Papers

We prove that the $SL(2,\mathbb{C})$ character varieties of compact oriented surfaces and the generic relative $SL(2,\mathbb{C})$ character varieties of punctured surfaces admit divisorial log terminal (dlt) log Calabi-Yau compactifications. To do this, we establish a general result giving sufficient conditions for a compactification of an affine variety arising from a filtration of its algebra of regular functions to be log Calabi-Yau. We then apply this result to show that the compactifications constructed by Kutteri-Tehrani-Frohman in the compact case and by Tehrani-Frohman in the punctured case are log canonical and log Calabi-Yau.

Understanding how fermionic exchange and Coulomb repulsion jointly shape two-electron correlations is essential for identifying genuine quantum signatures in multi-electron interference experiments. To address this interplay, we investigate Hanbury Brown and Twiss interference of two electrons generated by two independent needle-tip emitters within a full quantum-mechanical framework. In the absence of Coulomb interaction, the approach reproduces the results previously obtained within a quantum path formalism. For Coulomb-interacting electrons, we predict characteristic features absent in a semiclassical description: a pronounced Coulomb-dominated suppression region as well as Coulomb-induced phase offsets and fringe shifts. At the same time, outside of the Coulomb-dominated region, the spatial oscillation frequency is essentially governed by fermionic exchange symmetry. Our results establish quantitative parameter regimes for disentangling Coulomb interaction from fermionic exchange symmetry in such experiments.

Dynamical decoupling protocols provide a versatile toolbox for robust control of quantum systems, suppressing noise that would otherwise limit their performance. This tutorial aims to provide an intuitive interpretation of these protocols. Written from an experimentalist's perspective, we use the Bloch sphere picture to visualise quantum dynamics for a range of sequences. Here each dynamical decoupling operation can be viewed as an "effective field vector", whilst the phase and population of the two-level system is encoded into a "Bloch vector". The two-level dynamics are calculated by the cross product between these two vectors, such that the quantum state represented by the Bloch vector rotates around the effective field vector. Several prominent control schemes are simulated using this picture, starting with a Rabi oscillation, followed by the Ramsey sequence and a selection of pulsed and continuous dynamical decoupling techniques. These sequences are engineered to preserve control of the quantum system for as long as possible, with their efficacy often quantified by a coherence time. We clarify the definition of the longitudinal $T_1$, transverse $T_2$ and inhomogeneous $T_2^*$ coherence times, which are used across the literature. Each simulation has an accompanying animation to illustrate the protocol's two-level dynamics on the Bloch sphere. The custom simulation and Bloch sphere plotting scripts are also provided in an open access repository to allow the reader to reproduce, modify, and explore these results. Dynamical decoupling sequences have established themselves as an invaluable tool across the entire quantum technologies remit; by providing an open framework alongside this tutorial, we aim to make these protocols more accessible to new researchers discovering this thriving field.

In this work, we study $(r,t,x)$ quantum locally recoverable codes (qLRCs) with locality $r$, $t$ recovery sets per qudit, and intersection parameter $x$. We first show that, assuming the underlying classical codes have dual minimum distance at least two, a CSS code is an $(r,t,x)$-qLRC if and only if the underlying classical codes are $(r,t,x)$ classical LRCs (cLRCs) with common recovery sets. We then use subset-inclusion matrices to construct families of binary dual-containing $(r,t,x)$-cLRCs, which yield binary $(r,t,x)$-qLRCs via the CSS construction. For CSS $(r,t,x)$-qLRCs, we derive upper bounds on the dimension and rate, minimum-distance bounds in the pure case, and a Singleton-like dimension bound in the exact case. Finally, we show that these families attain high rates and nontrivial minimum distances.

X-ray absorption spectroscopy (XAS) probes the local chemical environment of the absorbing atom and is one of the most powerful characterization techniques for battery materials. Here we present a database of simulated oxygen K-edge XAS spectra for the layered cathode material LiNi0.8Mn0.1Co0.1O2 (NMC811), built on the atomic structures of our recent work[1]. All spectra were obtained using the excited electron and core-hole (XCH) method with the R2SCAN meta-GGA functional, as implemented in the Vienna Ab initio Simulation Package (VASP). The database covers benchmark binary oxides (TiO, Ti2O3, TiO2, Mn3O4, Mn2O3, MnO2) together with a realistic NMC811 supercell containing 60 transition metal sites at three states of charge. Because each spectrum is resolved at the level of individual oxygen sites, the database links O K-edge spectral features to specific oxygen environments defined by their local coordination and transition metal neighbors. All data are freely available and can serve as a reference for spectral fingerprinting, for direct comparison with experiments, and as training data for machine learning models.

