The friction on a quantized vortex in a truncated Gross-Pitaevskii field follows the temperature, not the normal density: a pre-registered test of alpha proportional to rho_n against the cutoff, and the correction of a published reading
The mutual friction alpha on a quantized vortex in a two-dimensional Bose field is commonly taken to scale with the normal fraction rho_n/rho. In an earlier paper (doi:10.5281/zenodo.23225293) we measured alpha(T) in the projected Gross-Pitaevskii equation at one cutoff and found alpha ~ 0.24 rho_n/rho; we read the coefficient as a mode-averaged transport cross-section of the vortex (1.4 healing lengths) and registered its cutoff dependence as the open question, with the Born phonon-scattering law (coefficient proportional to the cutoff) as the rival. Here the same energy-conserving instrument, with its T = 0 known answer reproduced to 0.3 % at each cutoff, is run at two further cutoffs (k_c xi = 2 pi/3 and 2 pi). The registered prediction of a cutoff-independent coefficient fails and so does its rival: the coefficient is 0.31 +- 0.03, 0.232 +- 0.014, 0.102 +- 0.011 at k_c xi = 2.1, 3.1, 6.3. What does not move is alpha itself: alpha/T = 0.054 +- 0.003 over six arms and three cutoffs (chi2 = 1.3 for 5 degrees of freedom; 0.060 +- 0.002 with the regression estimator, because the energy estimator, the primary one, is biased low by diffusion -- a flaw of the registered synthetic gate exposed by a second, Rust implementation of the estimators). The proportionality to rho_n was a degeneracy of a single cutoff, where rho_n is proportional to T; the 'geometric core size' reading is withdrawn. The temperature law was found after the data were read; it is registered as a hypothesis, with its prediction for the next arm fixed before that arm was read (reported in the paper). Pre-registrations, controls, scripts, analysis file and Lean companion (FrictionKinetic.lean) are in the software record.
The mutual friction alpha on a quantized vortex in a two-dimensional Bose field is commonly taken to scale with the normal fraction rho_n/rho. In an earlier paper (doi:10.5281/zenodo.23225293) we measured alpha(T) in the projected Gross-Pitaevskii equation at one cutoff and found alpha ~ 0.24 rho_n/rho; we read the coefficient as a mode-averaged transport cross-section of the vortex (1.4 healing lengths) and registered its cutoff dependence as the open question, with the Born phonon-scattering law (coefficient proportional to the cutoff) as the rival. Here the same energy-conserving instrument, with its T = 0 known answer reproduced to 0.3 % at each cutoff, is run at two further cutoffs (k_c xi = 2 pi/3 and 2 pi). The registered prediction of a cutoff-independent coefficient fails and so does its rival: the coefficient is 0.31 +- 0.03, 0.232 +- 0.014, 0.102 +- 0.011 at k_c xi = 2.1, 3.1, 6.3. What does not move is alpha itself: alpha/T = 0.054 +- 0.003 over six arms and three cutoffs (chi2 = 1.3 for 5 degrees of freedom; 0.060 +- 0.002 with the regression estimator, because the energy estimator, the primary one, is biased low by diffusion -- a flaw of the registered synthetic gate exposed by a second, Rust implementation of the estimators). The proportionality to rho_n was a degeneracy of a single cutoff, where rho_n is proportional to T; the 'geometric core size' reading is withdrawn. The temperature law was found after the data were read; it is registered as a hypothesis, with its prediction for the next arm fixed before that arm was read (reported in the paper). Pre-registrations, controls, scripts, analysis file and Lean companion (FrictionKinetic.lean) are in the software record.
