In Willems' behavioral systems theory, a dynamical system is identified with the set of all trajectories compatible with its laws of motion. In the linear time-invariant setting this trajectory set is a linear subspace, and its algebraic structure underpins the Fundamental Lemma: a single persistently exciting data trajectory generates the entire finite-horizon behavior. For nonlinear or stochastic systems, however, the admissible trajectory set is generally nonconvex, obstructing direct optimization over the behavior. In this paper, we lift the behavioral viewpoint from trajectories to probability measures on trajectories by representing a finite-horizon dynamical system with the set of all Borel probability measures supported on its admissible trajectories. For deterministic systems, this behavioral-measure set is convex and weakly closed even when the dynamics are nonlinear, because convex combinations of trajectory distributions remain dynamically admissible even when convex combinations of trajectories do not. Its extreme points are precisely the Dirac masses on individual admissible trajectories, so the classical deterministic theory is embedded as the extremal skeleton of the richer measure-valued object. On this foundation we establish two core deterministic results and outline a stochastic extension based on history-conditional kernel consistency.

Log-logistic distribution is a flexible distribution that can model a wide range of failure patterns in the field of electrical, electronic and mechanical engineering and is often used in reliability inference. However, the inference of the parameters and reliability function of the log-logistic distribution can be challenging, especially in small sample scenarios. In this paper, we propose a new inference framework based on the least squares estimator-based generalized pivotal quantities (LSE-GPQ) for the parameters and reliability functions of the log-logistic distribution, which can provide better coverage in small sample scenarios. We will compare the performance of our proposed method with traditional methods such as the MLE and parametric bootstrapping through simulation studies and real data applications.

We address several aspects of entanglement entropy of 2D interface CFT using the replica method. Unlike the case of boundary CFT, we consider the boundary OPE (BOPE) of the Rényi twist operator and find a boundary twist operator anchored on the interface. This approach gives the $O(1)$ contribution to the entanglement entropy in terms of the BOPE coefficients of the twist operator. We further analyze entanglement entropy of different intervals and compare our findings with previous holographic results.

In a wireless network, the spatial location of the transmitters has a large impact on the achievable rate at each user location. The optimal placement of -- for example -- cellular base stations is a difficult non-convex problem, and is usually addressed with simplified propagation models and simplified heuristics that may account for specifics such as the site topology, building locations, and user density. We propose a mathematically rigorous framework for optimal transmitter placement that explicitly integrates detailed site-specific maps, spatial material properties, and realistic signal attenuation. We introduce a novel aggregated network quality functional which captures the essential trade-off between maximizing network coverage and minimizing cost, and establish the problem's sub-modularity under certain practical conditions. To solve the resulting resource-constrained optimization problem for sparse, discrete transmitter configurations, we propose the Interference-Aware Submodular Placement Algorithm (IA-SPA) and prove theoretical performance guarantees on its gap from optimality. IA-SPA is general and can incorporate existing BS locations and prohibited areas (e.g. a lake), making it useful for either clean-slate or incremental deployments. We show the utility of our approach using a ray tracing-based simulation framework applied to 3D maps of San Francisco and Florence, where we compare to known base station deployments by AT&T, T-Mobile and Iliad. We demonstrate that our proposed placement strategy achieves significant increases in mean data rate (about 2x) and edge rate ($2-8$x) compared to existing tower deployments, using the same number of transmitters.

We prove a transfer theorem for hereditary classes of $(r+1)$-uniform hypergraphs. Let $\mathcal H$ be such a class, and for $H\in\mathcal H$ write $Δ(H)$ and $d(H)$ for the maximum degree and average degree of $H$, respectively. We show that, for every nearly logarithmic function $f$ in the sense defined below, a maximum-degree lower bound for the independence number of the form \[ α(H)\ge (1-o(1))\frac{f(Δ(H))}{Δ(H)^{1/r}}|V(H)| \qquad\text{as }Δ(H)\to\infty \] for all $H\in\mathcal H$ implies the corresponding average-degree lower bound \[ α(H)\ge (1-o(1))\frac{f(d(H))}{d(H)^{1/r}}|V(H)| \qquad\text{as }d(H)\to\infty . \] We combine this transfer theorem with known coloring and fractional-coloring bounds to obtain consequences for graphs excluding a fixed cycle, graphs with bounded clique number, locally $q$-colorable graphs, and locally sparse uniform hypergraphs.

