Understanding Jupiter's zonal winds is crucial to unraveling the dynamics of its atmosphere. Over the last decades, multiple data sources and techniques have been used to study zonal winds in Jupiter. Here, we develop a correlation-based method for the near-infrared data from the Cassini spacecraft to investigate zonal winds at different altitudes. The new method uses Jupiter's rotation to scan the planet as it rotates, allowing retrieval of winds from the analysis of light-curves of specific pixels over the Jovian disc. The method allows the retrieval of winds at multiple wavelengths from the Cassini/VIMS spectral data despite the low spatial resolution and the non-uniform cadence of the data. By applying this method to two VIMS data cubes acquired on 15 January 2001 at 09:42 UT and 16 January 2001 at 03:22 UT, we reveal the zonal winds at five main latitudes using information from three different wavebands, as well as the wind vertical structure at the equator, showing significant vertical wind shear in the troposphere. The vertical wind shear we derived is weaker than as reported in previous studies, highlighting the intricate interactions among multiple dynamical processes in Jupiter's atmosphere and reflecting the complexity of its atmospheric circulation. Despite the uncertainty due to the low spatial/temporal resolution and non-uniform cadence of the Cassini/VIMS-IR spectral data, the new method established in this study maximizes the value of the Cassini/VIMS in understanding Jupiter's zonal winds. Further observations are essential to explore the underlying mechanisms in Jupiter's atmosphere.

Designing functional protein sequences that satisfy multiple desired properties is a core research focus of protein engineering. Prior methods struggle with inability or inefficiency when dealing with numerous, often conflicting, properties. We propose Multi-Property Protein Diffusion (MP2D), a unified framework for multi-objective protein sequence optimization that integrates conditional discrete diffusion with constrained MCTS and global iterative refinement. MP2D formulates diffusion denoising as a constrained sequential decision-making process and employs MCTS to explore diverse denoising trajectories guided by Pareto-based rewards. A global iterative refinement strategy further enables repeated remasking and re-optimization of candidate sequences, while a dynamic Pareto constraint prevents candidate bloat and maintains balanced trade-offs across objectives. We evaluate MP2D on two challenging multi-objective protein design tasks: antimicrobial peptide and protein binder optimization, involving four to five conflicting properties. Experimental results demonstrate that MP2D consistently outperforms existing multi-objective baselines, achieving robust and balanced improvements across all objectives without retraining generative models. These results highlight MP2D as a practical and scalable solution for multi-objective functional protein design.

The quantum Pontus-Mpemba effect is a counterintuitive phenomenon in which a quantum system relaxes faster through a two-step evolution protocol than through a single, unquenched relaxation. This work proposes its realization in cavity quantum electrodynamics using the Jaynes-Cummings model with photon loss. The model captures the coherent interaction between a two-level atom and a single quantized mode of a lossy cavity, providing a minimal yet realistic setting to explore dissipative quantum dynamics. Restricting the analysis to the single-excitation sector, the dynamics feature damped vacuum Rabi oscillations for weak dissipation that transition to near-exponential atomic decay under strong dissipation. A sudden quench of the cavity decay rate generates distinct relaxation trajectories from the same initial atom-cavity state. The atomic excitation then displays a non-monotonic, accelerated decay, where a trajectory with a quenched dissipation relaxes faster than fixed-loss evolution. The effect originates from the interplay between coherent atom-photon exchange and cavity dissipation, establishing a clear and experimentally accessible realization of the quantum Pontus-Mpemba effect in both optical and circuit QED platforms.

The ambitious performance targets of modern wireless networks, including 6G and Industrial IoT (IIoT) systems, necessitate advanced hardware platforms utilizing millimeter-wave (mmWave) technology. High-frequency signals provide the bandwidth and low latency required for these systems, but rely on beamforming to overcome path loss and exploit channel sparsity. This kind of architecture provides all the specifications needed to build a SLAM (Simultaneous Localization and Mapping) system. This paper presents a dataset based on a validated, high-performance testbed integrating a Xilinx Zynq UltraScale+ RFSoC with a Sivers 60 GHz beamforming front-end. We demonstrate a novel methodology using segment-scrambled, quasi-orthogonal chirp waveforms to perform rapid exhaustive beamspace sampling. The system is integrated with Pynq Linux for real-time control and high-speed waveform upload. We present a high-density spatial dataset consisting of 350 measurement points across a 1.95 m x 3.60 m indoor grid. We exploited the system's ability to scan 63 transmit directions and construct a complete 63x63 beamspace intensity map in 200us. This dataset serves as a benchmark for spatial channel modeling, Digital Twins and Integrated Sensing and Communication (ISAC) research.