Context. With their precisely determined physical parameters, classical Cepheids in eclipsing binary systems are often used to constrain stellar evolution and pulsation theories. To this end, their position in the HR diagram, effective temperature and luminosity, and radius are commonly used when matching best-fitting evolutionary models. However, the pulsation period of Cepheids, the most precise observable, is rarely used in such studies. Aims. We explore how including the pulsation period as a constraint in matching evolutionary models affects the best-fitting solution. As the pulsation period follows the period-mean density relation, we examine whether it provides information consistent with that based on the stellar radius. Methods. We modeled four eclipsing binary systems with Cepheids from the LMC. We used $\chi^2$ minimization to find the best-matching evolutionary model from a grid of models computed with MESA. The evolutionary models are supplemented with pulsation periods computed with RSP, with nonlinear period corrections taken into account. Results. Depending on whether the radius or the pulsation period is used to select the best-fitting model, discrepant solutions are obtained. In solutions selected based on the pulsation period, the stellar radius is systematically too low compared with observations. Conversely, for solutions based on the radius, the pulsation period is systematically too long. The tension amounts to a few sigma for stars with precisely determined radii and part of it is traced to a nonlinear increase in radius for large-amplitude pulsators, which has not been studied in detail in the literature. Conclusions. When using Cepheids in eclipsing binary systems to constrain stellar models, we recommend using the pulsation period instead of the radius. A systematic study of nonlinear effects on stellar radius in large-amplitude pulsators is needed.

The Gari–Krümpelmann (GK) model provided one of the first semiphenomenological descriptions of the electromagnetic nucleon form-factors over a wide range of momentum transfer. It combined vector-meson dominance at low and intermediate momentum transfer with perturbative-QCD constraints at large momentum transfer. The present work extends this framework by including the dispersive two-pion continuum explicitly in the isovector channel. The two-pion contribution is taken from modern dispersive analyses and embedded into the intrinsic high-$Q^2$ structure of the GK ansatz. The physical $\rho$ dynamics is then interpreted as part of the correlated two-pion spectral function. Similarly, we include in the isoscalar spectral function the correlated $K\bar K$ and $\rho\pi$ continua, where the former also includes the $\phi$-meson pole. The direct coupling terms remain essential for the transition to the asymptotic regime. This gives a cleaner separation between long-range continuum dynamics, coherent vector-meson contributions, and direct quark-current dynamics. With just 7 parameters, we can describe the large body on form-factor data and differential cross sections from electron-proton scattering in the space-like region.

Using dimensional arguments and other simple estimates, it is pointed out that a branch of nuclei with mass number A>10^7 could be metastable if they have close to maximal isospin. This branch is continuous with neutron stars. The fastest decay mode of these hypothetical metastable nuclei (which we call monsters) is through neutron evaporation. We present an estimate of the lifetime and show it could be more than 100 ps for A>10^{39}. We speculate about their creation in binary neutron star mergers, where we argue that their lifetimes could be extended further, and their presence could be detectable in gravity wave experiments of the near future.

It is shown that a skyrmion string in a ferromagnetic film consisting of $N$ atomic layers exhibits flexural oscillations at eigenfrequencies $|\omega_{m}|=(4J'S/\hbar)\sin^{2}[\pi m/(2N)]$, where $J'$ is the interlayer exchange coupling, $S$ is the length of the atomic spin, and $m=1,...,N-1$. This result is confirmed numerically for individual skyrmion strings in a discrete microscopic spin-lattice model of up to ten atomic layers, as well as for skyrmion-string lattices at finite temperature.

Tracking charged particles with high precision is of vital importance for collider experiment like those operating at the Large Hadron Collider (LHC), CERN. The tracking detector in the CMS experiment is composed of multi-layer silicon based tracker with 3-dimensional position sensitivity. High precision position data from tracker operated in high magnetic field, is used to reconstruct the trajectories of charged particles and obtain their kinematic parameters ($p_t$,$\eta_0$, $\phi_0$) with high accuracy. In this paper, for Phase2 CMS tracker design, we present a novel track reconstruction algorithm for high luminosity (HL) era of the LHC. The algorithm identifies hits associated with each track and utilizes them to accurately determine the kinematic parameters of each track using machine learning (ML) architecture. The proposed algorithm has been applied on a large sample of hard interactions simulated at high luminosity (HL) era of the LHC using Pythia8 and Geant4 framework for equivalent geometry of the outer tracker of the CMS experiment. Performance of the proposed algorithm has been studied using the key indicators such as reconstruction efficiency, fake rate and resolution. Comparison with traditional methods demonstrates robust performance with excellent efficiency and resolution with minimal fake rate.

