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

In the paper's words

We develop a research program in which quantum jumps are reinterpreted as continuous, deterministic relaxation events in the Madelung hydrodynamic picture of quantum mechanics, where the wavefunction ψ=√ρ e^iS/ℏ becomes a compressible fluid with density ρ and velocity v=∇ S/m. Adding a minimal linear drag term to the momentum equation leaves the continuity equation untouched, so stationary eigenstate densities remain exact fixed points of the flow; we show by explicit linearization that perturbations about the harmonic-oscillator ground state decay at rate γ/2 for every wavenumber, with an overdamped crossover at γ=ℏ k²/m, and by a Lyapunov-energy argument that the damped flow is contractive. We compute closed-form level-to-level relaxation times t_n→ n-1=ln[(n+(1)/(2))/(n-(1)/(2))]/(2γ), giving t₁→0=0.549306/γ, a per-period phase-space contraction factor $0.881895$ at γ/ω=0.01, and a diffusive-versus-relaxative timescale hierarchy t_D≈2.2×10^-13 s≪τ_E=1.0×10^-9 s for stated parameters. [...]

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. This entry describes a cloud-accessible superconducting quantum prototype, Tianyan-287, with 105 qubits and operational fidelities of 99.90% (single-qubit gates), 99.56% (two-qubit gates), and 98.7% (readout).

   from 2. Background and Related Work

2. Input 1: $V(x)=\frac{1}{2}m\omega^2x^2$ and ground-state density $\rho_0(x)=\frac{1}{\sqrt{\pi}\sigma}e^{-x^2/\sigma^2}$ with $\sigma^2=\hbar/(m\omega)$.

   from 4.1 Exact quantum potential of the oscillator ground state

3. Then $\sqrt{\rho_0}\propto e^{-x^2/(2\sigma^2)}$ and $$\frac{d^2}{dx^2}\sqrt{\rho_0}=\left(\frac{x^2}{\sigma^4}-\frac{1}{\sigma^2}\right)\sqrt{\rho_0},$$ so $$Q_q=-\frac{\hbar^2}{2m}\left(\frac{x^2}{\sigma^4}-\frac{1}{\sigma^2}\right)=\frac{\hbar^2}{2m\sigma^2}-\frac{\hbar^2x^2}{2m\sigma^4}.$$ Substituting $\sigma^2=\hbar/(m\omega)$: $\frac{\hbar^2}{2m\sigma^2}=\frac{\hbar^2m\omega}{2m\hbar}=\frac{\hbar\omega}{2}$ and $\frac{\hbar^2}{2m\sigma^4}=\frac{\hbar^2m^2\omega^2}{2m\hbar}=\frac{m\omega^2}{2}$.

   from 4.1 Exact quantum potential of the oscillator ground state

4. Hence $$Q_q(x)=\frac{\hbar\omega}{2}-\frac{m\omega^2x^2}{2}.$$ Stationarity check: with $S_0=\text{const}$, the Hamilton–Jacobi equation requires $V+Q_q=E_0$; indeed $V+Q_q=\frac{m\omega^2x^2}{2}+\frac{\hbar\omega}{2}-\frac{m\omega^2x^2}{2}=\frac{\hbar\omega}{2}=E_0$.

   from 4.1 Exact quantum potential of the oscillator ground state

5. Input 3: under linear friction, the mechanical energy $E=\frac{1}{2}m\dot{x}^2+\frac{1}{2}m\omega^2x^2$ obeys $dE/dt=m\dot{x}\ddot{x}+m\omega^2x\dot{x}=\dot{x}(-m\gamma\dot{x})=-m\gamma\dot{x}^2$.

   from 4.3 Energy decay and level-to-level relaxation

6. We compute closed-form level-to-level relaxation times $t_{n\to n-1}=\ln[(n+\tfrac{1}{2})/(n-\tfrac{1}{2})]/(2\gamma)$, giving $t_{1\to0}=0.549306/\gamma$, a per-period phase-space contraction factor $0.881895$ at $\gamma/\omega=0.01$, and a diffusive-versus-relaxative timescale hierarchy $t_D\approx2.2\times10^{-13}\ \mathrm{s}\ll\tau_E=1.0\times10^{-9}\ \mathrm{s}$ for stated parameters.

   from Abstract

How it is done

1. We compute closed-form level-to-level relaxation times $t_{n\to n-1}=\ln[(n+\tfrac{1}{2})/(n-\tfrac{1}{2})]/(2\gamma)$, giving $t_{1\to0}=0.549306/\gamma$, a per-period phase-space contraction factor $0.881895$ at $\gamma/\omega=0.01$, and a diffusive-versus-relaxative timescale hierarchy $t_D\approx2.2\times10^{-13}\ \mathrm{s}\ll\tau_E=1.0\times10^{-9}\ \mathrm{s}$ for stated parameters.

   from Abstract

2. (An earlier draft of this work sketched preliminary double-slit convergence results; because those results could not be documented to the standard of the present analysis, they are withdrawn and listed as future work.) The paper proceeds in three pillars: derivation of the dissipative Madelung equations and closed-form relaxation laws (Sections 3–4); stability analysis of eigenstate fixed points (Section 4); and a falsification protocol (Sections 3.3 and 6).

   from 1. Introduction

3. No simulation results are reported in this paper.

   from 3.3 Numerical protocol (specified, not executed)

4. All inputs are either structural equations from Section 3 or explicitly stated assumptions.

   from 4. Analysis

What would falsify it

1. We connect the program to rigorous relaxation limits in quantum hydrodynamics, hydrodynamic attractors in kinetic theory, and measured qubit relaxation, and we specify a falsification protocol.

   from Abstract

2. It does not derive the Born rule: the claim that relaxation endpoints are sampled with Born-rule weights over initial conditions is carried as a labeled assumption (Assumption B) throughout, and its failure is one of the primary falsifiers of the program.

   from 1. Introduction

3. (An earlier draft of this work sketched preliminary double-slit convergence results; because those results could not be documented to the standard of the present analysis, they are withdrawn and listed as future work.) The paper proceeds in three pillars: derivation of the dissipative Madelung equations and closed-form relaxation laws (Sections 3–4); stability analysis of eigenstate fixed points (Section 4); and a falsification protocol (Sections 3.3 and 6).

   from 1. Introduction

Where the claim stops

1. The energy-decay derivation (D1) uses orbit averaging, valid only for $\gamma\ll\omega$; the strong-damping regime requires the full linearized analysis of Section 4.2, which is itself local in $\rho_0$ and assumes locally flat $V$.

   from 6. Discussion

2. The identification of excited-state population with excess energy (Section 4.3) is a projection approximation, not a derivation from the field equations.

   from 6. Discussion

3. (ii) Endpoint statistics stable across initial-condition ensembles but inconsistent with Born weights — this falsifies Assumption B while leaving H's dynamics intact.

   from 6. Discussion

Where to read

1. Abstract182 words · 1 min
2. 1. Introduction339 words · 1 min
3. 2. Background and Related Work913 words · 4 min
4. 3. Methods327 words · 1 min
5. 4. Analysis627 words · 3 min
6. 5. Results234 words · 1 min
7. 6. Discussion439 words · 2 min
8. 7. Conclusion218 words · 1 min
9. Appendix A. Divergence report141 words · 1 min
10. Appendix B. Claim attribution154 words · 1 min