In this work, we extend the standard pre-inflationary misalignment mechanism for axion-like particles (ALPs) by introducing a pre-oscillatory phase with constant equation of state $ω_ϕ\in[-1,1]$, generated by a tracking potential. During the radiation-dominated era, the potential undergoes a rapid transition to the conventional cosine potential. The resulting change in the potential energy across the transition can drive the ALP into a kinetic misalignment phase ($ω_ϕ=1$) prior to the onset of oscillations. Motivated by persistent cosmological tensions, such as those in $H_0$ and $S_8$, we also investigate an ALP coupling to a dark radiation sector (DR), allowing for its decay. Using a Bayesian analysis, we constrain the ALP parameter space with current cosmological data. Our analysis shows that ALP-induced DR does not resolve the existing tensions. Instead, the data place robust constraints on the model, favoring negative values of $ω_ϕ$ and constraining the symmetry-breaking scale to $f_ϕ\in[80,1.5\times10^{10}]~\mathrm{TeV}$, corresponding to ALP masses in the range $m_ϕ\in[10^{-20},10^{-2}]~\mathrm{eV}$.

Euclidean random matrices arise in a wide range of physical systems where interactions are determined by spatial configurations, including disordered media and cooperative phenomena in atomic ensembles. Unlike classical random matrix ensembles, their entries are strongly correlated through the geometry of the underlying random points, making their analytical treatment challenging. While global spectral properties such as the spectral density are relatively well understood, much less is known about extremal eigenvalues and the associated eigenvectors, despite their central role in applications. Here we address the problem of characterising the largest eigenvalue and the corresponding top eigenvector of large Euclidean random matrices, illustrating the formalism on the case of quadratic distance kernel. For vectors in any dimension $d\geq 1$ drawn independently from a common distribution, we show that both quantities can be computed within a unified replica-based framework, leading to a set of $d+2$ self-consistent equations. This approach yields an explicit expression for the average largest eigenvalue, fully determined by low-order moments of the underlying distribution, and an analytical characterisation of the distribution of top eigenvector's components in the large-$N$ limit. We find that the top eigenvector exhibits a non-trivial geometric structure, with components concentrating on a hypersurface determined by the same parameters controlling the largest eigenvalue. We further perform extensive numerical simulations that confirm these predictions. More broadly, our work provides a general framework to access extremal spectral properties of Euclidean random matrices.

In the present work, we study the causal structure of spherically symmetric black holes immersed in a Chaplygin-like dark fluid, emphasizing the impact of the fluid parameters on curvature and horizon formation. We show that the spacetime curvature is significantly stronger than in its similar counterpart, the Reissner-Nordstrom-de Sitter geometry with the same mass and charge, leading to modifications of the internal causal structure. For the presence of horizons the Chaplygin black hole possesses an upper bound $Q \approx 0.556219 M$, which is much smaller than that for Reissner-Nordstrom spacetime $Q_{\text{critical}} = M$ or of the Reissner-Nordstrom-de Sitter case $Q_{\text{critical}} = 3M/(2\sqrt{2})$, indicating that the black holes immersed in a Chaplygin-like dark fluid reach the extremal regime more easily. We derive a second critical condition for the Chaplygin cosmological parameter $B$, $B_c Q_c^4 = 4/3^9$, setting an upper bound on $B$ for a multi-horizon solution.