A quantized vortex in a thermal two-dimensional Bose gas is dragged by the normal fluid (friction α), pushed sideways by it (transverse coefficient α′) and kicked by it (diffusion η). We measure the three coefficients in the projected Gross-Pitaevskii equation — microcanonical, no damping term, no noise put in by hand — with sub-grid vortex tracking and estimators that are identities of the dissipative point-vortex model, machine-checked in Lean 4. Every hypothesis, gate and decision rule was committed before the corresponding run; the instrument failed its T = 0 control twice before it passed, an earlier gate verdict built on the flawed instrument is withdrawn, and both are reported. Results. At T/TBKT = 0.14, 0.27, 0.43: α = 0.0062(4), 0.014(2), 0.02(1), size- and estimator-independent, α ≈ 0.24 ρn/ρ. α′ is zero within 1% once the T = 0 compressible pair speed is subtracted; the Iordanskii value ρn/ρ is excluded at 18σ and the registered Iordanskii-like prediction is refuted. The Einstein relation is inconclusive by the registered rule (one admitted temperature at ηK/α = 2.1 ± 0.7; informally 1.4-3x, not 100x). Single pairs stop shrinking while the phonon band takes the impulse they shed (a self-generated counterflow with a published precedent in the truncated GPE); the decisive box-size control is not run. The single non-equilibrium pair that corrupts the superfluid-stiffness estimator in finite boxes decays by 4-27x given Lz-scaled time (six of six L = 192 trajectories admitted, one unfinished at t = 14 500). The pre-registered mode-by-mode dielectric relation fails in the thermal pair gas (coherence 0.06-0.28), while the vortex-charge structure factor equals the Kosterlitz-Thouless polarisability Σd2 from the measured pair sizes to 5-21%. The Josephson offset ηK − 1 = +0.3 survives at L = 192 in equilibrium, independent of fit window and torus geometry. Formal companions (Lean 4 / Mathlib, standard axioms): energy dissipation of any skew-plus-gradient vortex flow, the d2 law and lifetime bound, the centre-velocity identity, the wind stall, the zero-flux/Einstein equivalence, the pair-polarisation form and response ceiling, the charge-density/matching bound, the Kosterlitz-flow invariant, the bounded increments of quasi-periodic signals. Pre-registrations, amendments, gate records, data and code are in the software repository (concept DOI 10.5281/zenodo.22855581). No physics is claimed as new; the numbers, the refutations and the method are.
We present algorithms for the ab-initio determination of the temperature (T) dependence of the mutual-friction coefficients α and α′ and the normal-fluid density ρn in the two-dimensional (2D) Galerkin-truncated Gross-Pitaevskii system. Our algorithms enable us to determine α(T), even though fluctuations in 2D are considerably larger than they are in 3D. We also examine the implications of our measurements of α′(T) for the Iordanskii force, whose existence is often questioned.
Classical-field (projected Gross-Pitaevskii) simulations of two-dimensional Bose fluids are usually scripts whose correctness is checked, when at all, by a few conservation laws. We describe qf-pgpe, the quantum-fluids module of the rusty-SUNDIALS numerical library: a pure-Rust integrating-factor RK4 engine for the projected equation on a periodic grid, a vortex instrument (periodic theta-function imprint, sub-grid detection, tracking), the registered estimators of vortex friction and diffusion, a thermal-state toolkit, a vortex-phonon scattering instrument, and a Python extension (pyo3, numpy) with a drop-in stepper. Every module is a port of a Python original of an ongoing pre-registered research programme and carries a numeric cross-check against it: estimators agree to 2e-14 on real vortex tracks, a 31-run scattering matrix to 5e-11, measurements on a shared thermal field to 1e-14; the engine is checked against known answers and against rusty-SUNDIALS' own CVODE (global error falling as dt^4.00). The cross-checks found three defects, one of them in the Python programme: its primary estimator of the mutual friction is biased low by vortex diffusion (-8 to -22 %), a flaw the registered synthetic gate had not shown. On one thread of a 2013 laptop CPU the engine is 4.3, 4.2, 2.7, 2.5 times faster per step than the numpy engine at N = 64, 128, 256, 512 (3.2 to 1.5 times against the fastest Python engine tried, torch on CPU), and the drop-in extension makes an end-to-end thermal-state measurement 5.3 times faster with identical results. No claim is made about GPU codes or other published solvers, which were not measured. Version 2 adds the first reproduction of an external, published solver run (Kwon and Shin, flow past a penetrable obstacle: a different scheme and precision): the initial field is reproduced to 2.8e-8, the force on the obstacle to 2e-4 over 10 time units, and the vortex counts at 9 of 10 times up to t = 50, with the disagreements stated (force differences of 0.19 % of the maximum once the wake forms; one count differs by one). It also reports a JAX/XLA row (checksums equal to 1e-13; faster than the single-threaded Rust step at N >= 256 only because it uses threads; pinned to one core Rust is 1.1-1.8 times faster) and an optional, bit-identical intra-step threading mode of the engine (2.4 times at N = 512, measured on a loaded machine).
