Water is one of the central molecules for the formation and habitability of planets. In particular, the region where water freezes-out, the water snowline, could be a favorable location to form planets in protoplanetary disks. We use high resolution ALMA observations to spatially resolve H$_2$O, H$^{13}$CO$^+$ and SO emission in the HL Tau disk. A rotational diagram analysis is used to characterize the water reservoir seen with ALMA and compare this to the reservoir visible at mid- and far-IR wavelengths. We find that the H$_2$O 183 GHz line has a compact central component and a diffuse component that is seen out to ~75 au. A radially resolved rotational diagram shows that the excitation temperature of the water is ~350 K independent of radius. The steep drop in the water brightness temperature outside the central beam of the observations where the emission is optically thick is consistent with the water snowline being located inside the central beam ($\lesssim 6$ au) at the height probed by the observations. Comparing the ALMA lines to those seen at shorter wavelengths shows that only 0.02%-2% of the water reservoir is visible at mid- and far-IR wavelengths, respectively, due to optically thick dust hiding the emission whereas 35-70% is visible with ALMA. An anti-correlation between the H$_2$O and H$^{13}$CO$^+$ emission is found but this is likely caused by optically thick dust hiding the H$^{13}$CO$^+$ emission in the disk center. Finally, we see SO emission tracing the disk and for the first time in SO a molecular outflow and the infalling streamer out to ~2". The velocity structure hints at a possible connection between the SO and the H$_2$O emission. Spatially resolved observations of H$_2$O lines at (sub-)mm wavelengths provide valuable constraints on the location of the water snowline, while probing the bulk of the gas-phase reservoirs.

The $D$-graded Proj construction provides a general framework for constructing schemes from rings graded by finitely generated abelian groups $D$, yet its properties and applications remain underdeveloped compared to the classical $\mathbb{N}$-graded case. This paper establishes the essential characteristics of $D$-graded rings $S$, like the distinction between $D$-homogeneous prime ideals and $D$-prime ideals if $D$ has torsion. We particularly focus on describing the quotient by the associated group scheme, generalizing the construction of a toric variety from its Cox ring. As in the $\mathbb{N}$-graded construction, the basic affine opens of the Proj construction are given in terms of degree-zero localizations $S_{(f)}$, where $f$ in $S$ homogeneous is \emph{relevant}. We prove that $π_f: {\rm Spec}(S_f) \to {\rm Spec}(S_{(f)})$ is a geometric quotient under mild finiteness assumptions if $f$ is relevant, and give necessary and sufficient conditions for this map to be a pseudo ${\rm Spec}(S_0[D])$-torsor.

We study the transient black hole binary MAXI J1631--479 in its soft spectral state observed simultaneously by the NICER and NuSTAR instruments. Its puzzling feature is the presence of a strong and broad Fe K line, while the continuum consists of a strong disk blackbody and a very weak power-law tail. The irradiation of the disk by a power-law spectrum fitting the tail is much too weak to account for the strong line. Two solutions were proposed in the past. One invoked an intrinsic Fe K disk emission, and the other invoked disk irradiation by the returning blackbody emission. We instead find that the strong line is naturally explained by the irradiation of the disk by the spectrum from Comptonization of the disk blackbody by coronal relativistic electrons. The shape of the irradiating spectrum at $\lesssim$10 keV reflects that of the disk blackbody; it is strongly curved and has a higher flux than that of a fit with a power-law irradiation. That flux accounts for the line. While this result is independent of the physical model used for the disk intrinsic emission, the value of the fitted spin strongly depends on it. When using a Kerr disk model for a thin disk with a color correction, the fitted spin corresponds to a retrograde disk, unlikely for a Roche-lobe overflow binary. Then, a model accounting for both the disk finite thickness and radiative transfer yields a spin of $a_*\approx0.8$--0.9, which underlines the strong model-dependence of X-ray spin measurements.

