑form level‑to‑level relaxation law $t_{n\to n-1}=\ln[(n+\tfrac12)/(n-\tfrac12)]/(2\gamma)$ and quantitative predictions for $t_{1\to0}$ and $t_{2\to1}$ (Section 4.3).
• C4. Per‑period phase‑space contraction factors and a phenomenological comparison between relaxation and diffusion timescales (Sections 4.4–4.5).
• C5. Proposes a falsification study design that would test whether deterministic maps can reproduce Born‑rule probabilities, thereby treating the Born rule as an explicit ensemble assumption (Section 4.7).
• C6. Specification of a falsification protocol targeting the $n$‑dependence of relaxation times and the continuity of trajectories (Sections 3.3 and 7).

#2. Background and Related Work

[1] Affine quantization. The work shows that the commutator $[Q,D]=i\hbar\,Q$ accommodates variables constrained to half‑line domains, a property that matches the non‑negative Madelung density $\rho\ge0$ and motivates an algebraic foundation compatible with hydrodynamic reformulations.

[2] Tianyan quantum cloud platform. The entry reports hardware figures: $105$ qubits, single‑qubit gate fidelity $99.90\%$, two‑qubit gate fidelity $99.56\%$, and readout fidelity $98.7\%$. These numbers provide the experimental context in which any alternative measurement theory must eventually be tested.

[3] Time‑relaxation limit for quantum hydrodynamics. The work establishes a time‑relaxation limit for weak solutions of the quantum hydrodynamics system, proving that solutions of a dissipative quantum‑hydrodynamic system converge at long times. This supplies the mathematical setting our damped Madelung flow requires: a hydrodynamic formulation with dissipation that possesses a well‑defined long‑time limit.

[4] Hydrodynamic attractors for Gubser flow. Reference [4] analyzes hydrodynamic attractors for Gubser flow.

[5] Eigenvalues and eigenstates of the quantum Rabi model. Reference [5] presents the eigenvalues and eigenstates of the quantum Rabi model.

[6] Relaxation of excited qubit states in silicon quantum dots. The work reports that orbital relaxation times in silicon dots depend sensitively on dot size and magnetic field, providing empirical data against which the predicted $n$‑dependence of relaxation times can be compared.

[7] Hydrodynamic attractor in periodically driven ultracold quantum gases. Reference [7] reports hydrodynamic attractor behavior in periodically driven ultracold quantum gases.

[8] Prior version of this program. The QNFO record of this work states the same core results that Sections 4–5 here re‑derive and consolidate: eigenstate densities as exact fixed points of the damped flow, linearized decay at rate $\gamma/2$ with crossover at $\gamma=\hbar k^{2}/m$, the relaxation law $t_{n\to n-1}=\ln[(n+\tfrac12)/(n-\tfrac12)]/(2\gamma)$ with $t_{1\to0}=0.549306/\gamma$, and the limitation that the Lyapunov argument establishes attractivity of the fixed‑point set but not global convergence to a unique eigenstate.

[9] Hydrodynamic stability hypothesis. Reference [9] proposes a hybrid model in which a continuous fluid appears discrete after settling into stable resonant modes.

[10] Pre‑registered falsification of deterministic relaxation. The work reports a pre‑registered simulation testing whether deterministic measurement‑triggered relaxation toward eigenstate basins can reproduce the Born probabilities of a two‑level system; all seven configurations produced a verdict FAIL with a maximum deviation of $0.5$ against a tolerance of $10^{-2}$, because a deterministic map from a fixed initial state always yields a degenerate outcome channel with probabilities $0$ or $1$. This result forces the Born rule to be treated as an explicit ensemble assumption (Assumption B) rather than a derived consequence of the deterministic dynamics.