CORRECTION NOTICE (v5, 2026-09-06). The central result of v1-v4 is WITHDRAWN. This record has not been peer reviewed. Issued 2026-09-06 by the author (v5 of this record). This notice was added to the archived record; the manuscript itself is unaltered. This record has not been peer reviewed. What is wrong The force exponent reported here as alpha = 3.92 +/- 0.16 is an artifact of the fitting procedure, not a property of the data. The fit was unweighted least squares in linear space applied to a steeply falling profile, which places 95.8 percent of its objective on the single point at the shortest retained separation. Re-fitting the same data with nothing discarded, changing only the error model, gives 2.82 (relative-error weighting) or 2.89 (log-space regression). Across fifteen combinations of standard estimator and fit window the exponent spans 2.53 to 3.92, and the value published here is the largest of them. What the data show instead, and how firmly The retraction is certain. The replacement is not. On the open 24-site chain the same data give alpha = 2.6 +/- 0.2, where the interval is the spread of a plateau in the local logarithmic slope over six consecutive separations, not a statistical error (the bootstrap error on the published fit to the same data is 0.63); the plateau value was reproduced at 2.5921 by a separately written implementation run on the same recorded state (SIM-2026-073). This is a single geometry at a single size. On a 24-site periodic ring, analysed with chord distance as pre-registered, two probes that are both energy-density-like (Delta = 1) give 1.83 and 2.37 against a CFT two-point value of 2 (SIM-2026-074): one size, not converged, disagreeing with each other by 29 percent, and not attributable to a single operator channel. No corrected exponent is established. The exchange channel is an open question. Treat 2.6 +/- 0.2 as the provisional open-chain value, not as a result. Also withdrawn or voided alpha = 3.92 +/- 0.16 (24 sites) and alpha = 3.97 +/- 0.15 (16 sites, exact) from stress-tensor exchange. Was: headline result and its first corroboration. Now: withdrawn. Why: the same estimator artifact: both move in lockstep under it (3.92 to 2.56, 3.97 to 2.75 on dropping the contact point), because they share one observable, one fit window and one fitting routine. scalar exchange (alpha = 2) excluded. Was: stated as excluded at about 12 sigma. Now: withdrawn. Why: the 12 sigma used the covariance error 0.16; at the bootstrap error 0.63 computed before release the exclusion is about 2.2 sigma, and under any window starting at separation 3 the point estimate itself falls to about 2.56. alpha = 4 is universal; alpha = 4.00 at Ising, tricritical Ising and Potts. Was: confirmed at three classes. Now: withdrawn; the operator towers at all three classes stand. Why: the exponent was measured only at Ising and inherited by the other two; the towers are spectra, not forces. third independent evaluation alpha = 4.07 (chi = 3 MERA), and with it the statement that this was the first holographic force measurement from an optimized tensor network state. Was: supporting evidence and a priority claim. Now: both withdrawn. Why: the profile changes sign inside its own fit window and cannot be a fixed-sign power law; the priority claim rests on that evaluation alone and was never gated against prior work. the discriminating test: stress tensor (Z_j Z_j+1) versus spin primary (X_j). Was: registered killer test. Now: void. Why: at the critical point these are one operator, related by exact Kramers-Wannier self-duality. Measured on the periodic ring: identical expectation values to 1e-15 and identical force profiles to 1e-13 at 14 sites, 2.4e-9 at 24 sites (SIM-2026-074). Open boundaries break