For each integer $m \geq 1$, we construct a finite-dimensional family of rational maps, given by Blaschke-type products, whose restriction to the unit circle consists of $2m$-multimodal maps. We show that every post-critically finite $2m$-multimodal circle map satisfying natural dynamical conditions is topologically conjugate to a map in this family. Moreover, we prove that this realization is unique up to rotation: two maps in the family that are topologically conjugate on the circle differ by a rigid rotation. In particular, the family provides a canonical model realizing all post-critically finite combinatorics in this class. The proofs combine a detailed description of the critical geometry of these Blaschke-type maps with a Thurston-type fixed point argument for a pull-back operator on the parameter space.

The aim of this paper is to present a general way to calculate quasi-normal modes (QNM) of the Teukolsky equation for higher dimensional (d > 4) Kerr spacetime with compactified extra dimensions. In order to do so, we develop a formalism derived from spinors: we call it multispinor formalism. It is based on vectors of two-spinors and permits us to develop a formalism analogous to that of Newman-Penrose in 4d. From this we show how to derive the Teukolsky equation for gravitational perturbations and calculate the QNM. In order to keep calculations simple we fix, as an example, the dimension number to be six, but the work can be readily generalized to other spacetime dimensions.

Scaling symmetries have previously been examined for classical field theories described by singular Lagrangians; in this article, we apply these results to the first-order formulation of General Relativity. It is shown that the dynamical content of the Hilbert action may be formulated in terms of the conformal spacetime geometry, together with a dissipative sector, which is required in order to compensate the elimination of the notion of scale encoded by the conformal factor. Further, we consider the linearisation of the equations of motion of the scale-invariant action, demonstrating that the first-order metric perturbations satisfy a free wave equation, as expected. The second-order dynamics, describing gravitational backreaction, are found to be sourced by quadratic combinations of the first-order perturbations. However, these dynamics are non-conservative, as is made manifest by the presence of terms which couple the action sector with the geometrical degrees of freedom.

Spin-carrying single-photon emitters operating in the telecommunication C-band (1530-1565nm) are prime candidates for integrated spin-photon interfaces, offering seamless compatibility with existing fiber-optic infrastructure, an essential component for future quantum networks. In this context, erbium-dopants ($\text{Er}^{3+}$) are particularly compelling due to their exceptional emitter properties, including small spectral diffusion and long spin coherence times. However, their low C-band photon-emission rate and operation at cryogenic temperatures has limited the realization of this technology. In this work, we demonstrate fully integrated single-photon emission from an ion implanted $\text{Er}^{3+}$-embedded into a 4H-silicon-carbide-on-insulator (4H-SiCOI) microring resonator operating at room temperature. By optimizing the mode overlap between the resonator and the $\text{Er}^{3+}$-defect, we achieved a $\sim$70$\times$ Purcell enhancement and recorded small spectral diffusion of $\sim$54 MHz. We further characterize the $\text{Er}^{3+}$ single photon emission via photon correlation g$^{(2)}$-histograms and investigate its performance under varying magnetic-field, demonstrating Zeeman splitting on single emitters.

The irrational behavior of investors selling profitable assets too early while holding onto losing assets for too long is known as the disposition effect. Due to the development of the Internet, the information environment for individual investors has been greatly improved. As an important source of information for individual investors, whether social media can improve investors' behavioral biases and return to rational expectations is a question worth studying. Based on the post data and actual trading data of the social investment platform Xueqiu.com, this paper studies the impact of social media information on the disposition effect of individual investors. The research results show that social media information can significantly reduce the disposition effect. Furthermore, it is through negative information that social media information reduces the disposition effect. When presented with negative information, individual investors will gradually become more rational in adjusting their positions. At the individual level, factors such as investment experience, users followed, region, and gender can all influence the effectiveness of the information acquired by individual investors in reducing the disposition effect.

