We prove that modules over an Artinian Gorenstein local ring $R$ have rational Poincaré series sharing a common denominator if $R/\soc(R)$ is a Golod ring. If $R$ is a Gorenstein local ring with square of the maximal ideal being generated by at most two elements, we show that modules over $R$ have rational Poincaré series sharing a common denominator. By a result of \c Sega, it follows that $R$ satisfies the Auslander-Reiten conjecture. We provide a different proof of a result of Rossi and \c Sega concerning rationality of Poincaré series of modules over compressed Gorenstein local rings. We also give a new proof of the fact that modules over Gorenstein local rings of codepth at most three have rational Poincaré series sharing a common denominator, which is originally due to Avramov, Kustin and Miller.

Radiative near-field (RNF) beamforming is vulnerable to blockages that disrupt Fresnel zones. This letter proposes an obstruction-unaware wavefront shaping strategy inspired by catastrophe optics. By superimposing a calibrated quartic phase, we generate a Pearcey-like wave packet that exhibits structural stability against perturbations. We establish a fair comparison protocol where the quartic beam is calibrated in free space to avoid exploiting obstruction knowledge. Numerical results demonstrate up to 8.5~dB SINR gain over conventional focusing for multi-user scenarios near the depth-of-focus limit. Crucially, this gain stems from improved channel conditioning under partial blockage, which mitigates the severe noise amplification inherent to zero-forcing precoding.

We conduct a topological-numerical analysis of global dynamics in a discrete-time two-gene Andrecut-Kauffman model. This model describes gene expression regulation through nonlinear interactions. We use rigorous numerical methods to construct Morse decomposition of the system across a wide range of parameters. We obtain qualitative results by effectively computing the Conley indices of the constructed isolating neighborhoods that form the Morse decomposition. We introduce new symbols to convey the information provided by the Conley index in an easy to understand schematic way. We additionally conduct numerical simulations aimed at confirming the presence of complex dynamical phenomena, including multistability and the existence of chaotic attractors. The results demonstrate the usefulness of topological methods in understanding the global structure of dynamics in a gene regulatory model and highlight the richness of dynamics that can be observed in such a system when parameter values change.

The $hp$ local discontinuous Galerkin (LDG) method proposed by Castillo et al. [Math. Comp.,~71 (238): 455-478, 2002] has been shown to be an efficient approach for solving convection-diffusion equations. However, theoretical analysis indicates that, for solutions with limited spatial regularity, the method exhibits suboptimal convergence in $p$, suffering a loss of one order, comparing to numerical experiments. The purpose of this paper is to close the gap between theoretical estimates and numerical evidence. This is accomplished by establishing new approximation results for the associated Gauss-Radau projections of functions in suitable function spaces that can optimally characterize the regularity of singular solutions. We show that such a framework arises aturally and enables the study of various types of singular solutions, with full consistency between theoretical analysis and numerical results. This investigation sheds light on the resolution of the suboptimality in $p$ observed in the literature for several other types of DG schemes in different settings.

We present a computational framework to investigate steady state distributions and perform stability analysis for random ordinary differential equations driven by parameter uncertainty. Using the nonlinear Rosenzweig McArthur predator prey model as a case study, we characterize the non-trivial equilibrium steady state of the system and investigate its complex distribution when the parameter probability densities are multi-modal mixture models with partially overlapping or separated components. In consequence, this application includes both, uncertainties and superpositions, of the system parameters. In addition, we present the stability analysis of steady states based on the eigenvalue distribution of the system's Jacobian matrix in this stochastic regime. The steady state posterior density and stability metrics are computed with a recently published Monte Carlo based numerical scheme specifically designed for random equation systems (Hoegele, 2026). Particularly, the simplicity of this stochastic extension of dynamic systems combined with a broadly applicable computational approach is demonstrated. Numerical experiments show the emergence of multi-modal steady state distributions of the predator prey model and we calculate their stability regions, illustrating the method's applicability to uncertainty quantification in dynamical systems.