the duality, which is why the open-chain run of this test (SIM-2026-073) showed a spurious difference between the two arms. The test compared the energy density to itself. Neither label was correct: Z_j Z_j+1 is the energy density, not the stress tensor; X_j is its dual, not the spin primary. transport time T = (pi/2) l_AdS to 1.2 percent as a geometric prediction. Was: confirmed. Now: measurement stands (T = 2.239 at N = 16); geometric reading not established. Why: l_AdS cancels in the cited derivation, which also gives pi rather than the pi/2 the manuscript compares against; the chi = 3 estimate of l_AdS gives a 57 percent miss; the transport driver never applies a MERA. The true spin primary Z_j was never in the design, and the direct replacement test is blocked: Z_j is odd under the Z2 symmetry, so the energy-difference observable used throughout this work subtracts across symmetry sectors and is undefined for it. A replacement test using two Z2-even probes has been pre-registered (PREREGISTRATION-T.md, committed before its driver) and run; its result is deliberately not part of this correction and will appear in a later version. Not previously reported The published fit has a reduced chi-squared of 6.008 on 9 degrees of freedom, and a bootstrap standard deviation of 0.63 rather than the quoted 0.16. Both were computed before release and omitted. What is unaffected Of the claims in this record: the two-phase optimization method, the chi = 3 criticality threshold, the exact-diagonalization gates, the measured OPE coefficient C_sigma-sigma-epsilon = 0.50 +/- 0.03 against the exact 1/2, and the conformal towers at all three classes. Not part of this record, mentioned only because they are sometimes read together with it: the later C_TTT stress-tensor self-coupling measurements (SIM-2026-067/068/069, dated after this record) and the matched off-critical ring control (SIM-2026-074) are unaffected by this correction and make no claim about the force exponent. How the error was found and what this record contains The error was found by the author's own internal adversarial audit, not by a reviewer; the journal submission it was found after is closed, and no review of this work has taken place. Files in this record: the full re-analysis (SIM-2026-072.json, driver, figure), the pre-registered chain test (SIM-2026-073.json) and ring test (SIM-2026-074.json) with both pre-registrations as committed before the runs, the runnable drivers with their shared module and the negative test suite for their validation gates, and the error register (ERRATA.md). Concept DOI 10.5281/zenodo.19660302 always resolves to the current version. The abstract of the original manuscript follows unchanged, for the record. Every claim in it about the force exponent should be read against the correction above. We report results from classical simulation of critical spin chains that establish quantitative physical content in the MERA-AdS correspondence. First, the interaction energy between boundary-primary excitations follows a power law F(dr) ~ dr^(-alpha) with three consistent evaluations yielding alpha = 3.92 +/- 0.16 (24q), 3.97 +/- 0.15 (16q), and 4.07 +/- 0.18 (16q chi=3 MERA), all consistent with stress-tensor exchange (alpha = 2*Delta_T = 4) and inconsistent with scalar exchange (alpha = 2); the exponent is explained by the Ising CFT fusion rule eps x eps = [1]. Because Delta_T = 2 is fixed by conformal symmetry in every unitary 2D CFT, alpha = 4 is universal: we confirm this at two further fixed points - the tricritical Ising model (c = 7/10), where a scale-invariant MERA recovers the full M(4,5) primary spectrum (Delta_sigma = 0.092, Delta_eps = 0.204, Delta_T = 1.999), and the three-state Potts model (c = 4/5), where a