Christoph, Draganić, Girão, Hurley, Michel, and Müyesser conjectured that, when $d\mid n$, the expected number of cycles in a uniformly random cycle-factor of a directed $d$-regular graph on $n$ vertices is uniquely maximised by the disjoint union of $n/d$ copies of the complete looped digraph $K_d^\circ$, with value $(n/d)H_d$ [FOCS 2025]. We disprove this conjecture in the strongest possible range. For every $d\ge 3$ and every multiple $n=kd$ with $k\ge 2$, we construct a directed $d$-regular graph on $n$ vertices whose uniformly random cycle-factor has expected cycle count strictly larger than $kH_d$. We also show that the conjectured extremal picture is correct in degree $d=2$, giving a sharp dichotomy between degree two and all higher degrees.

$β$ Cep stars in eclipsing binary (EB) systems give us an opportunity to put observational constraints on their structure and stellar parameters. We present a comprehensive analysis of the $β$ Cep star in the EB V446 Cep, using \textit{TESS} photometry and HERMES spectra. We calculate the stellar and orbital parameters using light curve modelling and spectral disentangling. The EB has an orbital period of $3.808567 \pm 0.000012$ d and a mass ratio of $0.1550 \pm 0.0012$. We find the $β$ Cep star to have a mass of $10.68 \pm 0.06$ $\mathrm{M_{\odot}}$, a radius of $5.864 \pm 0.033$ $\mathrm{R}_{\odot}$, and a $T_{\rm eff}$ of $24220 \pm 180$ K. The secondary has a mass of $1.657 \pm 0.017$ $\mathrm{M_{\odot}}$, a radius of $1.530 \pm 0.014$ $\mathrm{R}_{\odot}$, and a $T_{\rm eff}$ of $9080 \pm 390$ K. We also extract the abundances of C, N, O, Mg, and Si for the $β$ Cep star, which are found to be consistent with galactic OB binaries. We identified 21 distinct pulsation frequencies, with the dominant mode at 10.24324 d$^{-1}$, which corresponds to a near-harmonic of the system's orbital frequency. The two stars in the EB have asynchronous rotation, with both stars rotating faster than the orbital frequency. We detect a companion to the EB using eclipse timing variations and period changes of the dominant pulsation frequency. We calculate the minimum mass of this tertiary companion to be $4.11 \pm 0.32$ $\mathrm{M_{\odot}}$ which is on an orbit of 2303$\pm$69 d around the EB. Using spectral energy distributions and MIST isochrones, we conclude that V446 Cep is either a co-evolving hierarchical 2+2 quadruple or a triple system where the third body is a compact object.

HD~20794 is a nearby, bright, metal-poor G-type dwarf hosting a compact planetary system, including a super-Earth near the habitable zone. Its low stellar activity and the availability of precise radial-velocity and photometric data make it an excellent benchmark for studying stellar structure and chemical abundances in low-metallicity planet-hosting stars. We present, to our knowledge, the first grid-based stellar evolution analysis of HD~20794 using \texttt{MESA}, focusing on its main-sequence and late main-sequence evolution. A set of 252 stellar models was computed for initial masses between $0.78$ and $0.80\,M_{\odot}$, varying convective efficiency, numerical resolution, and atmospheric boundary conditions. Models were selected through $χ^2$ minimization using observed constraints on effective temperature, surface gravity, luminosity, radius, and age. The best-fit models favor a mass of $0.80\,M_{\odot}$ and an age of about $9$~Gyr, reproducing all observed stellar properties within uncertainties. They also successfully recover the observed surface abundance pattern over a wide range of elements, including light elements, $α$-elements, and the odd-$Z$ species phosphorus and chlorine. Comparison with nucleosynthesis yields from massive stars suggests that the measured phosphorus and chlorine abundances are compatible with enrichment from core-collapse supernovae and have remained preserved during stellar evolution. Our results support standard stellar evolution theory, indicating that low-mass, metal-poor G dwarfs such as HD~20794 can retain their natal chemical signatures over Gyr timescales. This highlights their importance as probes of stellar evolution, Galactic chemical enrichment, and the chemical environments associated with long-lived planetary systems.