The formation of Fabry Perot and guided wave self hybridized exciton polaritons in two dimensional materials results in long range exciton energy transfer and strong exciton exciton interactions. Here, we demonstrate that the coupling strength between photonic modes and excitons is significantly boosted by embedding the active excitonic layer in a metal clad WS2 waveguide. The photonic modes in this waveguide exhibit modified dispersion properties for both Fabry Perot type and guided wave exciton polaritons compared to pure WS2 flakes, and show an increased couplingr strength. Our results provide a robust approach for controlling exciton photon interactions and their coupling strength in hybrid heterostructures.

Optomechanical systems offer a promising platform for observing dynamical signatures of quantum gravity through precision measurements of quantum harmonic oscillator dynamics. However, most existing analyses consider only the linear radiation-pressure interaction while neglecting higher-order optomechanical couplings and laser phase noise. These neglected contributions can be comparable in magnitude to the predicted quantum-gravity corrections and may therefore introduce spurious signals or mask the genuine physical effect. Here we reanalyze two experimentally realized platforms, a Fabry-Perot optomechanical system and a membrane-in-the-middle optomechanical system, by incorporating the complete nonlinear dynamics and realistic laser phase noise. Using measured device parameters, we derive revised protocols for generalized uncertainty principle tests and establish practical sensitivity bounds. Our results demonstrate that previous idealized estimates significantly overestimate the achievable resolution, underscoring the necessity of including higher-order interactions and implementing effective laser phase noise suppression in realistic assessments of optomechanical quantum gravity tests.

The spatial distribution of open star clusters (OSCs) younger than 30 million years old in the Local System was studied. It was shown for the first time that a significant number of OSCs belong to the recently discovered Vela Ridge gas and dust supercloud. The most intriguing property of this sample of OSCs is the presence of periodic perturbations in their vertical coordinates with a maximum amplitude of 47 pc and a wavelength of 1.1 kpc. Thus, the discovered chain of young OSCs is analogous to the Radcliffe Wave, but with a lower amplitude of vertical perturbations, a shorter wavelength, and is, on average, 2 million years older.

Quantum software patterns provide high-level abstractions for building quantum programs, but there is still little empirical evidence on how they are adopted in practice. In prior work, we extended an existing quantum-pattern atlas into a 61-pattern catalog, created a knowledge base that links framework components to those patterns, and built a tool that mines pattern implementations from open-source code. We applied this tool on 80 projects and find that all 23 patterns occur in practice. In this work, we extend the tool with two additional matching channels and a vocabulary expansion step, and execute a quantitative evaluation of its accuracy on Qrisp, a framework not present in the knowledge base, reaching a micro-F1 of 0.712 against 0.449 without the expansion step. We then construct composition graphs that record calls among the high-level framework components associated with patterns and store them in a graph database. We use these graphs to examine how pattern implementations are assembled inside each framework, why patterns co-occur, and how much of a pattern's detection count comes from components called directly by developers rather than introduced through internal framework calls. We release qpa, an open-source mining pipeline, together with the knowledge base, which maps 286 framework components across five sources to the pattern catalog, maintained with the support of an LLM ensemble that classifies newly extracted components, and the resulting pattern usage dataset, to support reproducible studies on the adoption and evolution of quantum patterns.

Rotation and magnetism are important phenomena to consider when trying to understand stars. By shaping the structure and chemical mixing in the interior of stars, they can induce significant changes in both their fundamental and observable properties. Despite this, our theoretical picture of how rotation and magnetism evolve throughout the stellar lifecycle remains incomplete. Gaps in our understanding are particularly stark for evolved stars, where slow rotation rates and weak magnetic fields make observational studies difficult. In response to this issue, the last 20 years have seen instrumentation to probe magnetism and rotation progressing in leaps and bounds. Through these new tools we have found that evolved stars rotate in ways that we cannot predict and have magnetic fields with properties that we did not expect. In the following, I will provide a brief overview of some recent observational results and their implications for our understanding of stars. I will focus on advances in asteroseismology and high-precision spectropolarimetry, as these techniques have advanced significantly over the last few decades. In keeping with the theme of Cool Stars, I'll cover stars with low to intermediate masses and their evolution from the subgiant to the white dwarf phase.