Z3-symmetric MERA at chi = 12 recovers Delta_sigma = 0.134, Delta_eps = 0.805, Delta_T = 2.001 - giving alpha = 4.00 at all three universality classes. Second, the holographic transport time matches the AdS_3 null geodesic prediction T = (pi/2) l_AdS to 1.2% at N = 16 sites on a time-grid-converged measurement, with the residual shrinking from 2.8% at N = 12. Third, we introduce a two-phase optimization method (ALM + Z2 projection) for Z2-symmetric MERA ground states at 24 qubits. Fourth, extending the MERA bond dimension from chi=2 to chi=3 overcomes the expressibility barrier, achieving 0.15% energy error at 16 qubits and enabling the first holographic force measurement from an optimized tensor network state. The OPE coefficient is also measured directly from the fixed-point tensors: C_sigma-sigma-eps = 0.50 +/- 0.03 against the exact Ising value 1/2. A hardware path is designed for Quantinuum Helios (57 CNOTs, 0.57% logical error). Version 4 changes (relative to v3): Transport: N=16 residual restated as the time-grid-converged value of -1.21% (dt = 1e-3), with -2.78% at N=12 showing the finite-size trend; the 0.72% figure in v3 was an artifact of the dt = 0.05 integration grid and is now labeled as such. The N=24 transport point carries a new reliability caveat (double-lobe trace structure). The l_AdS comparison table is recomputed against the converged transport time (best fit 1.425, deviation 1.21%). Effective-mass derivation rewritten in a consistent AdS3 convention m^2 l^2 = Delta(Delta-2); m_eff = 0.693 = 1/l_AdS follows from Delta = 1 exactly. NEW: universality section extended to a third fixed point - a Z3-symmetric MERA at chi = 12 recovers the three-state Potts (c = 4/5) tower, giving alpha = 4.00 at all three universality classes (c = 1/2, 7/10, 4/5). NEW: the OPE coefficient C_sigma-sigma-eps = 0.50 +/- 0.03 measured directly from the fixed-point tensors, with C_sigma-sigma-sigma = 0 at machine precision as the Z2 structural check. Supplemental table row SIM-007 annotated (record lost; script preserved); data availability expanded; Zenodo DOI added to the front matter.
We report systematic validation of the Davis Law of Critical Velocity governing vortex nucleation in two-dimensional Gross-Pitaevskii dynamics: C=Kτ⇒β=csvc=f(Keff)(species-independent, geometry-indexed) Through GPU-accelerated Gross-Pitaevskii simulations across three atomic species (⁸⁷Rb, ²³Na, ⁷Li) spanning a 12:1 mass ratio, we demonstrate that the dimensionless critical velocity β = v_c/c_s is species-independent—a universal geometric invariant within each curvature regime, determined solely by obstacle geometry. Key Results Universal Critical Velocity: Hard obstacles (tanh-wall): β_hard = 0.425 ± 0.000 across all three species Semi-hard obstacles (Gaussian, V₀ = 50μ): β_soft = 0.486 ± 0.000 across all species Excellent agreement with Frisch-Pomeau-Rica result (β ≈ 0.4) Perfect Species Independence: The coefficient of variation is 0.0% for both obstacle types across a 12:1 mass ratio, definitively proving that β depends only on obstacle geometry, not atomic properties. Davis Law Support: The relation C = τ/K from the Davis Field Equations correctly predicts that obstacle curvature controls the critical velocity. High-curvature (hard) obstacles nucleate vortices at lower β than low-curvature (soft) obstacles. Quantum vs Classical Wake Dynamics: In the post-critical wake, coherent oscillations occur with phase φ ≈ 0° (symmetric breathing) rather than φ ≈ π (alternating von Kármán), implying suppression of the classical anti-phase shedding mode. The Geometric Trichotomy Regime Obstacle Type K_eff Mechanism β = v_c/c_s Sonic Hard (tanh-wall) High Local Mach = 1 at obstacle rim 0.42–0.49 Sonic Semi-hard (Gaussian) Medium