We propose a framework for determining whether the causal dependence of an outcome $Y$ on a covariate $X$ changes at a given time point, given confounders $\boldsymbol{Z}$. For instance, in financial markets, the effect of a market indicator on asset returns may causally change over time. While many existing measures of association can be used to detect changes in joint and marginal distributions, in the absence of strong assumptions on the data generating process none are suitable for detecting changes in the causal mechanism or in the strength of causal relationship. In this work we approach the problem from a fully non-parametric perspective, and treat the causal mechanism as well as the distribution of the data as unknown. We introduce a quantity based on the integrated difference between kernel mean embeddings of certain conditionals copula, which is provably equal to zero if the causal dependence does not change and strictly positive else. A near-linear time estimator for the quantity is proposed, with rates of convergence explicitly spelled out. Extensive experiments demonstrate that the proposed statistic achieves high accuracy on multiple synthetic and real-world datasets. We additionally show how the proposed statistic can be used for change point detection when the goal is to detect changes in causal dependence occurring at an unknown times.

Let $x'=S(t,x)$ be a differential equation in the cylinder, linear piecewise in $x$ and with trigonometric coefficients in $t$. In this paper, we provide an upper bound on the number of limit cycles in terms of the number of regions of the piecewise equation and the degree of the coefficients, that is, an analogue of Hilbert's 16th problem in this context.

Generative Recommendation (GR) reformulates recommendation as a next-token generation problem and has shown promise in industrial applications. However, extending GR to industrial advertising is non-trivial because the system must optimize not only user interest but also commercial value. Existing GR pipelines remain largely semantics-centric, making it difficult to align value signals across tokenization, decoding, and online serving. To address this issue, we propose UniVA, a Unified Value Alignment framework for advertising recommendation. We first introduce a Commercial SID tokenizer that injects value-related attributes into SID construction, yielding value-discriminative item representations. We then develop a Generation-as-Ranking SID Decoder jointly optimized by supervised learning and eCPM-aware reinforcement learning, which fuses value scores into next-item SID generation to perform generation and ranking in one decoding process. Finally, we design a value-guided personalized beam search that reuses generation-as-ranking logits as online value guidance and applies a personalized trie tree to constrain decoding to request-valid SID paths. Experiments on the Tencent WeChat Channels advertising platform show that UniVA achieves a 37.04\% improvement in offline Hit Rate@100 over the baseline and a 1.5\% GMV lift in online A/B tests.

We investigate self-correcting gossip protocols with errors. In distributed computing, protocols with errors have been widely investigated in temporal epistemic logics. Instead, we propose a dynamic epistemic logic. We show how to correct transmission errors due to faulty messages without a central authority coordinating protocol execution, how this affects optimality, and how this compares to bounded memory and full information protocols.

We systematically study the mass spectra of $S$-wave singly-heavy tetraquark states $Qq\bar{q}\bar{q}$ ($Q=c,b$; $q=u,d,s$) in a mass splitting model. We adopt the assumption that the $X(4140)$ is the lowest $J^{PC}=1^{++}$ $cs\bar{c}\bar{s}$ tetraquark and use this state as a reference to determine the mass splittings. According to the obtained results, we also estimate the rearrangement decay widths of the tetraquarks within a simple scheme. We find that the recently observed states $T^a_{c\bar{s}0}(2900)^{++/0}$ and $T_{cs0}^*(2870)^0$ by the LHCb Collaboration can be consistently interpreted as the second highest $I(J^P)=1(0^+)$ $cn\bar{s}\bar{n}$ ($n=u,d$) and the higher $I(J^P)=0(0^+)$ $cs\bar{n}\bar{n}$ tetraquark states, respectively. We predict several narrow tetraquark candidates: the lowest $cn\bar{s}\bar{n}$ and $cs\bar{n}\bar{n}$ with $I(J^P)=0(0^+)$ and $0(1^+)$, and their bottom counterparts. The obtained information from mass spectrum and rearrangement decay properties will help search for the new singly-heavy tetraquark states.

The Ciliberto-Di Gennaro conjecture predicts that a nodal hypersurface of degree $d\geq 3$ with at most $2(d-2)(d-1)$ nodes is either factorial, or contains a plane and has at least $(d-1)^2$ nodes, or contains a quadric surface and has $2(d-2)(d-1)$ nodes. This conjecture is classically known for $d=3,4$. In 2022 the author proved this conjecture for $d\geq 7$ by the author. Kvitko announced a proof for $d=6$ in 2025. In this paper we prove the conjecture for the remaining open value of $d$, namely $d=5$.