The Pauli exclusion principle imposes an important structural constraint in cluster descriptions of light nuclei and is commonly taken into account using methods such as the Resonating Group Method (RGM), the Orthogonality Condition Model (OCM), and the Orthogonalizing Pseudopotential (OPP) approach. The latter provides a practical implementation for suppressing Pauli-forbidden states in few-body calculations through the introduction of a large auxiliary coupling constant $λ_0$ and an associated limiting procedure. Exact $λ_0$-eliminated formulations have appeared in the literature. It is shown that the OPP method may be interpreted as the singular $λ_0\to\infty$ limit of the Feshbach-Schur projection, and that the Schur complement provides a natural operator-level framework for understanding this connection. In contrast to earlier approaches, the elimination is derived explicitly as a closed Schur-complement operator identity. When formulated in terms of operators and Green's functions, the Feshbach-Schur projection eliminates Pauli-forbidden components algebraically and avoids the explicit introduction of large pseudopotential parameters. This reformulation clarifies the implementation of Pauli projection in cluster models and provides a convenient framework for few-body calculations.

Error-correcting codes are a method for representing data, so that one can recover the original information even if some parts of it were corrupted. The basic idea, which dates back to the revolutionary work of Shannon and Hamming about a century ago, is to encode the data into a redundant form, so that the original information can be decoded from the redundant encoding even in the presence of some noise or corruption. One prominent family of error-correcting codes are Reed-Solomon Codes which encode the data using evaluations of low-degree polynomials. Nearly six decades after they were introduced, Reed-Solomon Codes, as well as some related families of polynomial-based codes, continue to be widely studied, both from a theoretical perspective and from the point of view of applications. Besides their obvious use in communication, error-correcting codes such as Reed-Solomon Codes are also useful for various applications in theoretical computer science. These applications often require the ability to cope with many errors, much more than what is possible information-theoretically. List-decodable codes are a special class of error-correcting codes that enable correction from more errors than is traditionally possible by allowing a small list of candidate decodings. These codes have turned out to be extremely useful in various applications across theoretical computer science and coding theory. In recent years, there have been significant advances in list decoding of Reed-Solomon Codes and related families of polynomial-based codes. This includes efficient list decoding of such codes up to the information-theoretic capacity, with optimal list-size, and using fast nearly-linear time, and even sublinear-time, algorithms. In this book, we survey these developments.

Femtosecond pulsed laser systems constitute powerful tools for the high-precision structuring of materials at micro/nano-scale resolutions. A critical parameter influencing the efficacy of ultrafast laser-material interactions is the laser-induced damage threshold (LIDT), which is defined as the minimum laser fluence required to induce irreversible modification to the material surface. While extensive studies have addressed single-pulse damage mechanisms, the response of thin metallic films to double-pulse femtosecond irradiation, particularly when the film thickness is of the order of the optical penetration depth, remains, generally, unexplored. In this work, we present a rigorous theoretical investigation into the spatiotemporal evolution of energy deposition, thermalization processes and optical parameter changes under double-pulse excitation conditions. The analysis considers key parameters including the inter-pulse delay and the film thickness to evaluate their influence on the LIDT for a range of technologically relevant metals: Au, Ag, Cu, Al, Ni, Ti, Cr, Pt, W, Mo and Stainless Steel (100Cr6). ). A comparative analysis highlights the potential of controlled double-pulse irradiation schemes to manipulate energy coupling efficiency, improve the spatial selectivity of laser-induced modifications and compile a comprehensive LIDT database for commonly used industrial materials. The approach is aimed to provide a robust foundation for the design and optimization of advanced laser micromachining and nanofabrication protocols across a broad spectrum of metallic systems.

Global climate projections rely on computationally demanding Earth System Models (ESMs), which are typically limited to coarse spatial resolutions due to their high cost. To obtain high-resolution projections for regions of interest, it is common to use Regional Climate Models (RCMs), which are driven by data produced by ESMs as boundary conditions. While more efficient than running ESMs at fine resolution, RCMs remain expensive and restrict the size of ensemble simulations. Inspired by recent advances in probabilistic machine learning for weather and climate, we introduce a data-driven climate downscaling method based on stochastic interpolants. Our approach efficiently transforms coarse ESM output into high-resolution regional climate projections at a fraction of the computational cost of traditional RCMs. Through extensive validation, we demonstrate that our method generates accurate regional ensembles, enabling both improved uncertainty quantification and broader use of high-resolution climate information.