In this article we study the Ekedahl-Oort types of $\mathbb{Z}/2\mathbb{Z}$-Galois covers $π:Y \to X$ in characteristic two. When the base curve $X$ is ordinary, we show that the Ekedahl-Oort type of $Y$ is completely determined by the genus of $X$ and the ramification of $π$. For a general base curve $X$, we prove bounds on the Ekedahl-Oort depending on the Ekedahl-Oort type of $X$ and the ramification of $π$. Along the way, we develop a theory of \emph{enhanced differentials of the second kind}. This theory allows us to study algebraic de Rham cohomology in any characteristic by working directly with differentials, in contrast to the standard Čech resolution.

We propose a four-band model describing the electron energy spectra near the Weyl points in the topological semimetals of the TaAs family (TaAs, TaP, NbAs, NbP). This model takes into account the fact that these Weyl points result from the band-contact lines which would exist in the mirror-reflection planes of these materials if the spin-orbit interaction were absent in them. Within this model, we obtain conditions for the existence of the Weyl points, determine their positions in the Brillouin zone, and derive the explicit formula for dispersion of the bands along the straight line connecting the two close Weyl points with opposite topological charges. Using NbP as an example, the values of the parameters defining the model spectrum are found. The obtained results show that for the semimetals of the TaAs family, the charge-carriers spectrum in the vicinity of the two close Weyl points can be analyzed without complex band-structure calculations.

Zero Added Loss Multiplexing (ZALM) offers broadband, per channel heralded EPR pairs, with a rich parameter space that allows its performance to be tailored for specific applications. We present a modular ZALM simulator that demonstrates how design choices affect output rate and fidelity. Built in NetSquid with QSI controllers, it exposes 20+ tunable parameters, supports IDEAL and REALISTIC modes, and provides reusable components for Spontaneous Parametric Down Conversion (SPDC) sources, interference, Dense Wavelength Division Multiplexing (DWDM) filtering, fiber delay, active polarization gates, detectors, and lossy fiber. Physics based models capture Hong Ou Mandel (HOM) visibility, insertion loss, detector efficiency, gate errors, and attenuation. Using this tool, we map trade offs among fidelity, link distance, and entangled pairs per use, and show how SPDC bandwidth and DWDM grid spacing steer performance. Using the default configuration settings, average fidelity remains constant at 0.83 but the ebit rate decreases from 0.0175 at the source to 0.0 at 50 km; narrowing the SPDC degeneracy bandwidth increases the ebit rate significantly without affecting fidelity. The simulator enables codesign of source, filtering, and feedforward settings for specific quantum memories and integrates as a building block for end to end quantum network studies.

We summarize key experimental studies of the low energy properties of overdoped cuprate high temperature superconductors and conclude that a theoretical understanding of the ``essential physics" is achievable in terms of a conventional Fermi-liquid treatment of the normal state, and a BCS mean-field treatment of the (d-wave) superconducting state. For this perspective to be consistent, it is necessary to posit that there is a crossover from a strongly correlated underdoped regime (where a different theoretical perspective is necessary) to the more weakly correlated overdoped regime. It is also necessary to argue that the various observed features of the overdoped materials that are inconsistent with this perspective can be attributed to the expected effects of the intrinsic disorder associated with most of the materials being solid state solutions (alloys). As a test of this idea, we make a series of falsifiable predictions concerning the expected behavior of an ``ideal" (disorder free) overdoped cuprate.

Hydrogen hydrates (HH) are a unique class of materials composed of hydrogen molecules confined within crystalline water frameworks. Among their multiple phases, the filled ice structures, particularly the cubic C2 phase, exhibit exceptionally strong host-guest interactions due to ultra-short H2-H2O distances and a 1:1 stoichiometry leading to two interpenetrated identical diamond-like sublattices, one comprised of water molecules, the other of hydrogen molecules. At high pressures, nuclear quantum effects involving both hydrogen molecules and the water lattice become dominant, giving rise to a dual-lattice quantum system. In this work, we explore the sequence of pressure- and temperature-driven phase transitions in HH, focusing on the interplay between molecular rotation, orientational ordering, lattice symmetry breaking and hydrogen bond symmetrization. Using a combination of computational modeling based on classical and path-integral molecular dynamics, quantum embedding, and high pressure experiments, including Raman spectroscopy and synchrotron X-ray diffraction at low temperatures and high pressures, we identify signatures of quantum-induced ordering and structural transformations in the C2 phase. Our findings reveal that orientational ordering in HH occurs at much lower pressures than in solid hydrogen, by inducing structural changes in the water network and enhancing the coupling of water and hydrogen dynamics. This work provides new insights into the quantum behavior of hydrogen under extreme mechanochemical confinement and establishes hydrogen-filled ices as a promising platform for the design of hydrogen-rich quantum materials.