#3. Methods

#3.1 The Madelung formulation

Writing the wavefunction as $\psi(\mathbf{x},t)=\sqrt{\rho(\mathbf{x},t)}\,e^{iS(\mathbf{x},t)/\hbar}$ and substituting into the Schrödinger equation $i\hbar\,\partial_t\psi=\hat{H}\psi$ with $\hat{H}=-\frac{\hbar^{2}}{2m}\nabla^{2}+V$ yields the Madelung system

where $\mathbf{v}=\nabla S/m$ and the quantum potential is

Taking the gradient of (M2) gives the Newton‑like equation

#3.2 The dissipative extension

We introduce a minimal linear drag term in the Newtonian form:

The continuity equation (M1) remains unchanged, guaranteeing probability conservation. Since stationary eigenstates satisfy $\mathbf{v}=0$, the drag term vanishes there, preserving eigenstate densities as exact fixed points. Throughout the analysis we use dimensionless units $\hbar=m=\omega=1$ unless SI values are explicitly stated; the electron mass $m_e=9.109\times10^{-31}\,\mathrm{kg}$, Boltzmann constant $k_{B}=1.381\times10^{-23}\,\mathrm{J\,K^{-1}}$, temperature $T=300\,\mathrm{K}$, and $\hbar=1.055\times10^{-34}\,\mathrm{J\,s}$ appear in the numerical instances.

The drag coefficient $\gamma$ is phenomenological; its microscopic origin is not derived here and is listed as an open question in Section 6.

#3.3 Numerical protocol (specified, not executed)

No simulations are reported. The proposed protocol would:

• Propagate the damped Madelung fluid through a double‑slit potential and verify continuous deterministic trajectories and Born‑rule endpoint statistics under an explicitly specified ensemble over initial conditions.
• Propagate a spin‑fluid analogue through a Stern–Gerlach field gradient and verify convergence to two spatially separated attractor basins.
• Falsify the continuity hypothesis if any trajectory discontinuity or step‑size‑sensitive statistics are observed, and falsify Assumption B if endpoint statistics deviate from Born‑rule weights at fixed dynamics.

The ensemble specification is mandatory because a deterministic map from a fixed initial state cannot produce fractional probabilities, as demonstrated in [10].

#4. Analysis

All inputs are either structural equations from Section 3 or explicitly stated assumptions. Every number is computed step by step.

#4.1 Exact quantum potential of the oscillator ground state

Input 1. Harmonic potential $V(x)=\tfrac12 m\omega^{2}x^{2}$ and ground‑state density $\rho_{0}(x)=\frac{1}{\sqrt{\pi}\sigma}\,e^{-x^{2}/\sigma^{2}}$ with $\sigma^{2}=\hbar/(m\omega)$.

Hence

Substituting $\sigma^{2}=\hbar/(m\omega)$ gives

so

With $S_{0}=\text{const}$ the Hamilton–Jacobi equation requires $V+Q_{q}=E_{0}$; indeed

Thus the ground state is an exact fixed point of the flow, with or without drag. The same argument applies to any eigenstate density $\rho_{n}$ with $\mathbf{v}=0$; therefore eigenstate densities are exact fixed points of the damped flow.

#4.2 Linearized attractor dynamics

Input 2. Perturb $\rho=\rho_{0}(1+\eta)$, $\mathbf{v}=\delta\mathbf{u}$, with $\eta\ll1$. To first order the quantum‑potential perturbation is $\delta Q_{q}=-(\hbar^{2}/4m)\nabla^{2}\eta$. For a plane wave $\eta\propto e^{i(kx-\Omega t)}$, $\delta Q_{q}=(\hbar^{2}k^{2}/4m)\eta$. The linearized equations are

Fourier substitution yields

Cancelling common factors gives the quadratic dispersion relation

with solutions

Every mode therefore decays as $e^{-\gamma t/2}$; the oscillation frequency $\omega_{k}=\sqrt{\hbar^{2}k^{4}/(4m^{2})-\gamma^{2}/4}$ is real (underdamped) when $\gamma\langle \hbar k^{2}/m$ and imaginary (overdamped) otherwise. Since $\operatorname{Im}\Omega=-\gamma/2\lt 0$ for all $k$ when $\gamma\gt 0$, the ground‑state density is linearly stable for every wavenumber.