The effects of black hole's spin in effective field theories (EFTs) of gravity are explored through observables of wave scattering on black hole backgrounds in the geometric optics approximation. The considered observables are polarisation rotation angle, wavenumber kick (or deflection angle), and (Shapiro/Wigner–Smith) time delay, each of which are related to gravitational Faraday rotation, gravitational spin Hall effect, and infrared causality. The observables are computed from scattering amplitudes through the Magnusian formalism, where the Magnusian is promoted to a matrix to account for helicity information. It is found that (1) the gravitational spin Hall effect in EFTs of gravity are qualitatively different from that of general relativity due to "noncommutative" wavenumber kicks, and (2) black hole's spin slightly enhances the causality constraints on EFT coefficients.

We develop an analytical theory of anomalous transport in a noninteracting Anderson model with heavy-tailed hopping amplitudes. The broad distribution of hopping amplitudes gives rise to an extended intermediate-time regime with a subdiffusive effective exponent, despite the absence of interactions or genuine many-body effects. By solving the transport equations analytically, we derive the time dependence of the mean-square displacement and identify a continuous crossover from an intermediate anomalous regime to asymptotically diffusive transport. As the localization transition is approached, the spatial extent of the subdiffusive window diverges parametrically, while the crossover to conventional diffusion remains finite in units of $\Gamma_0^{-1}$. This produces an increasingly broad anomalous transport regime in space that can closely resemble Griffiths-type transport observed near the many-body localization transition. Our results demonstrate that rare hopping processes alone provide a microscopic single-particle mechanism for robust transport anomalies, establishing an analytical benchmark for distinguishing interaction-induced effects from disorder-driven dynamics.

Unresolved binaries are common in spectroscopic surveys, and their blended light biases the parameters inferred for them. Methods to detect them exist, but applying one to a new data release is limited mostly by operating know-how that is rarely written down. We package the double-lined spectroscopic binary decomposition of El-Badry et al. (2018b) for reuse on APOGEE spectra, with the executable operations as nine tool servers built on the Model Context Protocol (MCP) and the operating decisions as a Skill. Run over the 238,205 DR19 dwarfs under a fixed driver script, the classifier flags 41,466 SB2 candidates, about fifteen times the 2,645 identified in DR13, though not at matched purity. The 8.1% control false-positive rate implies that close to 40% are single stars, so we release the sample as a candidate list. The joint visit fit prefers two stars for 46% of the SB2 with multiple visits. Of these, 4,149 have velocity changes above 10 km/s. This cut is also passed by 7% of flagged single stars. The pipeline velocities identify another 3,943 velocity variables. We compare the eccentricities of the best-sampled binaries and find no significant difference between the close twins (q > 0.95) and matched non-twins, which excludes an eccentricity excess of the kind measured at wide separations. An ablation of two operating decisions illustrates that a fresh agent recovers a removed decision only when its absence leaves a measurable trace in the fit. We release the tools, the Skill, and the DR19 catalog with its multi-epoch supplement.

Transient spatially localized activity underlies a broad range of biological processes, yet the statistical property of fluctuations that controls its nucleation remains unclear. We systematically investigate noise-induced dynamics in a spatially extended FitzHugh-Nagumo excitable system and identify three regimes: a quiescent phase, a spatially extended alternating phase, and a pattern-forming phase characterized by transient localized excitation patches. Surprisingly, we find that temporal noise correlations are not required for patch formation: Gaussian white noise produces localized patches when its amplitude is sufficiently large, whereas weaker fluctuations can achieve the same effect when temporal correlations allow them to persist. Our results identify the instantaneous amplitude and the persistence as joint stochastic control parameters for transient pattern formation in excitable media. These distinct quantities are linked through the integrated noise strength, which quantifies the accumulated stochastic forcing available to nucleate an excitation. In addition, the inhibitor response time subsequently determines whether the excitation remains localized or spreads throughout the system.

We initiate a study of centrally extended ($\textit{anomalous}$) symmetries in Kähler geometry, focusing on the simplest, and most ubiquitous, Abelian case. In particular, we provide a local description of the geometry admitting such isometries. A long-standing no-go theorem asserts that there is an obstruction to gauging such symmetries in the purely Kähler framework. Utilizing the language of sypersymmetry, we then show that these symmetries may be gauged within the setup of generalized Kähler geometry. Our results may be applied to quotients, T-dualities, etc.