This study compares transponder-based, Point-to-Point, and DSCM-based Point-to-Multipoint (PtMP) access-metro architectures. Findings demonstrate that PtMP IPoWDM significantly optimizes efficiency across diverse geotypes, slashing CAPEX by 92.0% and power by 99.2% compared to the traditional benchmark over a ten-year horizon.

In this article, we review the status of the charmonium production in low energy fixed target proton-nucleus (p-A) and nucleus-nucleus (A-A) collisions as measured by different experimental collaborations at CERN-SPS, Fermilab and HERA accelerator facilities. The interplay of different cold and hot medium effects influencing the production of these $c\bar{c}$ bound states at low collision energies is discussed in detail. Prospect for upcoming charmonium measurements close to kinematic production threshold, in the CBM experiment at FAIR SIS100 and NA60+ experiment at CERN-SPS facilities are also investigated.

Smart glasses with AI assistants are increasingly used in daily life. However, current systems lack awareness of the user's internal cognitive state, leaving them unable to proactively anticipate users' needs without access to cognitive load. Existing methods for assessing cognitive load either rely on impractical sensors for lightweight eyewear or utilize eye gaze-based models that suffer from poor interpretability, and require task-specific fine-tuning, often failing to generalize across individuals. We propose GazeMind, a gaze-guided LLM agent framework for cognitive load assessment on smart glasses. It encodes eye-tracking data into structured representations for LLM-based reasoning and provides interpretable cognitive load predictions. Importantly, GazeMind generalizes across scenarios without LLM fine-tuning through a novel task-guidance reasoning approach and achieves personalized adaptation by incorporating user-specific characteristics and historical references. To support evaluation, we introduce CogLoad-Bench, the largest gaze-based cognitive load dataset with 152 participants, 40+ hours of multimodal data, and 10K+ real-time annotations across controlled and real-world tasks. Experiments show that GazeMind achieves state-of-the-art performance, outperforming baselines by over 20% across all metrics.

Pedestrian arrival times exhibit complex temporal organization across multiple scales, shaped by working hours, transportation schedules, and collective behaviors - features often neglected in conventional pedestrian arrival models. Using a dataset comprising over 23 million pedestrian movements at a Dutch railway station, we show that arrival processes cannot be fully characterized by inter-arrival time statistics alone. Instead, we demonstrate that pedestrian arrivals exhibit clear multifractal scaling, revealing scale-dependent correlations across a broad range of timescales. To quantify these properties, we apply a framework based on generalized fractal dimensions, which captures the heterogeneous structure of arrivals beyond standard point-process descriptions. This approach enables the identification of distinct temporal regimes associated with external forcing and provides a quantitative basis for constructing more realistic synthetic arrival processes. Beyond pedestrian dynamics, this approach offers methodological relevance for understanding non-trivial arrival processes in other physical or biological systems.

In-memory graph-based approximate nearest neighbor (ANN) search has superior search performance but incurs significant memory footprint. Disk-based methods reduce memory usage but suffer from high disk access latency. A common challenge is how to achieve low-latency search while significantly reducing memory footprint. In this paper, we propose SkipDisk, a disk-memory hybrid ANN search that significantly reduces memory footprint while achieving search latency comparable to or lower than in-memory method HNSW. By analyzing existing disk-based methods, we observed that disk access remains the primary bottleneck, and existing lower bound based filtering methods are two loose to effectively reduce disk access. Therefore, we design SkipDisk to achieve tight lower bound with low memory footprint to reduce the search latency. First, we design a dedicated pivot for each point to improve the lower bound of the triangle inequality for effective filtering. We further design an estimation-based approach based on this lower bound. Second, to reduce the memory footprint, we employ a three-level data pruning strategy to preserve informative data in memory. Third, to further reduce search latency, we design an asynchronous I/O strategy based on the decoupling of in-memory search and disk access by storing neighbor nodes in memory. Experiments show that our method achieves a latency of 85 of HNSW's latency with approximately 10 memory footprint, and a latency to 63 of HNSW's with a slightly higher memory footprint of around 20.