Same mechanism, different K_eff ~0.486 Quantum Truly soft (V₀ ≪ μ) Low Holonomy budget τ = 2π exhausted ~0.95–√2 The functional relationship β = f(K_eff) is universal—independent of atomic mass, scattering length, and density—with distinct numerical values corresponding to distinct geometric classes. Unifying Principle The various measured thresholds are unified by a single principle: Geometry selects the regime (soft vs hard) Compressibility selects the prefactor (local c_s and amplification A) Barrier physics selects the delay (sonic onset vs nucleation threshold) This validates the Davis Law prediction that coherence thresholds are determined by constraint geometry (C = τ/K), not microscopic physics. Evidence Status Claim Status Evidence β is species-independent Proven 0.0% CV across 12:1 mass ratio β_hard = 0.425 ± 0.000 Proven Identical value across Rb-87, Na-23, Li-7 β_soft = 0.486 ± 0.000 Proven Identical value across Rb-87, Na-23, Li-7 K_eff controls regime Proven Curvature scaling law shows monotonic β(K_eff) C = τ/K governs transition Supported Qualitative agreement; mechanism confirmed Wake mode is symmetric Proven Phase φ ≈ 0° measured (not 180° classical VK) Frisch-Pomeau-Rica agreement Proven β_hard = 0.425 matches FPR β ≈ 0.4 Included Files Simulation Code multi_species_trichotomy.py — Main simulation testing β across Rb-87, Na-23, Li-7 curvature_scaling_law.py — Tests C = τ/K scaling, K_eff operational definition Figure Generation plot_davis_landau_law.py — Publication figures (universal β, v_c vs c_s) mach_summary_for_paper.py — M_rim vs v/c_s showing sonic onset mechanism phase_comparison_figure.py — VK vs GP symmetric modes diagnostic stability_islands_extended.py — Stability islands analysis Data trichotomy_data.json — Pre-computed multi-species results superfluid_test_results_corrected.json — Validation data Relation to Prior Work This paper extends the Davis Framework for geometric physics: The Incompressibility of Topological Charge (Yang-Mills Mass Gap) — DOI: 10.5281/zenodo.17846521 Holonomy-First Navier-Stokes Regularity — DOI: 10.5281/zenodo.18216597 Field Equations of Semantic Coherence — DOI: 10.5281/zenodo.14784553 The critical velocity emerges as the threshold where topological defect creation (vortex nucleation) becomes energetically favorable—the "cost of distinguishability" in the information-geometric framework. Keywords Bose-Einstein condensate, critical velocity, superfluidity, Gross-Pitaevskii equation, vortex nucleation, Landau criterion, sonic onset, geometric trichotomy, Davis Law, quantum hydrodynamics, obstacle geometry, Mach number, effective curvature, species independence, universal constant License Creative Commons Attribution 4.0 International (CC BY 4.0) Related Identifiers References: DOI 10.5281/zenodo.17846521 (Yang-Mills Mass Gap) References: DOI 10.5281/zenodo.18216597 (Navier-Stokes Regularity) References: DOI 10.5281/zenodo.14784553 (Field Equations of Semantic Coherence)
On the basis of a recently proposed scenario of the transformation of the Kolmogorov cascade into the Kelvin-wave cascade, we develop a theory of low-temperature cutoff. The theory predicts a specific behavior of the quantized vortex line density, L, controlled by the frictional coefficient, alpha(T)<<1, responsible for the cutoff. The curve ln L(lnalpha) is found to directly reflect the structure of the cascade, revealing four qualitatively distinct wave number regions. Excellent agreement with a recent experiment by Walmsley et al. [Phys. Rev. Lett. 99, 265302 (2007)] -- in which L(T) has been measured down to T ~ 0.08 K -- implies that the scenario of low-temperature superfluid turbulence is now experimentally validated and allows to quantify the Kelvin-wave cascade spectrum.