In this work, we study photon propagation and the radiative properties of a thin accretion disk around a ModMax black hole and examine the effects of the charge $Q$ (i.e. $Q^2 = Q^2_e +Q^2_m$ is the total dyonic charge, with $Q_e$ and $Q_m$ denoting the electric and magnetic charges, respectively) and the nonlinearity parameter $v$. We determine the event-horizon, photon-sphere, and shadow radii and find that increasing $Q$ decreases these radii, whereas increasing $v$ shifts them toward their Schwarzschild values. Using the EHT shadow measurements of M87$^\star$ and Sgr~A$^\star$, together with the available mass and distance measurements, we perform an MCMC analysis to constrain the ModMax parameters. The strongest upper limits are found to be $Q<0.391$ and $v<4.153$ at the 95% credible level. We also investigate a Novikov-Thorne thin accretion disk and generate simulated disk images using backward ray tracing. The observed flux increases with $Q$, while increasing $v$ produces a slight decrease in the disk brightness. These results show how the ModMax parameters affect the black-hole shadow and thin-disk emission and provide observational constraints on $Q$ and $v$.

Although considered a niche technique, muon spectroscopy provides a unique and complementary insight into a range of different materials from hard condensed matter to biological samples and everything in between. In matter, the muon has a mass of $\frac{1}{9}~m_p$ or $207~m_e$, and is a local probe of quantum states that can provide a focus on the bulk properties of materials. While often interpreted in a classical framework, the muon is itself a quantum particle and it is increasingly common for researchers to take account of this when thinking about muon spectroscopy experiments. In this perspective, we focus on the power of using this quantum treatment of muon spectroscopy, which is a key future direction for the technique.

This study investigates whether the macroscopic statistical maturity of cryptocurrencies implies dynamical equivalence with an equity-market benchmark. We analyze one-minute data (2020–2025) for Bitcoin, Ethereum, XRP, and E-mini S&P~500 futures using the Complexity–Entropy Causality Plane (CECP) and directed horizontal visibility graphs (DHVG). While conventional stylized facts overlap across the analyzed assets, cryptocurrencies exhibit weaker local ordinal organization and a larger divergence between their high-degree in- and out-visibility distributions relative to matched surrogate baselines under the primary specification. This high-degree separation is strongest at the one-minute scale and weakens under 5- and 10-minute aggregation, while the direction of the underlying asymmetry remains more asset- and preprocessing-dependent. We therefore conclude only that macroscopic similarity can coexist with diagnostic-specific structural differences in the assets and period examined.

When a liquid drop impacts a solid surface, rebound and splashing are known to be suppressed by additives that induce elastic effects; however, the influence of elasticity on the maximum spreading radius remains debated. The difficulty lies in isolating elastic resistance from the enhanced spreading caused by viscous shear thinning. Here, we experimentally decouple these effects by investigating Boger fluid drops (constant-viscosity solutions with high elasticity) of polyethylene oxide (PEO) and polyacrylamide (PAAM). We demonstrate that elastic properties do not alter the maximum spreading ratio, even at high concentrations up to 1000 ppm. We formulate an energy balance that accounts for the inertial, capillary, viscous, and elastic contributions, and yields a dimensionless criterion ($\Gamma$) that estimates the degree of elastic effects. We show that $\Gamma \ll 1$ for all impact conditions tested, demonstrating that elastic effects are energetically negligible during droplet spreading, opposite to what Weissenberg and Deborah numbers would predict.

Collective motion in self-propelled particle systems has been widely studied using the Vicsek model, which relies on pairwise alignment interactions. We introduce a generalized Vicsek model that incorporates higher-order (triadic) alignment interactions. Using agent-based simulations and mean-field theory, we demonstrate that pure triadic alignment induces a discontinuous phase transition, evidenced by hysteresis, a double-well free-energy landscape, and a Binder cumulant minimum that deepens with system size, whereas the standard pairwise model exhibits a continuous transition at the same system sizes. We further show that higher-order interactions require higher particle densities to sustain collective order and produce sharper fluctuation peaks near the transition with lower critical noise. These results establish that the microscopic structure of the alignment interaction, whether pairwise or many-body, is an independent control parameter for the order of the phase transition in active matter, with implications for understanding collective behavior in biological and synthetic systems.