Hoare-style verification provides a principled foundation for reasoning about the correctness of quantum programs, but existing approaches do not allow fully automatic verification. While automata-based verification scales well when specifications are given directly as automata, prior frameworks incur exponential blow-up when translating high-level set-based assertions into automata, which severely limits practicality. We introduce an extended set-based specification language and a specification-to-automata translation algorithm whose complexity is linear in the number of qubits, enabled by controlled automaton construction and qubit reordering. The resulting compact automata enable fully automatic Hoare-style verification of fixed-qubit quantum programs at previously infeasible scales, while substantially improving expressiveness without compromising efficiency.

We develop a new theory of an optical force exerted on a carbon nanotube (CNT) with a nonlocal conductivity. The optical force is expressed in terms of the surface current density and the axial electric field on the CNT surface. To determine these quantities, we employ an integral-equation-based approach in terms of the current density. The analysis is constructed for a finite-length cylindrical CNT by rigorously accounting for edge effects. In addition to numerical solutions of the integral equation, we obtain an approximate analytical expression for the optical force acting on the CNT, which shows good agreement with numerical simulations. We also demonstrate the existence of some frequency ranges in which the optical force becomes negative, corresponding to the optical pulling effect. Such a pulling behavior is shown to originate from the nonlocality of the conductivity and to vanish in the local limit. This work advances theoretical understanding of optomechanical interactions in finite-length low-dimensional conductors and clarifies the role of spatial dispersion in the emergence of optical pulling forces.

Let $\{U(m)\}_{m\in \N}$ and $\{V(n)\}_{n\in \N}$ be linear recurrence sequences. It is a well-known Diophantine problem to determine the finiteness of the set of natural numbers $n$ such that the ratio $U(n)/V(n)$ is an integer. We study the finiteness problem for the set $(m, n)\in \mathbb{N}^2$ such that there exist non-zero positive integers $d_{m, n}$ satisfying $\log |d_{m, n}|=o(n)$, and $d_{m, n}U(m)/V(n)$ is an element from a finitely generated subring of $\C$. In particular, we prove that for $m\neq n $, there exists a polynomial $P$ such that $d_{m, n}P(n)U(m)/V(n)$ is a multi-recurrence and $V(n)/P(n)$ is a linear recurrence and for $m=n$ both $d_{m, n}P(n)U(m)/V(n)$ and $V(n)/P(n)$ are linear recurrences. To prove our results, we employ Schmidt's subspace theorem, and the concept of moving hyperplanes, moving polynomials, and moving points.

Isomer depletion via the process of nuclear excitation by inelastic electron scattering is investigated theoretically. A comprehensive study on low-energy nuclear excitations by inelastic electron scattering is performed to analyze the impact of the nuclear and ion charge, the nuclear transition energy, and the nuclear transition multipolarity on the cross section of the process. We apply the analysis to the case of isomer depletion, in which an excitation from the isomeric state to a nuclear level above the isomeric state can lead to decay to a nuclear level below the isomer itself and hence lead to the release of the energy stored in the isomer. For this purpose, the isomer depletion of $\mathrm{{}^{93m}{Mo}}$, $\mathrm{{}^{152m}{Eu}}$, and $\mathrm{{}^{178m}{Hf}}$, which represent the most important scenarios of isomer depletion, are studied. Our results demonstrate the capability of the process of nuclear excitation by inelastic electron scattering for isomer depletion.

We give a proof of the Thom-Sebastiani theorem for mixed Hodge modules using a compatibility with Verdier specialization.

We study the impact of lithography imperfections on quasi-bound states in the continuum (quasi-BICs) supported by a one-dimensional metasurface of Ge$_2$Sb$_2$Te$_5$ (GST) bars with trapezoidal deviations from rectangular cross-sections. Several mechanisms of quality ($Q$) factor scaling, including the impact of material losses, dispersion, and geometric imperfections are established. We demonstrate that transition to identical isosceles trapezoids, despite preserving the required $C_2$ symmetry, reduces the $Q$ factor in the amorphous phase due to absorption changes accompanying the resonance shift. Further, the $Q$ factor remains robust for both GST phases under random element-to-element variations of the trapezoid angle, while analytical and numerical estimations in the absence of material losses show inverse-quadratic scaling of the Q factor with the disorder amplitude. We reveal that in the GST-based metasurface, the $Q$ factor is tolerant to geometric imperfections for insignificant dispersion near the BIC wavelength, but changes in case of substantial dispersion. The phase shifting and established robustness of BICs in GST can be useful for applications where stable moderate $Q$ factors are essential.