Numerical instance. Take $\gamma=10^{9}\,\mathrm{s^{-1}}$, $k=10^{7}\,\mathrm{m^{-1}}$, $m=m_{e}=9.109\times10^{-31}\,\mathrm{kg}$.

Amplitude decay time:

Crossover frequency:

Since $\gamma=10^{9}\lt 1.16\times10^{10}$, the mode is underdamped, with

#4.3 Energy decay and level‑to‑level relaxation

Input 3. For a particle obeying the damped Newtonian equation $m\ddot{x}+m\gamma\dot{x}+m\omega^{2}x=0$, the mechanical energy $E=\tfrac12 m\dot{x}^{2}+\tfrac12 m\omega^{2}x^{2}$ satisfies

Averaging over an orbit for weak damping ($\gamma\ll\omega$) gives $\langle\dot{x}^{2}\rangle=E/m$, so

Input 4. Eigenenergies $E_{n}=\hbar\omega\left(n+\tfrac12\right)$. Solving $E_{n}e^{-2\gamma t}=E_{n-1}$ for the relaxation time yields

For $n=1$:

For $n=2$:

Thus relaxation accelerates with excitation because the logarithmic factor shrinks as $n$ grows; in the large‑$n$ limit $t_{n\to n-1}\sim 1/(2\gamma n)$.

Two‑mode instance. With $\omega=10^{12}\,\mathrm{rad\,s^{-1}}$, $\gamma=10^{9}\,\mathrm{s^{-1}}$:

#4.4 Per‑period phase‑space contraction

Input 5. For weak damping $\gamma/\omega=0.01$, the period $T=2\pi/\omega$ gives a contraction factor for the phase‑space area element:

The radial contraction per period is $e^{-\gamma T}=e^{-0.062832}=0.939101$. After $k=10$ periods:

#4.5 Timescale comparison: relaxation versus diffusion (phenomenological)

Input 6. Using the Einstein relation $D=k_{B}T/(m\gamma)$ as an order‑of‑magnitude estimate (valid in the overdamped limit, which does not hold for the parameters of Section 4.2), we compute:

Thermal energy:

Momentum damping factor:

Diffusivity:

Diffusion time across $L=1\,\mu\mathrm{m}=10^{-6}\,\mathrm{m}$:

Hence $t_{D}\ll\tau_{E}$ with ratio $\tau_{E}/t_{D}=4.5\times10^{3}$. The diffusion estimate is presented as a phenomenological bound; its validity in the underdamped regime remains an open question.

#4.6 Hardware fidelity statistics (from [2])

Input 7. Fidelities: $F_{1}=99.90\%$, $F_{2}=99.56\%$, $F_{r}=98.7\%$. In decimal form $f_{1}=0.9990$, $f_{2}=0.9956$, $f_{r}=0.9870$.

Arithmetic mean:

Qubit‑weighted composite (weights $1$, $2$, $105$):

#4.7 Sensitivity of relaxation times to the damping coefficient

Input 8. The relaxation law $t_{n\to n-1}= \ln[(n+\tfrac12)/(n-\tfrac12)]/(2\gamma)$ shows inverse proportionality to $\gamma$. Varying $\gamma$ by an order of magnitude yields:

• $\gamma=10^{8}\,\mathrm{s^{-1}}$: $t_{1\to0}=5.49\times10^{-9}\,\mathrm{s}$.
• $\gamma=10^{10}\,\mathrm{s^{-1}}$: $t_{1\to0}=5.49\times10^{-11}\,\mathrm{s}$.

Thus the model predicts a strong, linear sensitivity of relaxation latency to engineered dissipation.

#4.8 Born‑rule ensemble assumption (Assumption B)

The deterministic damped Madelung dynamics do not generate fractional outcome probabilities. The pre‑registered falsification study [10] shows that any deterministic map from a fixed initial state yields only $0$ or $1$ outcome probabilities, violating the Born rule. Consequently the Born rule is introduced as an explicit ensemble assumption: the measure over initial conditions in each attractor basin is taken to equal $|\langle\psi_{\mathrm{in}}|\phi_{n}\rangle|^{2}$. This assumption is not derived from the fluid dynamics and constitutes the primary falsification target.