Energetic electrons create distinct cathodoluminescence (CL) angular distributions upon interaction with dielectric and plasmonic nanostructures, providing valuable information on coherences in the excitation pathways. A counterintuitive prediction is that the CL signals arising from the excitation associated with different lateral regions of an extended electron wave are mutually incoherent and do not interfere, while the signals originating from different structures within the electromagnetic field of a narrow electron beam are mutually coherent. We present conclusive experimental evidence of these effects by examining the angular CL emission profile from defocused electron-beam excitation of a thin silicon nitride film, which is shown to follow an incoherent sum of CL excitations within the electron beam spot. In contrast, CL interferences are observed for separated plasmonic scatterers excited within the evanescent field of a single electron. Coherence may be recovered through correlations between emitted light and post-selected electron states, for which we propose a measurement geometry that erases which-path information of the electron trajectory.

We analyze the prompt emission of the exceptionally bright GRB 220426A observed by Fermi/GBM, whose time-resolved spectra are among the narrowest measured in any GRB. Here we show that observations during the first $\sim 5$~s are consistent with the signal being emitted while the jet was still in the initial radiation-dominated acceleration phase. The time-resolved spectra allow the effective launch radius, $r_0$, to be inferred with unusual precision. We find $r_0 \sim \mathrm{few} \times 10^{10}\,\mathrm{cm}$, increasing linearly with time. These findings match the theoretical expectation for the recollimation shock, implying that this GRB shows the first clear evidence for the existence and evolution of a recollimation shock. Using this interpretation, the linear increase observed is sustained over a period longer than expected from a pure jet breakout. Therefore, the jet collimation must have persisted even after breakout. We suggest that the collimation was maintained by a finite, dense, wind-like circumburst medium, which would reproduce the observed behavior of the prompt emission. We conclude that late-stage progenitor mass-loss can shape the earliest prompt emission and that the photospheric emission provides a way to probe both the jet collimation and the innermost region of the circumburst medium (CBM) surrounding the progenitor, independently of the constraints from interacting supernovae.

We establish that supersymmetric monodromy defects in ABJM theory, also known as vortex loops, define integrable boundary states of the underlying alternating SU(4) spin chain. Exploiting the KT relation within the framework of the algebraic Bethe ansatz, we derive a closed-form expression for the leading-order one-point functions of non- protected operators in the presence of these defects. As a special case, our result reproduces the spin-chain overlap governing the three-point functions of two maximal giant gravitons and a single tiny graviton. We furthermore investigate the quantization of the theory in the defect background and compute the first quantum correction to the one-point functions in the simplest setting. Finally, we discuss possible extensions of our overlap formula.

In $(3+1)$ spacetime dimensions, we numerically calculate the suppression exponent of the false vacuum decay probability in the presence of a thermal bath and a black hole.

We investigate broad hydrogen line ratios in little red dots (LRDs) using five high-redshift ($z>2$) sources from JWST/NIRSpec medium/high-resolution spectra in the DAWN JWST Archive and fifteen low-redshift sources ($z=0.1$–$0.9$) from the literature, all with broad H$\beta$ detected at $>5\sigma$. After carefully measuring the broad-line fluxes while accounting for absorption features and neighboring emission lines, we find that the broad H$\alpha$/H$\beta$ ratios are very high, ranging from 6 to 30, well above the Case B recombination value. Using plane-parallel Cloudy photoionization models with the total line emission from both sides of the slab, we investigate the physical origin of the broad hydrogen line ratios beyond Case B, jointly modeling the Balmer and Paschen line ratios for the subset of one high-$z$ and two low-$z$ LRDs with detected broad Paschen lines. We find that one low-$z$ LRD is reproduced by a high gas number density ($\log (n_{\mathrm H}/{\rm cm^{-3}})\sim10$–11) on the broad-line H$\alpha$/H$\beta$–Pa$\beta$/Pa$\gamma$ plane, whereas the remaining two LRDs additionally require substantial dust extinction of $E(B-V)\gtrsim0.2$, even after accounting for non-Case B effects. Since the narrow H$\alpha$/H$\beta$ ratios do not indicate such large dust extinction, these results demonstrate that the obscuring dust is spatially associated with the broad-line region. Even without Paschen-line measurements, two and eight LRDs in the high- and low-redshift samples, respectively, exhibit H$\alpha$/H$\beta>13$, which cannot be reproduced by the non-Case B models, suggesting that dust obscuration might be common among LRDs, occurring in at least about half of the population. Such dust may represent a lower-column-density counterpart of the dusty torus in AGNs, reconciling the weak hot-dust emission.