Planar metasurfaces can profoundly control electromagnetic scattering. At microwave frequencies, such devices are typically implemented using multilayer cascades of patterned metallic sheets, whose design often requires time-consuming full-wave optimization. Here, we extend analytical models originally developed for sparse loaded-wire metagratings to accurately describe densely packed Jerusalem-cross meta-atoms embedded in standard printed circuit board (PCB) dielectric stacks. The model captures both near- and far-field coupling within and between layers, enabling efficient prediction of the dual-polarized response. Using this framework, we identify highly transmissive meta-atoms whose phase is controlled by the leg lengths of the Jerusalem crosses (microscopic design stage). This (phase)-(leg-length) "lookup table" allows rapid synthesis of Huygens' metasurfaces (macroscopic design stage), demonstrated through a full-wave-validated metalens exhibiting low-reflection beam manipulation. Notably, we implement a judicious scaling method to further extend the model to predict wideband meta-atom responses. In the companion paper (Part II), a hybrid machine-learning approach leverages this semianalytical framework to enhance accuracy without requiring the conventional exhaustive full-wave training, enabling ultrafast inverse design across the full parameter space. Overall, the presented methodology -- the standalone semianlytical scheme (Part I) and the machine-learning enhanced version (Part II) -- establishes an effective open-source toolkit for versatile, rapid, and highly accurate synthesis of fabrication-ready dual-polarized transmissive Huygens' meta-atoms and metasurfaces.

For twenty years, a persistent discrepancy between experimental measurements and theoretical calculations of the muon anomalous magnetic moment have provided tantalising hints of new physics. In recent years, improvements to the experimental precision have appeared to make the tension stronger and stronger. However, at the same time, our lattice calculation overturned the theoretical consensus, completely eliminating the tension. I will present the latest results from the Budapest-Marseille-Wuppertal (BMW) and DMZ collaborations, with a hybrid determination of the hadronic vacuum polarisation contribution to a precision of 0.45%

We analyse the dynamics of a qubit coupled to a dissipative impurity by comparing local and global derivation schemes of a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation within the Born-Markov and full secular (FS) approximations. We show that the local approach correctly captures a crossover in the dynamics of the qubit coherence, while the FS approximation restricts the validity of the global approach to regimes with well-separated energy scales. Our results clarify the domains of validity of the two approaches and show that the local scheme provides a better GKSL description of the qubit dynamics in the experimentally relevant parameter regime.

The stability of non-isolated equilibria to quasilinear parabolic problems of the form $u' = A(u)u + f(u)$ is established in interpolation spaces (and thus extending previous results relying on maximal regularity). The approach allows full flexibility in choosing the interpolation methods and requires only low regularity assumptions on the semilinear part $f$. Applications to concrete problems are presented, including the capillarity-driven Hele--Shaw problem and the fractional mean curvature flow.

We calculate the genus zero cobordism-valued Gromov-Witten invariants of a point by refining the string equation on $\overline{\mathcal{M}}_{0,n}$ from the Chow ring to algebraic cobordism. This gives inductive formulas for cobordism-valued psi-class intersections on $\overline{\mathcal{M}}_{0,n}$, and in particular the cobordism classes $[\overline{\mathcal{M}}_{0,n}]$, and for their images in $K$-theory. Explicit formulas are given up to $n = 8$.

The philosophical foundations of statistics involve issues in theoretical statistics, such as goals and methods to meet these goals, and interpretation of the meaning of inference using statistics. They are related to the philosophy of science and to the philosophy of probability. We review the core and partly interrelated themes and place them in context.

Multimodal intent recognition aims to infer human intents by jointly modeling various modalities, playing a pivotal role in real-world dialogue systems. However, current methods struggle to model hierarchical semantics underlying complex intents and lack the capacity for self-evolving reasoning over multimodal representations. To address these issues, we propose HIER, a novel method that integrates HIerarchical semantic representation with Evolutionary Reasoning based on Multimodal Large Language Model (MLLM). Inspired by human cognition, HIER introduces a structured reasoning paradigm that organizes multimodal semantics into three progressively abstracted levels. It starts with modality-specific tokens capturing localized semantic cues, which are then clustered via a label-guided strategy to form mid-level semantic concepts. To capture higher-order structure, inter-concept relations are selected using JS divergence scores to highlight salient dependencies across concepts. These hierarchical representations are then injected into MLLM via CoT-driven prompting, enabling step-wise reasoning. Besides, HIER utilizes a self-evolution mechanism that refines semantic representations through MLLM feedback, allowing dynamic adaptation during inference. Experiments on three challenging benchmarks show that HIER consistently outperforms state-of-the-art methods and MLLMs with 1-3% gains across all metrics. Code and more results are available at https://github.com/thuiar/HIER.

