We demonstrate that the chiral $\mathbb{Z}_p$ toric code -- the quintessential model of topological order -- hosts additional, emergent topological phases when perturbed: descendant fractional quantum Hall-like states, which we term \textit{Hall-on-Toric}. These hierarchical states feature fractionalized $\mathbb{Z}_p$ charges and increased topological ground-state degeneracy. The Hall-on-Toric phases appear in the vicinity of the transitions between deconfined $\mathbb{Z}_p$ phases with different background charge per unit cell, in a fixed non-trivial flux background. We confirm their existence through extensive infinite density matrix renormalization group (iDMRG) simulations, analyzing the topological entanglement entropy, entanglement spectra, and a generalized Hall conductance. Remarkably, the Hall-on-Toric states remain robust even in the absence of $U(1)$ symmetry. Our findings reinforce the foundational interpretation of star and plaquette defects as magnetic and electric excitations, and reveal that this perspective extends to a much deeper level.

Magnetism has witnessed remarkable progress in recent decades, largely driven by its potential for next-generation storage devices. However, the classification of magnetic orders, even for fundamental concepts such as ferromagnetism and antiferromagnetism, remains a topic of active evolution, particularly with the discovery of unconventional magnetic materials and advances in antiferromagnetic spintronics. Here, we present a unified classification of magnetic order utilizing the state-of-the-art spin space group (SSG) theory. Based on whether the net spin magnetization is constrained to zero by SSG, we systematically categorize magnetic orders into ferromagnetism (including ferrimagnetism) and antiferromagnetism. We further introduce an oriented SSG description, i.e., an SSG with a fixed magnetic orientation, thereby unifying the SSG and magnetic space group frameworks. This approach clearly reveals the symmetry-breaking pathway induced by spin-orbit coupling. The proposed group framework completes the intrinsic logic of magnetic symmetry and identifies a distinct magnetic phase, termed spin-orbit magnetism, in which the net spin magnetization is induced by spin-orbit coupling. Our work provides a comprehensive symmetry-based perspective for classifying magnetic order, offering fresh insights into unconventional magnets and broad applicability in spintronics and quantum material design.

Free energy calculations are at the heart of physics-based analyses of biochemical processes. They allow us to quantify molecular recognition mechanisms, which determine a wide range of biological phenomena from how cells send and receive signals to how pharmaceutical compounds can be used to treat diseases. Quantitative and predictive free energy calculations require computational models that accurately capture both the varied and intricate electronic interactions between molecules as well as the entropic contributions from motions of these molecules and their aqueous environment. However, accurate quantum-mechanical energies and forces can only be obtained for small atomistic models, not for large biomacromolecules. Here, we demonstrate how to consistently link accurate quantum-mechanical data obtained for substructures to the overall potential energy of biomolecular complexes by machine learning in an integrated algorithm. We do so using a two-fold quantum embedding strategy where the innermost quantum cores are treated at a very high level of accuracy. We demonstrate the viability of this approach for the molecular recognition of a ruthenium-based anticancer drug by its protein target, applying traditional quantum chemical methods. As such methods scale unfavorable with system size, we analyze requirements for quantum computers to provide highly accurate energies that impact the resulting free energies. Once the requirements are met, our computational pipeline FreeQuantum is able to make efficient use of the quantum computed energies, thereby enabling quantum computing enhanced modeling of biochemical processes. This approach combines the exponential speedups of quantum computers for simulating interacting electrons with modern classical simulation techniques that incorporate machine learning to model large molecules.

This paper develops a probabilistic approximation scheme for a class of nonstandard, fully nonlinear second-order partial integro-differential equations (PIDEs) associated with nonlinear Levy processes under Peng's G-expectation framework. The PIDE features a supremum over a family of alpha-stable Levy measures, possibly degenerate diffusion coefficients, and a non-separable uncertainty set, which places it outside the scope of existing numerical theories for PIDEs. We construct a recursive, piecewise-constant approximation of the viscosity solution and establish explicit error estimates for the scheme. As a key application, our results yield quantitative convergence rates for the universal robust limit theorem under sublinear expectations. This provides a unified treatment of Peng's robust central limit theorem and law of large numbers, as well as the alpha-stable limit theorem of Bayraktar and Munk, together with explicit Berry-Esseen-type bounds.

We introduce and study a new topological notion of the size for subsets of the real line, called \emph{super-density}. A set $A\subset\mathbb{R}$ is super-dense if for every non-empty open interval $I$ and every nowhere constant continuous function $φ\colon I\to\mathbb{R}$, we have $φ(I\cap A)\cap A\neq\emptyset$. We first establish basic properties of super-dense sets. Our main topological result characterizes them within the framework of Baire category: a set with the Baire property is super-dense if and only if it is co-meager. We then investigate the implications for the theory of normal numbers. We prove that the set of non-normal numbers is super-dense, whereas the set of normal numbers is not. Consequently, no nowhere constant continuous function can map all non-normal numbers to normal numbers. Conversely, we explicitly construct a computable nowhere constant continuous function that maps all normal numbers to non-normal numbers. Finally, we provide a constructive algorithm that, given any countable family of nowhere constant continuous functions, produces a real number $x$ such that $x$ and all its images under these functions are non-normal. As a corollary, we obtain the existence of a non-normal number $x$ such that $e^{αx}$ is non-normal for every non-zero algebraic $α$.