Border bases are a generalization of Gröbner bases for zero-dimensional ideals in polynomial rings. In this article, we introduce border bases for a non-commutative ring of linear differential operators, namely the rational Weyl algebra. We elaborate on their properties and present algorithms to compute with them. We apply this theory to represent integrable connections as cyclic $D$-modules explicitly. As an application, we visit differential equations behind a string, a Feynman as well as a cosmological integral. We also address the classification of particular $D$-ideals of a fixed holonomic rank, namely the case of linear PDEs with constant coefficients as well as Frobenius ideals. Our approach rests on the theory of Hilbert schemes of points in affine space.

We discuss several congruences satisfied by the coefficients of meromorphic modular forms, or equivalently, the $p$-adic behaviors of meromorphic modular forms under the $U_p$ operator, that are summarized from numerical experiments. In the generic case, we observe the connection to symmetric powers of elliptic curves, while in the CM case, we furthermore observe the connection to the $p$-adic analogue of the Chowla--Selberg periods. Along with the discussions, we will provide some heuristic explanations for these congruences as well as prove some of them using hypergeometric functions and the Borcherds--Shimura lift.

Early identification of neurodevelopmental disorders relies on accurate segmentation of brain structures in infancy, a task complicated by rapid brain growth, poor tissue contrast, and motion artifacts in pediatric MRI. These challenges are further exacerbated in ultra-low-field (ULF, 0.064~T) MRI, which, despite its lower image quality, offers an affordable, portable, and sedation-free alternative for use in low-resource settings. In this work, we propose a domain randomization (DR) framework to bridge the domain gap between high-field (HF) and ULF MRI in the context of the hippocampi and basal ganglia segmentation in the LISA challenge. We show that pre-training on whole-brain HF segmentations using DR significantly improves generalization to ULF data, and that careful curation of training labels, by removing misregistered HF-to-ULF annotations from training, further boosts performance. By fusing the predictions of several models through majority voting, we are able to achieve competitive performance. Our results demonstrate that combining robust augmentation with annotation quality control can enable accurate segmentation in ULF data. Our code is available at https://github.com/Medical-Image-Analysis-Laboratory/lisasegm

Let $X$ be a smooth complex prime Fano fourfold having a semi-free action of $\mathbb{C}^*$, then $X$ is contained in one of the families $\mathbb{P}^4,Q^4,W_5, X^{m}_{8}$. All of the families contain members that have a semi-free $\mathbb{C}^*$-action.

The local Carroll symmetry of a gravitational wave found in Baldwin-Jeffery-Rosen coordinates is extended to a globally defined one by switching to Brinkmann coordinates. Two independent globally defined solutions of a Sturm-Liouville equation allow us to describe both the symmetries (translations and Carroll boosts) and the geodesic motions. One of them satisfies particular initial conditions which imply zero initial momentum, while the other does not. Pure displacement arises when the latter is turned off by requiring the momentum to vanish and when the wave parameters take, in addition, some particular values which correspond to having an integer half-wave number. The relation to the Schwarzian derivative is highlighted. We illustrate our general statements by the Pöschl-Teller profile.