We ask whether the Hamiltonian, interaction geometry, and symmetries of a quantum many-body system can be inferred from a few low-lying eigenstates without knowing which sites interact with each other. Directly solving the eigenvalue equations imposes constraints that yield a highly degenerate subspace of candidate operators, where the local Hamiltonian is hidden among an extensive family of conserved quantities, obscuring the interaction geometry. Here we introduce O-Sensing, a protocol designed to extract the Hamiltonian and symmetries directly from these states. Specifically, O-Sensing employs parsimony-driven optimization to extract a maximally sparse operator basis from the degenerate subspace. The Hamiltonian is then selected from this basis by maximizing spectral entropy (effectively minimizing degeneracy) within the sampled subspace. We validate O-Sensing on Heisenberg models on connected Erdős--Rényi graphs, where it reconstructs the interaction geometry and uncovers additional long-range conserved operators. We establish a learnability phase diagram across graph densities, featuring a pronounced ``confusion'' regime where parsimony favors a dual description on the complement graph. These results show that sparsity optimization can reconstruct interaction geometry as an emergent output, enabling simultaneous recovery of the Hamiltonian and its symmetries from low-energy eigenstates.

We introduce a class of algebras over a field $\mathbb{F}$ related to directed graphs in which all edges are labeled by nonzero elements of the field $\mathbb{F}$. If all labels are different from $1$, these algebras are axial algebras. We determine their fusion laws, prove them to be simple in almost all cases, and determine their automorphism group under some conditions on the degrees and girth of the graph. A construction of a class of these graphs with prescribed automorphism group enables us to construct for each group $G$ infinitely many simple (axial) algebras (with a fixed fusion law) such that the automorphism group of the algebra is isomorphic to $G$.

This study investigates the influence of a perturbed eddy-viscosity model on linear stability and shape sensitivity of the global vortex rope mode arising in a hydro-turbine flow under fully turbulent conditions. The framework is based on the Reynolds-averaged Navier--Stokes equations with a standard $k$-$\varepsilon$ turbulence closure, linearized around a base-flow state. This base state is tuned to match the vortex-rope bifurcation predicted from three-dimensional unsteady simulations. The shape sensitivity of the global mode is derived, accounting for perturbations of both the base flow and the linear operator. We show that although the perturbed eddy-viscosity model has only a marginal effect on the eigenvalues and eigenmodes of interest, it substantially alters the resulting shape sensitivities. These differences arise primarily through the base-flow contribution to the total sensitivity, which dominates the sensitivity to shape deformations. Although both models identify coherent-velocity production and advection as the leading contributors, the linearized model captures additional mechanisms associated with eddy-viscosity perturbations. Comparison with experiments shows that only the perturbed eddy-viscosity model reproduces the correct trends in shape sensitivity, whereas the frozen model fails to do so. These findings highlight the importance of consistently linearizing turbulence models for sensitivity-based control of turbulent global instabilities.

Regression discontinuity designs (RDD) are widely used for causal inference. In many empirical applications, treatment effects vary substantially with covariates, and ignoring such heterogeneity can lead to misleading conclusions, which motivates flexible modeling of heterogeneous treatment effects in RDD. To this end, we propose a Bayesian nonparametric approach to estimating heterogeneous treatment effects based on Bayesian Additive Regression Trees (BART). The key feature of our method lies in adopting a general Bayesian framework using a pseudo-model defined through a loss function for fitting local linear models around the cutoff, which gives direct modeling of heterogeneous treatment effects by BART. Optimal selection of the bandwidth parameter for the local model is implemented using the Hyvärinen score. Through numerical experiments, we demonstrate that the proposed approach flexibly captures complicated structures of heterogeneous treatment effects as a function of covariates.

We introduce linearly decomposable (LD) generalized pairs, which serve as a workable substitute for rational decompositions in the non-NQC setting. Using LD generalized pairs, together with a refinement of special termination and Kollár-type gluing theory, we prove the existence of flips for log canonical generalized pairs without assuming the klt condition, the NQC condition, or $\mathbb Q$-factoriality. Together with the cone and contraction theorems, this yields the existence of the minimal model program for arbitrary log canonical generalized pairs.