Every homogeneous manifold of negative curvature is known to be isometric to a Lie group with a left invariant metric. We define an SNC-algebra to be a Lie algebra which admits an inner product of strictly negative curvature. In the author's joint paper in 2022, we classified SNC-algebras in dimension four. In this article, we classify SNC-algebras in dimension five, as well as we calculate Ricci curvature of SNC-algebras in dimension four.

We compute the dimension of the motivic Galois group of a 1-motive M defined over the field of complex numbers, expressing it explicitly in terms of the rank of the multiplicative group generated by the points defining M. As an application, we obtain a new formulation of the Grothendieck--André periods Conjecture in the setting of 1-motives.

Kinetic Inductance Detectors (KIDs) are a promising technology for future space missions, where exposure to high-energy particles may affect detector performance. In this work, we irradiated two types of KID arrays, absorber coupled and antenna coupled, with high-energy protons at 120 mK. We used a total dose equivalent to approximately 10 years of operation at the L2 Lagrange point. A comparison between pre-irradiation and post-irradiation measurements (24 hours after a 5.7 krad total dose) was done, while keeping the detectors at 120 mK. We find that there is no significant change in the quasi-particle lifetime τ_qp and the dark responsivity dθ/dPdark, but we do observe an increase in the noise and NEP that is tentatively attributed to limitations in the post radiation data analysis.

We study edge-colorings of the complete $p$-graph on $n$ vertices that contain no three edges $A,B,C$ of distinct colors such that the symmetric difference of $A$ and $B$ is contained in $C$. For $p\ge3$ and $n\ge p+1$, we show that every such coloring contains at most $1+\floor{n/p}$ colors and characterize the extremal colorings, generalizing a theorem of Erdős, Simonovits and Sós. %\cite{erdos1975}. When $p=3$, the condition $A\triangle B\subseteq C$ implies $|A\triangle B|=2$, and the three edges necessarily form a copy of $F_4\coloneqq\{abc,abd,bcd\}$ or $F_5\coloneqq\{abc,abd,cde\}$. For $n\ge5$, we show that every rainbow $F_5$-free edge-coloring is rainbow cancellative. For rainbow $F_4$-free colorings, we construct colorings with $m(n)+1$ colors for all $n\ge4$, where $m(n)$ is the size of a maximum partial Steiner triple system of order $n$ and satisfies $m(n)=n^2/6+O(n)$, improving the linear lower bound by Budden and Stiles. %\cite{budden}. Moreover, for $n=2^s-1$, we obtain $\ar(n,F_4)\ge m(n)+n^2/42+o(n^2)=4n^2/21+o(n^2)$ via a construction based on independent sets in the Grassmann graph. We also prove that $\ar(n,F_4)\le (5n^2-8n)/21$ for $n\ge4$, improving the quadratic coefficient in the upper bound of Budden and Stiles from $1/4$ to $5/21$.

This paper documents a collaborative research process involving peacebuilders and data scientists in Kenya and Sudan to develop AI-based text classifiers for monitoring online polarization and hatespeech. The method describes a participatory annotation process in which practitioners and domain experts contributed to problem definition, annotation design, iterative validation, and model evaluation. Fine-tuned BERT-based classifiers were trained on collaboratively annotated datasets and evaluated against held-out test sets. In each case, the models produced enhanced contextual alignment, reduced misclassification driven by cultural nuance, and increased practitioner ownership of AI tools. The resulting models (Kenya-polarization and Sudan-hate speech) are open-source and accessible via HuggingFace. The study contributes empirical evidence that participatory AI development can simultaneously improve technical robustness, contextual validity, and normative alignment in sensitive humanitarian domains.