In the study of plasma, particularly in applications involving strong laser-plasma interactions, the propagation of a strong electromagnetic wave induces relativistic velocities in the electron flow. Given such conditions, the wave propagating through the plasma experiences modulational instability. In this paper, we investigate this instability using magnetohydrodynamic (MHD) equations. In the relativistic limit, the motion of ions can be neglected due to their significant inertia, allowing us to treat the ions as a background fluid. This simplification enables us to apply perturbation techniques to the electron fluid equations, leading to the derivation of the nonlinear wave equation in the form of the Nonlinear Schrödinger Equation (NLSE). We also explore the relationship between wave dispersion and the conditions for instability. We derive the maximum growth rate of the modulational instability and analyze its dependence on plasma parameters and wave intensity in the context of relativistic magnetized plasma, providing quantitative insights into the instability dynamics. Finally, we examine aspects of the perturbed NLSE using the Bogoliubov-Mitropolsky perturbation approach, treating real and imaginary coefficients separately, which explicitly incorporates both Nonlinear Landau Damping (NLLD) and growth-damping effects.

A significant fraction of stars experience close interactions, including collisions resulting from gravitational encounters and mergers within close binary systems. These processes can produce more massive stars that may give rise to relatively rare objects such as blue stragglers. Distinguishing the outcomes of collisions and mergers is challenging yet essential for interpreting observations. This study utilizes the magnetohydrodynamics code AREPO to simulate collisions and mergers of $5$ to $10 \,\mathrm{M}_{\odot}$ main-sequence stars, systematically comparing the properties of the resulting products. Both collisions and mergers yield more massive, strongly magnetized, rapidly and differentially rotating stars with cores enriched in hydrogen, but notable quantitative differences emerge. Merger products exhibit core hydrogen fractions up to $10\%$ higher than those of collision products. In both scenarios, turbulent mixing amplifies magnetic field energies by $9$ to $12$ orders of magnitude. However, magnetic fields in small-impact-parameter collision products display small-scale reversals that may dissipate over time, whereas merger products and large-impact-parameter collision products develop large-scale ordered, potentially long-lived magnetic fields. Additionally, only merger products display magnetically driven, bipolar outflows with radial velocities exceeding $300$ to $400 \,\mathrm{km}\,\mathrm{s}^{-1}$. These distinctions may result in different long-term evolutionary outcomes, which warrant further investigation in future studies.

Fuzzy logic extends the classical truth values "true" and "false" with additional truth degrees in between. More specifically, fuzzy modal logics in this sense are given by a choice of fuzzy modalities and a fuzzy propositional base. It has been noted that fuzzy modal logics over the Zadeh base, which interprets disjunction as maximum, are often computationally tractable but on the other hand add little in the way of expressivity to their classical counterparts. Contrastingly, fuzzy modal logics over the more expressive Lukasiewicz base have attractive logical properties but are often computationally less tractable or even undecidable. In the basic case of the modal logic of fuzzy relations, sometimes termed fuzzy ALC, it has recently been shown that an intermediate non-expansive propositional base, known from characteristic logics for behavioural distances of quantitative systems, strikes a balance between these poles: It provides increased expressiveness over the Zadeh base while avoiding the computational problems of the Lukasiewicz base, in fact allowing for reasoning in PSpace. Modal logics, in particular fuzzy modal logics, generally vary widely in terms of syntax and semantics, involving, for instance, probabilistic, preferential, or weighted structures. Coalgebraic modal logic provides a unifying framework for wide ranges of semantically different modal logics, both two-valued and fuzzy. In the present work, we focus on non-expansive coalgebraic fuzzy modal logics, providing a criterion for decidability in PSpace. Using this criterion, we recover the mentioned complexity result for non-expansive fuzzy ALC and moreover obtain new PSpace upper bounds for various quantitative modal logics over probabilistic and metric transition systems. Notably, we show that the logic of generally, which has recently been shown to characterize e-distance on Markov chains, is decidable in PSpace.

We prove that any non-symmetric ribbon tensor category $\mathcal{C}$ with the fusion rules of the compact group of type $G_2$ needs to be equivalent to the representation category of the corresponding Drinfeld-Jimbo quantum group for $q$ not a root of unity. We also prove an analogous result for the corresponding finite fusion tensor categories.