This paper is motivated by a cutting-edge application in neuroscience: the analysis of electroencephalogram (EEG) signals recorded under flash stimulation. Under commonly used signal-processing assumptions, only the phase angle of the EEG is required for the analysis of such applications. We demonstrate that these assumptions imply that the phase has a projected isotropic normal distribution. We revisit this distribution and derive several new properties, including closed-form expressions for its trigonometric moments. We then examine the distribution of the mean resultant and its square -- a statistic of central importance in phase-based EEG studies. The distribution of the resultant is analytically intricate; to make it practically useful, we develop two approximations based on the well-known resultant distribution for the von Mises distribution. We then study inference problems for this projected isotropic normal distribution. The method is illustrated with an application to EEG data from flash-stimulation experiments.

The Collins-Soper (CS) kernel may be obtained through the TMD soft function by formulating the Wilson line in terms of 1-dimensional auxiliary fermion fields on the lattice. Our computation takes place in the region of the lattice that corresponds to the "spacelike" region in Minkowski space, i.e., Collins' scheme. We explore two methods for obtaining the CS kernel. The "ratio method"; which would allow us to obtain the soft function as well as the CS kernel. And the "double ratio"; which allows us to achieve a high degree of statistical precision, but only produces the CS kernel. The matching of our result to Minkowski space is achieved through the mapping of the complex auxiliary field directional vector to the Wilson line rapidity. We present a preliminary extraction of the CS kernel using the "double ratio", and discuss the methodology employed.

When colloidal particles are vertically confined to a gap of between 1.3-1.6 particle diameters, they pack into buckled crystals of particles in either "up" or "down" states. Neighboring particles tend to occupy opposite states, analogous to the behavior of antiferromagnetic spins. The particles sit on a nearly-triangular lattice, and the spins of trios of adjacent particles are geometrically frustrated. Two levels of translational order exist in this system: that of the underlying triangular lattice in the horizontal plane, and that of the emergent frustrated spin lattice in the vertical dimension. We study the topological defects of both levels of translational order, and we find that both types of defects play a role in crystal grain boundary structure and spin domain coarsening. We classify the spin defects and outline the basic rules for their motion, and we observe interactions between dislocations and spin defects. Finally, we map the phase space of spin coarsening in the buckled monolayer, characterizing which types of defects drive the dynamics. Understanding defect formation, motion, and interaction in the buckled monolayer is the first step in predicting the material properties and aging of this geometrically frustrated, self-assembled system.

Label-free vibrational microscopy provides chemically specific access to cellular structure, yet quantitative volumetric chemical dynamics in living cells remain largely inaccessible, particularly on subsecond timescales relevant to intracellular transport and structural reorganization. This limitation arises because most high-speed vibrational techniques rely on raster scanning, which constrains volumetric throughput to approximately one volume per second (vps). Although mid-infrared photothermal (MIP) imaging offers a pathway toward spatially parallel chemical detection, existing implementations have remained far below video-rate volumetric operation, reflecting a fundamental trade-off between imaging speed and signal-to-noise ratio. Here, we overcome this trade-off in MIP tomography and realize video-rate volumetric chemical imaging using mid-infrared photothermal optical diffraction tomography (MIP-ODT), achieving high photothermal sensitivity while maintaining quantitative measurement fidelity. High per-angle detectability supports volumetric reconstruction without temporal averaging, yielding a signal-to-noise ratio exceeding 70 under video-rate acquisition conditions. Consequently, volumetric imaging at 19.2 vps is achieved, representing a nearly 400-fold improvement over prior implementations. Using this capability, we performed video-rate three-dimensional tracking of lipid droplets in living cells and quantified anomalous diffusion from full volumetric trajectories, revealing heterogeneous intracellular transport behaviors that are obscured in two-dimensional measurements. We further demonstrate high-speed hyperspectral volumetric chemical imaging across a 300 cm-1 spectral window within 1 s through rapid MIR wavenumber sweeping, paving the way for real-time three-dimensional organelle-specific chemical phenotyping.