#5. Results

We report only quantities computed in Section 4, plus clearly labeled projections.

Result 1 (fixed points and stability). Eigenstate densities are exact fixed points (Section 4.1); linearized perturbations decay at rate $\gamma/2$ for all $k$, with overdamped crossover at $\gamma=\hbar k^{2}/m$ (Section 4.2). For $\gamma=10^{9}\,\mathrm{s^{-1}}$, $k=10^{7}\,\mathrm{m^{-1}}$: $\tau_{\mathrm{lin}}=2.0\times10^{-9}\,\mathrm{s}$, crossover $1.16\times10^{10}\,\mathrm{s^{-1}}$ (underdamped), $\omega_{k}\approx5.8\times10^{9}\,\mathrm{rad\,s^{-1}}$.

Result 2 (energy decay). $E(t)=E_{0}\,e^{-2\gamma t}$ (Eq. D1), valid for $\gamma\ll\omega$ under orbit averaging.

Result 3 (relaxation times). $t_{1\to0}=0.549306/\gamma$; $t_{2\to1}=0.255413/\gamma$; general law $t_{n\to n-1}=\ln[(n+\tfrac12)/(n-\tfrac12)]/(2\gamma)$, decreasing with $n$. For $\omega=10^{12}\,\mathrm{rad\,s^{-1}}$, $\gamma=10^{9}\,\mathrm{s^{-1}}$: $\tau_{E}=1.0\,\mathrm{ns}$, $\Delta E=0.658\,\mathrm{meV}$.

Result 4 (contraction). At $\gamma/\omega=0.01$: per‑period phase‑space contraction factor $0.881911$; after $10$ periods the factor is $0.533488$.

Result 5 (timescale hierarchy). Phenomenological diffusion estimate $D=4.55\,\mathrm{m^{2}\,s^{-1}}$, $t_{D}=2.2\times10^{-13}\,\mathrm{s}$, $\tau_{E}/t_{D}=4.5\times10^{3}$. The diffusion estimate is a bound, not a model prediction.

Result 6 (hardware statistics). Average fidelity $99.39\%$; qubit‑weighted composite fidelity $98.73\%$.

Result 7 (projection). Assuming an engineered measurement‑induced effective damping $\gamma=10^{6}\,\mathrm{s^{-1}}$, the first‑level relaxation time is

Uncertainty arises solely from the assumed $\gamma$.

Result 8 (protocol status). The double‑slit and Stern–Gerlach tests are specified but not executed; no simulation outcomes are reported.

#6. Discussion

Limitations.

• The energy‑decay derivation (D1) relies on orbit averaging and holds only for $\gamma\ll\omega$; strong damping requires the full linearized analysis of Section 4.2.
• The Lyapunov‑type argument guarantees attractivity of the fixed‑point set but does not ensure convergence to a unique eigenstate; degenerate spectra or additional attractors could yield mixed outcomes.
• Identifying excited‑state population with excess mechanical energy (Section 4.3) is a projection approximation, not a derivation from the field equations.
• The drag coefficient $\gamma$ is phenomenological; its microscopic origin, system‑size dependence, and relation to environmental decoherence remain open questions.
• The diffusion estimate in Section 4.5 uses the Einstein relation outside its strict overdamped regime; its quantitative relevance to the underdamped dynamics is uncertain.

Failure modes.

1. Measurement perturbations that inject energy faster than dissipation removes it, leading to persistent oscillations rather than relaxation.
2. Endpoint statistics that remain stable across initial‑condition ensembles but deviate from Born‑rule weights, falsifying Assumption B while leaving the dynamics intact.
3. Observed discontinuities or step‑size‑sensitive trajectories, directly falsifying the continuous‑relaxation hypothesis.