MER2026 marks the fourth edition of the MER series of challenges. The MER series provides valuable data resources to the research community and offers tasks centered on recent research trends, establishing itself as one of the largest challenges in the field. Throughout its history, the focus of MER has shifted from discriminative emotion recognition to generative emotion understanding. Specifically, MER2023 concentrated on discriminative emotion recognition, restricting the emotion recognition scope to fixed basic labels. In MER2024 and MER2025, we transitioned to generative emotion understanding and introduced two new tasks: fine-grained emotion recognition and descriptive emotion analysis, aiming to leverage the extensive vocabulary and multimodal understanding capabilities of Multimodal Large Language Models (MLLMs) to facilitate fine-grained and explainable emotion recognition. Building on this trajectory, MER2026 continues to follow these research trends and contains four tracks: MER-Cross shifts the focus from individual to dyadic interaction scenarios; MER-FG centers on fine-grained emotion recognition; MER-Prefer aims to predict human preferences regarding different emotion descriptions; MER-PS focuses on emotion recognition based on physiological signals. More details regarding the dataset and baselines are available at https://zeroqiaoba.github.io/MER-Challenge.

The theory of bremsstrahlung $e \to eγ$ by extremely high energy electrons passing through ordinary matter has been qualitatively incomplete. We revisit the suppression of bremsstrahlung by the Landau-Pomeranchuk-Migdal (LPM) effect, here accounting for quantum disruption of that effect from pair production. Our analysis covers the full range of ultra-relativistic electron and photon energies (subject to a few simplifying approximations).

Weber's conjecture (1886) governs three aspects of lattice-based cryptography: the solvability of the Principal Ideal Problem, the freeness of modules over rings of integers, and the tightness of worst-case-to-average-case reductions in Ring-LWE (R-LWE) and Module-LWE (MLWE). Existing verifications for $k \ge 9$ rely on Generalized Riemann Hypothesis (GRH). In this paper, we present the first unconditional proof for $k \le 12$. Our method combines the Fukuda-Komatsu computational sieve, inductive structure of the cyclotomic $\mathbb{Z}_2$-tower, and Herbrand's theorem.

Non-invertible symmetries of a quantum field theory (QFT) are a natural generalization of unitary symmetries, but in which the product of operators does not satisfy a group multiplication law. We show that such symmetry operations on states define a collection of quantum gates for a parallel quantum computation scheme that includes post-selection / projection as a gate. Structures such as gate complexity and more geometric complexity measures generalize to this setting. We provide a class of distance / distinguishability measures that extend the standard notion of distance for Lie groups to both continuous and discrete non-invertible symmetries, as well as more general linear combinations of unitary quantum gates. We illustrate these considerations by computing the distance between non-invertible symmetries in some 4D and 2D QFTs. We find that the simple objects of a symmetry category can be highly complex computationally.

Although Uranus and Neptune are commonly classified as ice giants, their exact compositions remain poorly constrained. Recent studies of outer Solar System bodies challenge the traditional view that these planets are primarily ice-dominated, suggesting that refractory material plays a more significant role. Determining the proportions of ice and rock within Uranus and Neptune is essential for understanding their formation and the evolutionary history of the Solar System. In this work we computed interior structure models for both planets and explored, within a Bayesian framework, the range of compositions that satisfy the available observational constraints. We quantified the resulting ice and rock fractions and analyzed their impact on the inferred internal structure. Our results suggest that the envelopes of both Uranus and Neptune are systematically enriched in refractory material, with median rock fractions of approximately 60% within the heavy-element component, similar to Pluto, Kuiper belt objects, and comets. In contrast, the deep interiors of the two planets exhibit distinct compositions: Neptune is best fit by relatively rock-rich mantles (median rock fraction of ~ 55%), whereas Uranus is inferred to have more ice-rich mantles (median rock fraction of ~ 41%), consistent with a more strongly stratified structure. These results point to compositional differences between Uranus and Neptune that may reflect divergent formation and evolutionary pathways.