A transmit pinching-antenna system (T-PASS) framework is proposed, in which a single pinched waveguide is employed to jointly serve one wired user equipment (UE) and multiple wireless UEs. The signal radiated by the pinching antennas (PAs) is used to serve the wireless UEs, whereas the residual guided signal at the waveguide termination is used to serve the wired UE. To facilitate T-PASS transmission and mitigate inter-user interference, a hybrid non-orthogonal multiple access (NOMA) scheme is introduced. Wireless UEs are scheduled by time-division multiple access (TDMA), and, in each slot, the scheduled wireless UE is paired with the wired UE through power-domain NOMA. Within this framework, the PA positions, PA radiation coefficients, power allocation, and TDMA time-slot allocation are jointly optimized to maximize a weighted sum rate (WSR). i) For the two-user case with one wired UE and one wireless UE, the optimal PA position and successive interference cancellation (SIC) decoding order are derived. Closed-form optimal power allocation is obtained, and a near-optimal PA radiation coefficient is determined through a low-complexity one-dimensional search. ii) For the multiuser case with one wired UE and multiple wireless UEs, four protocols with different PA-position and PA-radiation configurations are proposed. For each protocol, a low-complexity element-wise alternating optimization algorithm is developed to optimize the PA positions and radiation coefficients, while closed-form solutions are derived for the optimal power allocation and time-slot allocation. Numerical results are presented to show that: i) under typical T-PASS configurations, the wired UE is selected as the strong user in the optimal SIC decoding order; ii) the proposed T-PASS framework achieves a significantly higher WSR than conventional wireless-only PASS.

We implement a variational quantum algorithm for Gibbs state preparation of a transverse-field Ising model on IonQ's quantum computers. To this end, we train the variational parameters via classical simulation and perform state tomography on the quantum devices to evaluate the fidelity of the prepared Gibbs state. As a main result, we find that fidelity decreases (non-monotonically) as a function of the inverse temperature $β$ of the system. Fidelity also decreases as a function of the size of the system. Interestingly, we find that a Gibbs state prepared for a specified $β$ is a better representative of a Gibbs state prepared for a $\textit{lower}$ $β$; or in other words, thermal fluctuations in the quantum hardware lead to digital heating, that is, an increase in the temperature of the prepared Gibbs state above what was intended.

Computing the vacuum and energy spectrum in non-Abelian, interacting lattice gauge theories remains an open challenge, in part because approximating the continuum limit requires large lattices and huge Hilbert spaces. To address this difficulty with near-term quantum computing devices, we adapt the variational quantum eigensolver to non-Abelian gauge theories. We outline scaling advantages when using a spin-network basis to simulate the gauge-invariant Hilbert space and develop a systematic state preparation ansatz that creates gauge-invariant excitations while alleviating the barren plateau problem. We illustrate our method in the context of SU(2) Yang-Mills theory by testing it on a minimal toy model consisting of a single vertex in 3+1 dimensions. In this toy model, simulations allow us to investigate the impact of noise expected in current quantum devices.

We report a phase-sensitive nonlinear x-ray response in the charge-density-wave material 1T-TaS2, revealed through x-ray parametric down-conversion into the ultraviolet. Extending nonlinear x-ray wave mixing beyond conventional crystalline systems to a correlated quantum material, we employ reciprocal-lattice phase matching to isolate distinct Fourier components of the nonlinear susceptibility. By selecting a fundamental reciprocal-lattice vector and a stacking-sensitive half-integer reciprocal-lattice vector, we probe the response across the nearly commensurate and incommensurate charge-density-wave phases. Tuning the ultraviolet photon energy through Ta O-shell resonances uncovers pronounced Fourier-component and phase-dependent resonant structure, indicating that the stacking-related nonlinear susceptibility couples differently to Ta-centered resonant states than the average lattice response. Remarkably, the nonlinear signal is strongly enhanced in the nearly commensurate phase despite weaker Bragg diffraction, demonstrating that the nonlinear susceptibility provides information inaccessible to linear probes. These results establish nonlinear x-ray spectroscopy as a phase-sensitive and orbital-selective probe of electronic reconstruction in quantum materials.

We study the Equi-Baire one property families of Möbius transformations on the Riemann sphere. For a loxodromic map $f$, we show its iterates $\{f^n\}$ form an orbitally Equi-Baire one family on the attracting basin. For a one-parameter subgroup $\{f_t \}$, we prove it is Equi-Baire one on all compact sets of $\widehat{\mathbb{C}}$ if and only if the subgroup is relatively compact in $\mathrm{SL}(2,\mathbb{C})$. This provides a dynamical characterization of the Equi-Baire one condition for Möbius families.