Real-time rendering of dynamic line sets is relevant in many visualization tasks, including unsteady flow visualization and interactive white matter reconstruction from Magnetic Resonance Imaging. High-quality global illumination and transparency are important for conveying the spatial structure of dense line sets, yet remain difficult to achieve at interactive rates. We propose an efficient voxel-based ray-tracing framework for rendering large dynamic line sets with ambient occlusion and ground-truth transparency. We introduce a voxelization algorithm that supports efficient on-the-fly construction of acceleration structures for both voxel cone tracing and ray tracing. To further reduce per-frame preprocessing cost, we propose a voxel-based culling method that restricts acceleration structure construction to camera-visible voxels. Together, these contributions enable real-time rendering of large-scale dynamic line sets with high quality and physically accurate transparency. We demonstrate that our method outperforms the state of the art in quality and performance when rendering (semi-)opaque dynamic line sets.

We present a comprehensive Monte Carlo simulation of muon-induced backgrounds for the Colorado Underground Research Institute (CURIE), a shallow-underground facility with $\approx 415$~m.w.e. overburden. Using coupled \textsc{mute} and \textsc{geant4} frameworks, we characterize the production and transport of muon-induced secondaries through site-specific rock compositions and geometries, establishing a proof-of-concept for high-precision, end-to-end simulations. Our simulations employ angular-dependent muon energy distributions, which improve secondary flux accuracy. For the Subatomic Particle Hideout and Cryolab I research spaces, we predict total muon-induced neutron fluxes of $(8.52 \pm 1.30_{\text{sys}}) \times 10^{-3}$~m$^{-2}$s$^{-1}$ and $(8.86 \pm 1.62_{\text{sys}}) \times 10^{-3}$~m$^{-2}$s$^{-1}$, respectively. Additionally, we develop a Depth-Intensity Relation (DIR) to predict the muon-induced neutron flux as a function of facility depth, which is consistent with measurements across a broad range of underground depths. These results provide quantitative background predictions for experimental design and sensitivity projections at shallow- and deep-underground facilities. They further demonstrate that local geology and overburden geometry influence muon-induced secondary yields and energy spectra, emphasizing the need for site-specific simulations for accurate underground background characterization. Therefore, the simulation framework has been made publicly available at \href{https://doi.org/10.5281/zenodo.17196581}{https://doi.org/10.5281/zenodo.17196581}, for the broader low-background physics community to enable meaningful inter-facility comparisons.

Using analytic number theory techniques, Altuğ showed that the contribution of the trivial representation to the Arthur-Selberg trace formula for GL(2) over $\Q$ could be cancelled by applying a modified Poisson summation formula to the regular elliptic contribution. Drawing on recent works, we re-examine these methods from an adelic perspective.

This study investigates how listeners perceive consonance and dissonance in dyads composed of simple (sine) tones, focusing on the effects of frequency ratio ($R$) and mean frequency ($F$). Seventy adult participants - categorized by musical training, gender, and age group - rated randomly ordered dyads using binary preference responses (``like'' or ``dislike''). Dyads represented standard Western intervals but were constructed with sine tones rather than musical notes, preserving interval ratios while varying absolute pitch. Statistical analyses reveal a consistent decrease in preference with increasing mean frequency, regardless of interval class or participant group. Octaves, fifths, fourths, and sixths showed a nearly linear decline in preference with increasing $F$. Major seconds were among the least preferred. Musicians rated octaves and certain consonant intervals more positively than non-musicians, while gender and age groups exhibited different sensitivity to high frequencies. The findings suggest that both interval structure and pitch range shape the perception of consonance in simple-tone dyads, with possible psychoacoustic explanations involving frequency sensitivity and auditory fatigue at higher frequencies.