The ultra-rare decay $K^+\toπ^+ν\barν$ is a golden mode in flavor physics. The Standard Model prediction for its branching ratio is below $10^{-10}$. This decay mode is highly sensitive to new physics models at mass scales up to $\mathcal{O}(100\,\mathrm{TeV})$. The NA62 experiment at CERN SPS is designed to measure this decay mode. A preliminary result of the branching ratio measurement using data collected in 2023--2024 is presented. With the new dataset, the NA62 experiment doubled its signal sample while reducing the background in proportion. Combining the data collected in 2016--2024, the branching ratio is measured to be $\mathcal{B}(K^+\toπ^+ν\barν) = \left(9.6^{+1.9}_{-1.8}\right)\times10^{-11}$. The result is compatible with the Standard Model prediction with a precision better than $20\%$.

We derive an upper bound on the maximum balanced bipartite entanglement entropy of ground states of many-body Hamiltonians defined on a graph, agnostic to any particular model, that possesses a nontrivial automorphism group. We show that the entropy is bounded by the logarithm of a weighted sum of multiplicities of irreducible representations of the bipartition-preserving automorphism subgroup. This bound complements the known degeneracy-based bound, with neither universally dominating the other. For the complete graph $K_n$, the new bound yields an exponential improvement from linear to logarithmic scaling in the system size, consistent with the exact value of the entropy.

We study the volume conjecture of the colored Jones invariants with sequences of colors corresponding to the deformation of the hyperbolic structure of a link complement. In particular, we investigate certain limits of the colored Jones invariants of the figure-eight knot and the Borromean rings and show that the limits are related to the volumes of hyperbolic cone manifolds whose singular sets are the links.

Hadronic form factors fulfill dispersion relations and superconvergence sum rules for their spectral density as genuine imprints of QCD. We show several instances where these conditions are flagrantly violated due to the lack of information in the region above the largest known resonance mass and below the onset of perturbative QCD. We propose to use radial Regge trajectories to fill this gap and examine the consequences of such a minimal spectral hadronic ansatz. We illustrate the results with the pion charge form factor.

A new class of symmetries of two Higgs doublet models was recently discovered, the result of an unorthodox transformation on scalar and gauge fields and spacetime coordinates. It was explicitly shown that it is possible to choose Yukawa matrix textures which respect those symmetries up to two-loops. In this work we will establish the fermion field transformations for the two Higgs doublet models to be considered in the context of the new symmetries established. New regions of parameter space, invariant under renormalization to all orders of perturbation theory, are discovered, including scalar-fermion interactions.

In a recent work [Phys. Rev. E 109, L042102 (2024)], interesting dimensional crossovers [from two- to one-dimensional (2D to 1D) scaling] were found in the growth of Kardar-Parisi-Zhang (KPZ) interfaces on rectangular substrates, with lateral sizes $L_y > L_x$. Here, we extend this study to other universality classes for interface growth -- specifically, the Edwards-Wilkinson (EW), the Mullins-Herring (MH), and the Villain-Lai Das Sarma (VLDS) classes. From extensive simulations, we demonstrate that, in all systems with sufficiently large aspect ratio $\mathcal{R}=L_y/L_x$, the roughness $W$ scales with time $t$ in the growth regime as $W \sim t^{β_{\text{2D}}}$ for $t \ll t_c$ and $W \sim t^{β_{\text{1D}}}$ for $t \gg t_c$, where $t_c \sim L_x^{z_{2\text{D}}}$ in most cases. For the VLDS class, this crossover is also observed in the height distribution (HD), which approaches its characteristic probability density function for the 2D case at short times ($t \ll t_c$) and then crosses over to the asymptotic 1D HD. Dimensional crossovers are also found in the steady state regime, both in the roughness scaling as well as in the VLDS HD, which interpolate between the 2D and 1D ones as $\mathcal{R}$ increases. The particular case $L_x = L_y^δ$, with $0 < δ< 1$, is also discussed in detail and reveals interesting features of the investigated systems. For instance, there exist a `special' exponent $δ^* = z_{1\text{D}}/z_{2\text{D}}$ such that the temporal crossover cannot be observed for $δ> δ^*$. Moreover, this leads the saturation roughness to display a nonuniversal scaling: $W_s \sim L_y^Λ$, with $Λ= (1-δ) α_{1\text{D}} + δα_{2\text{D}}$.