What would falsify the claims. The predicted acceleration of relaxation with excitation (the $n$‑dependence of $t_{n\to n-1}$) is a sharp structural signature. Measured relaxation times in silicon‑dot qubits that increase with excitation level, or that are independent of $n$, would falsify the linear‑drag model in its current form. A proposed test following the protocol of Section 3.3—preparing a superposition, applying a controlled measurement‑like kick, and measuring the post‑kick relaxation‑time distribution—would directly target this prediction.

Arguing against ourselves.

• The linear drag term is introduced ad hoc; without a microscopic derivation the model could be fitted post hoc to any observed relaxation time.
• The framework explains stability but not selection: without deriving the basin measure, the model predicts relaxation to some eigenstate but not with the Born‑rule frequencies.
• The time‑relaxation limit established in [3] concerns weak solutions of the quantum hydrodynamics system, not convergence to discrete eigenstate basins; thus the mathematical anchor is analogical rather than deductive. Progress requires either a microscopic derivation of $\gamma$ and the basin measure or a systematic experimental program to fix these phenomenological parameters.

#7. Practical Implications and Engineering Pathways

#(a) Primary users and decision impact

Quantum hardware engineers and error‑correction architects would use the derived relaxation times to design measurement circuits that deliberately introduce controlled damping, thereby steering qubit states toward desired eigenstates with predictable latency. Quantum algorithm developers could incorporate the deterministic relaxation model into simulation tools, replacing stochastic collapse modules with continuous dissipative dynamics, which may simplify resource estimation for measurement‑heavy algorithms.

#(b) Technology readiness

The work is purely analytical and has not undergone peer review; according to NASA’s Technology Readiness Level definitions, it resides at TRL 2 (technology concept formulation). No hardware or software prototype has been built.

#(c) Minimal next experiment

A proposed numerical experiment would implement the damped Madelung equations for a one‑dimensional double‑slit potential, using a spatial grid fine enough to resolve the quantum potential. The experiment would measure (i) continuity of individual fluid trajectories and (ii) the endpoint distribution over detectors. Observation of any trajectory discontinuity or a detector distribution that deviates from the Born‑rule weights would falsify the hypothesis. The required outcome to refute the main claim is a statistically significant detection of a step‑size‑dependent artifact or a non‑Born‑rule histogram.

#(d) Engineering constraints

• Material and device constraints. Implementing a linear drag term in a superconducting circuit requires a controllable coupling to a dissipative bath; the coupling strength must be tunable to achieve the target $\gamma$.
• Computational constraints. Numerical integration of the damped Madelung equations demands high‑resolution grids to avoid artificial attractors; the required memory scales with the inverse cube of the spatial step size.
• Measurement constraints. Verifying continuous trajectories experimentally would need time‑resolved state tomography with sub‑nanosecond resolution, which is presently at the edge of available hardware.

#(e) Projected cost and performance bounds (projection)

Assuming a superconducting platform can engineer $\gamma=10^{6}\,\mathrm{s^{-1}}$ via a calibrated resistive shunt, the projected first‑level relaxation time is $t_{1\to0}\approx0.55\,\mu\mathrm{s}$. If the same hardware can sustain this dissipation without degrading coherence beyond the reported readout fidelity ($98.7\%$), the overall measurement latency could be reduced by roughly a factor of two relative to the current readout cycle ($\sim1\,\mu\mathrm{s}$).

The present work therefore provides (i) a derivation of the damped Madelung equations that preserve eigenstate densities as exact fixed points (Sec 4.1), (ii) a linear stability analysis yielding the universal decay rate $\gamma/2$ and the overdamped crossover condition $\gamma=\hbar k^{2}/m$ (Sec 4.2), (iii) closed‑form level‑to‑level relaxation times $t_{n\to n-1}=\ln[(n+\tfrac12)/(n-\tfrac12)]/(2\gamma)$ with explicit numerical examples (Sec 4.3), (iv) quantitative phase‑space contraction factors and a diffusion‑versus‑relaxation timescale hierarchy (Sec 4.4–4.5), and (v) a concrete falsification protocol based on the $n$‑dependence of relaxation times and trajectory continuity (Sec 3.3, 7). These results establish a deterministic, continuous alternative to the conventional collapse postulate and delineate the empirical program required to assess its validity.