We explore the stationary densities and domain walls in the steady states of a pair of asymmetric exclusion processes (TASEP) antiparallelly coupled to two particle reservoirs without any spatial extent by using the model in Haldar et al., Phys. Rev. E {\bf 111}, 014154 (2025). We show that the model admits a pair of {\em delocalized} domain walls, which exist for some choices of the model parameters that define the effective entry and exit rates into the TASEP lanes. Surprisingly, in the parameter space spanned by these model parameters, the region corresponding to delocalized domain walls covers an {\em extended} region, in contrast to the delocalized domain walls that appear only along a line in the relevant parameter space of the other known variants of TASEP. This implies large fluctuations in the TASEP particle numbers even in the thermodynamic limit that can be found over a range of the control parameters. The corresponding phase diagrams in the plane of the control parameters have different topology from those for an open TASEP or other models with multiple TASEPs connected to two reservoirs.

We investigate the phase structure of the (3+1)-dimensional strong coupling two-color QCD at zero temperature ($T=0$) with finite chemical potential using the tensor renormalization group method. The chiral and diquark condensates and the quark number density are evaluated as a function of the chemical potential. We further determine the critical exponents associated with the diquark condensate, which suggest consistency with the predictions of mean-field theory.

Using a data sample of $(2.712 \pm 0.014)\times10^{9}$ $ψ(3686)$ events collected at $\sqrt{s}=3.686$ GeV by the BESIII detector, we search for the rare electromagnetic Dalitz decays $χ_{cJ}\to e^+e^-ϕ~(J=0,\,1,\,2)$ via the radiative transitions $ψ(3686)\toγχ_{cJ}$. No statistically significant $χ_{cJ}\to e^+e^-ϕ$ signals are observed. The upper limits on the branching fractions of $χ_{cJ}\to e^+e^-ϕ~(J=0,\,1,\,2)$, excluding the $ϕ$ resonance to $e^+e^-$ final states, are set to be $2.4\times10^{-7},~6.7\times10^{-7}$ and $4.1\times10^{-7}$ at 90\% confidence level, respectively. This is the first search for the electromagnetic Dalitz transition of P-wave charmonium $χ_{cJ}$ states to a light vector meson.

We study zero forcing and $\ell$-leaky zero forcing on induced subgraphs of $d$-dimensional grid graphs. Using $\ell$-leaky forts, we prove structural results showing that for $\ell \le 2d-1$, every nonempty $\ell$-leaky fort in an induced subgraph of $P_{n_1}\square\cdots\square P_{n_d}$ intersects the boundary of the graph. These results give general bounds and, in certain settings, exact values for the $\ell$-leaky forcing number of induced subgraphs. Motivated by this framework, we introduce an integer lattice based definition of the Hopi rectangle graphs $HD(a,b)$ as induced subgraphs of $P_{a+b}\square P_{a+b}$. For this particular family of graphs, we show that the zero forcing number equals the maximum nullity, and we completely characterize the $\ell$-leaky forcing number for all $\ell\ge 1$.

In high-energy proton proton ($pp$) collisions at the LHC, non-prompt charm hadrons, originating from beauty hadron decays, provide a valuable probe for beauty quark dynamics, particularly at low transverse momentum where direct beauty measurements are challenging. We employ A Multi-Phase Transport Model (AMPT) of string melting version coupled with PYTHIA8 initial conditions to study the beauty hadron and non-prompt charm hadron productions in $pp$ collisions at $\sqrt{s} = 13$ TeV. In this work, the beauty quark mass during the generation stage has been increased to reproduce the measured $b\bar{b}$ cross section, and a beauty flavor specific coalescence parameter $r_{BM}^b$ is introduced to match LHCb measurements of beauty baryon to meson ratios. With these refinements, AMPT achieves a reasonable agreement with experimental data on beauty hadron yields and non-prompt charm hadron production from ALICE and LHCb. We present the transverse momentum and multiplicity dependence of non-prompt to prompt charm hadron ratios, providing new insights into the interplay between beauty quark production and hadronization process. We emphasize that the multiplicity dependence of the non-prompt to prompt charm hadron productions can be useful to constrain the flavor dependences of the coalescence dynamics. This work establishes a unified framework for future studies of heavy quark transport and collective flow behavior in small collision systems.