We study the task of lifting arbitrary quantum states and channels to purifications and Stinespring dilations, respectively, in both the probabilistic exact and deterministic approximate settings. We formalize this task through a general framework of quantum purification machines that, given a finite number of copies or uses of a black-box input, aim to output a corresponding purification or Stinespring dilation. In the probabilistic exact setting, we show that universality is not necessary to rule out such transformations: the simple requirement that a machine purifies two inputs of different rank with non-zero probability already implies that it cannot be described by a linear positive map. This simple argument captures a fundamental obstruction of quantum theory and recovers the impossibility of universal probabilistic purification from finitely many copies. In the approximate setting, we allow for general machines that are not required, in general, to produce a pure output. Using the minimum average error as our figure of merit, we derive analytical expressions for the performance of several physically motivated strategies as well as a general upper bound on the achievable error, which is tight in a specific regime. Our analysis reveals a trade-off: strategies that produce a pure output - among which we prove the optimal to be a strategy that produces as a fixed output a maximally entangled purification of the fully depolarizing channel - perform optimally between those considered for large environment dimension, while append-environment strategies that generally produce non-pure outputs perform better at small environment dimension.

Let $p$ be a prime, $k$ an algebraic closure of $\mathbb{F}_p$ and $Γ$ the Galois group ${\rm Gal}(k/\mathbb{F}_p)$. Let $\mathcal{C}$ be a finite category and $\mathcal{O}_{\mathcal{C}}$ the $p$-orbit category of $\mathcal{C}$ defined by Linckelmann. We formulate a version of the Galois Alperin weight conjecture (GAWC) for finite category algebras stating that there exists a $Γ\times {\rm Aut}(\mathcal{C})$-equivariant bijection between the set of isomorphism classes of simple $k\mathcal{C}$-modules and that of the weights of $k\mathcal{O}_{\mathcal{C}}$. We reduce the GAWC for finite categories to finite groups. For $\mathcal{C}$ an EI-category, we give a partition of weights of $k\mathcal{O}_{\mathcal{C}}$ with respect to blocks of $k\mathcal{C}$ and then formulate a blockwise Galois Alperin weight conjecture (BGAWC) for $\mathcal{C}$. Similarly, we reduce the BGAWC for finite EI-categories to finite groups.

Localized one-particle states of a quantum field theory--whether in flat space or on a curved background--are expected to exhibit geodesic motion in an appropriate semiclassical regime. This expectation is often invoked heuristically: in this work we develop two precise implementations and test them in detail in global AdS$_3$. First, we define a covariant ''center-of-mass'' trajectory from the expectation value of the stress tensor operator and show, using only $\nabla_μ\langle T^{μν}\rangle=0$, that it obeys the geodesic equation in the monopole (sufficiently localized) approximation in a general spacetime. This provides a QFT-in-curved-spacetime generalization of the Mathisson-Papapetrou-Dixon framework in classical general relativity. Second, we construct position operators from the Klein--Gordon inner product and mode completeness, and compute their expectation values in generic single-particle wave packet states. We then build explicit normalizable wave packets of a free scalar field in empty AdS$_3$ with tunable energy and angular momentum, and demonstrate analytically and numerically that both prescriptions reproduce the expected radial, circular, and elliptical-like timelike and null geodesics. Our discussion also isolates a natural ultra-relativistic regime in which the wave packet trajectory exhibits a controlled crossover from timelike to null geodesic behavior. We identify precise limits where the localized geodesic interpretation of the wave packet breaks down. On the CFT side, we show that bulk localization--specifically the radial data--is captured by how the state is distributed over global descendants of the dual primary.