#Conclusion

We have developed a hydrodynamic‑relaxation research program in which quantum jumps are reinterpreted as continuous, deterministic relaxation events in a damped Madelung fluid. The established results are: (i) adding linear drag $-m\gamma\mathbf{v}$ to the Madelung momentum equation leaves the continuity equation unchanged and preserves eigenstate densities as exact fixed points (Section 4.1); (ii) linearization about the harmonic‑oscillator ground state shows every perturbation mode decays at rate $\gamma/2$, with an overdamped crossover at $\gamma=\hbar k^{2}/m$ (Section 4.2); (iii) the damped flow yields closed‑form level‑to‑level relaxation times

giving $t_{1\to0}=0.549306/\gamma$ and $t_{2\to1}=0.255413/\gamma$ (Sections 4.3–4.4); (iv) per‑period phase‑space contraction factors and a diffusion‑versus‑relaxation timescale hierarchy are quantified (Sections 4.4–4.5); and (v) a concrete falsification protocol based on the $n$‑dependence of relaxation times and the continuity of trajectories is specified (Sections 3.3 and 7). These findings provide a deterministic, continuous alternative to the conventional collapse postulate and delineate a clear experimental program to assess its validity.

In summary, the present work supplies a mathematically explicit, analytically tractable framework that connects dissipative Madelung dynamics to observable measurement‑induced relaxation. Future efforts will (a) derive the phenomenological damping coefficient $\gamma$ from microscopic open‑system models, (b) extend the analysis to multi‑mode and interacting quantum fields, and (c) execute the pre‑registered numerical experiments described in Section 3.3. By confronting the structural predictions—accelerating relaxation with excitation, deterministic continuous trajectories, and basin statistics matching Born weights—with empirical data, the program will either substantiate a reformulation of quantum measurement as a hydrodynamic relaxation process or be falsified, thereby clarifying the role of attractor dynamics in quantum foundations.

#Provenance and AI Involvement

• This paper was drafted, reconciled, reviewed, revised and checked by the QNFO automated research pipeline (qnfo-research-exec 0.30.0-refs-extend). Language models were tools in that process; they are not authors. The named author is accountable for the content.
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#References

[1] A Simple Factor in Canonical Quantization yields Affine Quantization Even for Quantum Gravity. arXiv:2108.04083v1. https://arxiv.org/abs/2108.04083v1 [2] Tianyan: Cloud services with quantum advantage. arXiv:2512.10504v2. https://arxiv.org/abs/2512.10504v2 [3] The time-relaxation limit for weak solutions to the quantum hydrodynamics system. arXiv:2412.16800v2. https://arxiv.org/abs/2412.16800v2 [4] Hydrodynamic attractors for Gubser flow. arXiv:2001.10756v2. https://arxiv.org/abs/2001.10756v2 [5] Eigenvalues and Eigenstates of Quantum Rabi Model. arXiv:2104.12771v2. https://arxiv.org/abs/2104.12771v2 [6] Relaxation of excited spin, orbital, and valley qubit states in single electron silicon quantum dots. arXiv:1301.0260v2. https://arxiv.org/abs/1301.0260v2 [7] Hydrodynamic attractor in periodically driven ultracold quantum gases. arXiv:2501.19240v2. https://arxiv.org/abs/2501.19240v2 [8] https://papers.qnfo.org/papers/quantization-as-hydrodynamic-relaxation-eigenstates-as-attractor-basins-in-a-dis. QNFO: Quantization as Hydrodynamic Relaxation: Eigenstates as Attractor Basins in a Dissipative Madelung Fluid. [9] https://papers.qnfo.org/papers/hydrodynamic-stability-hypothesis. QNFO: Hydrodynamic Stability Hypothesis. [10] https://papers.qnfo.org/papers/relaxation-equation-mechanism. QNFO: A Pre-Registered Falsification of Deterministic Measurement-Triggered Relaxation.

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