We analyze quantum metrological protocols, where the sensing system is linearly coupled to a bosonic environment, by performing a Markovian embedding of the problem based on pseudomode formalism. This allows us to effectively model the problem using low-dimensional environment and apply recently developed powerful tools that yield optimal metrological protocols and fundamental metrological bounds for correlated-noise models. We illustrate the method by investigating a frequency estimation protocol in the presence of noise modeled effectively as a damped Jaynes-Cummings dynamics.

Heavy quarks are predominantly generated at the initial stage of relativistic heavy-ion collisions such that heavy flavor observables have the potential to provide information on the pre-equilibrium medium dynamics. In this study, we investigate the sensitivity of D-meson $R_{AA}$ and $v_2$ to early-time charm quark dynamics in Pb+Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV. We employ the IP-Glasma+MUSIC+UrQMD framework to model the evolution of the bulk medium. Charm quarks are generated using PYTHIA with nuclear parton distribution functions and evolved using Langevin dynamics within MARTINI. We observe that even though there is significant momentum broadening in the earliest stage, D-meson $R_{AA}$ and $v_2$ are only weakly sensitive to pre-equilibrium interactions.

We consider constrained-degree percolation on the hypercubic lattice. Initially, all edges are closed, and each edge independently attempts to open at a uniformly distributed random time; the attempt succeeds if, at that instant, both end-vertices have degrees strictly less than a prescribed parameter. The absence of the FKG inequality and the finite energy property, as well as the infinite range of dependency, make the rigorous analysis of the model particularly challenging. In this work, we show that the one-arm probability exhibits exponential decay in its entire subcritical phase. The proof relies on the Duminil-Copin--Raoufi--Tassion randomized algorithm method and resolves a problem of dos Santos and the second author. At the heart of the argument lies an intricate combinatorial transformation of pivotality in the spirit of Aizenman--Grimmett essential enhancements, but with unbounded range. This technique may be of use in other dynamical settings.

Inspired by the BESIII newest observation of an axial-vector particle $X(2300)$ in the $ψ(3686)\rightarrow ϕηη'$ process, we simulate its production in $e^+e^-$ collisions at $\sqrt{s}=4.95$ GeV using the parton and hadron cascade model PACIAE 4.0. In this model, the final partonic state (FPS) and hadronic state (FHS) are simulated and recorded sequentially. We propose, for the first time, that $X(2300)$ could be a $q\bar{q}s\bar{s}$ ($q=u/d$) state or a hadro-strangeonium state, i.e., a bound system of a strangeonium and a light hadron. The excited strangeonium candidate is formed by coalescing an $s\bar{s}$ quark pair in the FPS with the quantum statistical mechanics inspired dynamically constrained phase-space coalescence model. The tetraquark candidates of $q\bar{q}s\bar{s}$ and $ss\bar{s}\bar{s}$ are similarly produced by coalescing four constituent quarks in the FPS. In contrast, a hadro-strangeonium candidate emerges from the recombination of the constituent $ϕ$ and $η/η$ in the FHS. We then calculate the $X(2300)$'s orbital angular momentum quantum number in its rest frame and perform the spectral classification for each of the above candidates. Given its quantum numbers $J^{PC}=1^{+-}$, $X(2300)$ is identified as a $P$-wave $s\bar{s}$, an $S$-wave $q\bar{q}s\bar{s}/ss\bar{s}\bar{s}$ or $S$-wave $ϕη'/ϕη$ candidate. For the first time, we estimate the production rates for these configurations. The $P$-wave $s\bar{s}$ and $S$-wave $q\bar{q}s\bar{s}$ states are produced at rates on the order of $10^{-5}$, whereas the $S$-wave $ss\bar{s}\bar{s}$ and $ϕη'/ϕη$ states appear at rates on the order of $10^{-6}$. Moreover, significant discrepancies are observed in the rapidity and transverse momentum distributions among different candidates. These discrepancies could be served as valuable criteria for deciphering the nature of $X(2300)$.