Double machine learning (DML) delivers valid inference on low-dimensional causal parameters while permitting flexible nuisance estimation, but its computational cost becomes prohibitive once cross-fitted learners must be trained on massive observational data. Applying DML to a uniformly drawn subsample alleviates this burden, yet such a reduction disregards the geometry of the covariate space and can exacerbate treated-control imbalance as well as overlap deficiency. We propose Uniform Design Double Machine Learning (UD-DML), a design-based subsampling strategy for average treatment effect (ATE) estimation. UD-DML first constructs a low-discrepancy skeleton in a PCA-rotated covariate space under the mixture-discrepancy criterion, and then assigns, to each skeleton point, the nearest treated and control units via KD-tree search. The resulting matched subsample is, by construction, both representative of the full covariate distribution and balanced across treatment arms; cross-fitted DML is subsequently applied to it. We establish discrepancy-based guarantees for representativeness and balance, and prove that the UD-DML estimator is $\sqrt{r}$-asymptotically normal under mild conditions, where the selected subsample size $r \ll n$. The dominant nuisance-fitting cost is thereby reduced from the $n$-scale to the $r$-scale. Monte Carlo experiments show that UD-DML attains lower RMSE, narrower confidence intervals and more reliable coverage than uniform subsampling, with the largest gains in low-overlap and misspecified regimes. An application to a large observational dataset further demonstrates its practical feasibility.

Human mobility prediction forecasts a user's next Point of Interest (POI) from historical trajectories, supporting applications from recommendation to urban planning. Recent studies have recognized the problem with long-tail POIs in human mobility prediction, which are POIs with few visit records, making new visits to such POIs difficult to predict. Our analysis shows that many predictions fail even for visits to popular POIs. The underlying cause is often transition-level sparsity: the corresponding source-destination transition appears rarely, or never appears, in the training set. We therefore argue that a core bottleneck in human mobility prediction lies in transition-level long-tail generalization. We formulate this problem as compositional generalization and propose a tRansition rEconstruction framework for Compositional generAlization in next-POI prediction (RECAP). RECAP reconstructs long-tail transitions from two generalizable signals: multi-hop transitivity in the global transition graph and revisit evidence from a user's historical trajectory. It further uses warm-transition holdout training to discourage memorization of frequent transitions and encourage generalization from transferable signals. Experiments on multiple real-world datasets show that RECAP consistently improves prediction accuracy, with clear gains on tail transitions.

Let $\varepsilon>0$ be given. For prime power moduli $q=p^k$ with $k\geq 2$ and $p\neq 3$, and assuming the Ramanujan--Petersson conjecture for $\GL_2$ Maass forms, we prove that the Rankin--Selberg coefficients $\{λ_f(n)^2\}_{n\geq 1}$ have a level of distribution $θ=2/5+3/305-\varepsilon$ in arithmetic progressions $n \equiv a \bmod q$.

A quantum state can be written in phase space, but the resulting object is not generally the probability density of a positive stochastic process on ordinary phase space. We spell this out for Wigner dynamics. If a positive phase-space process is required only to reproduce the Born density after integrating over momentum, the requirement fixes only an integrated current; the local drift and diffusion remain underdetermined. If one instead requires all Weyl-ordered expectation values, the phase-space object is fixed to be the Wigner function. For non-quadratic potentials the Wigner--Moyal generator contains higher-order, signed momentum-transfer terms, so it is not the Fokker--Planck generator of a positive Brownian diffusion. The exact Wigner function must therefore be reconstructed, in a stochastic representation, as a weighted empirical measure \[ \FW(\z,t)=\E_{\Pp}[W_tδ(\z-\z_t)], \qquad \z=(q,p), \] rather than the unweighted density of sampled carrier trajectories. With classical Hamiltonian flow as the carrier, all nonclassical correction beyond classical transport sits in the Moyal residual and can be represented by signed weights or branching events. The same split defines a residual diagnostic that vanishes for quadratic Hamiltonians and measures what classical carrier transport misses in anharmonic dynamics. The formulation also gives a forward--reverse relation for signed Wigner path measures. The ratio of forward and reversed contributions separates into a positive magnitude factor and a sign factor. This sign records the parity of the Wigner interference contribution; it is not a thermodynamic entropy production.