With the rapid development of distributed optimization (DO) theory, the distributed stochastic gradient methods (DSGMs) occupy an important position. Although the theory of different DSGMs has been widely established, the main-stream results of existing work are still derived under the condition of light-tailed stochastic gradient noises. Increasing examples from various fields, indicate that, the light-tailed noise model is overly idealized in many practical instances, failing to capture the complexity and variability of noises in real-world scenarios, such as the presence of outliers or extreme values from data science and statistical learning. To address this issue, we propose a new DO framework that incorporates stochastic gradients under sub-Weibull randomness. We study a distributed composite stochastic mirror descent scheme with sub-Weibull gradient noise (DCSMD-SW) for solving a convex distributed composite optimization (DCO) problem over the time-varying multi-agent network. By investigating sub-Weibull randomness in DCSMD for the first time, we show that the algorithm is applicable in some common heavier-tailed noise environments while also guaranteeing good convergence properties. We comprehensively study the convergence performance of DCSMD-SW. Satisfactory high-probability convergence rates are derived for DCSMD-SW without any smoothness requirement. The work also offers a unified analytical framework for several critical cases of both algorithms and noise environments.

The group testing problem consists of determining a sparse subset of defective items from within a larger set of items via a series of tests, where each test outcome indicates whether at least one defective item is included in the test. We study the approximate recovery setting, where the recovery criterion of the defective set is relaxed to allow a small number of items to be misclassified. In particular, we consider one-sided approximate recovery criteria, where we allow either only false negative or only false positive misclassifications. Under false negatives only (i.e., finding a subset of defectives), we show that there exists an algorithm matching the optimal threshold of two-sided approximate recovery. Under false positives only (i.e., finding a superset of the defectives), we provide a converse bound showing that the better of two existing algorithms is optimal.

We present new exact charged black hole solutions in (2+1) dimensions within the framework of $f({Q})$ gravity, where ${Q}$ denotes the non-metricity scalar. By considering a cubic $f({Q})$ form we derive classes of charged and uncharged spherically symmetric solutions, and we identify conditions under which these reduce to the well-known Banados-Teitelboim-Zanelli (BTZ) black hole. Notably, our analysis reveals a novel charged solution that is asymptotically Anti-de Sitter (AdS) but cannot be continuously deformed into a GR counterpart, and thus arising purely from the higher-order nature of the non-metricity corrections. Furthermore, we explore the geometrical properties of the solutions, demonstrating the existence of multiple horizons and a softer central singularity compared to GR. Additionally, we calculate various thermodynamic quantities, such as Hawking temperature, entropy, and heat capacity, with results confirming thermal stability. Finally, a detailed study of the geodesic motion and the effective potentials unveils stable photon orbits, as well as the effect of cubic non-metricity corrections on orbital dynamics.

We present a quantitative investigation of one- and two-body light-mediated processes that occur to few erbium atoms in an optical tweezer, when exposed to near-resonant light. In order to study the intertwined effects of recoil heating, cooling and light-assisted collisions, we develop a first-principles Monte Carlo algorithm that solves the coupled dynamics of both the internal and external degrees of freedom of the atoms. After validating our theoretical model against experimental data, we use the predictive power of our code to guide our experiment and, in particular, we explore the performance of different transitions of erbium for light-assisted collisions in terms of their efficiency and fidelity for single-atom preparation.

For nonempty subsets $X$ and $Y$ of a group $G$, we say that $(X,Y)$ is a tiling of $G$ if every element of $G$ can be uniquely expressed as $xy$ for some $x\in X$ and $y\in Y$. In 1966, Rothaus and Thompson studied whether the symmetric group $S_n$ with $n\geq3$ admits a tiling $(T_n,Y)$, where $T_n$ consists of the identity and all the transpositions in $S_n$. They showed that no such tiling exists if $1+n(n-1)/2$ is divisible by a prime number at least $\sqrt{n}+2$. In this paper, we establish a new necessary condition for the existence of such a tiling: the subset $Y$ must be partition-transitive with respect to certain partitions of $n$. This generalizes the result of Rothaus and Thompson, as well as a result of Nomura in 1985. We also study whether $S_n$ can be tiled by the set $T_n^*$ of all the transpositions, which finally leads us to conjecture that neither $T_n$ nor $T_n^*$ tiles $S_n$ for any $n\geq4$.

In this paper, we focus on the nonconvex-nonconvex bilevel optimization problem (BLO), where both upper-level and lower-level objectives are nonconvex, with the upper-level problem potentially being nonsmooth. We develop a two-timescale momentum-accelerated subgradient method (TMG) that employs two-timescale stepsizes, and establish its local convergence when initialized within a sufficiently small neighborhood of the feasible region. To develop a globally convergent algorithm for (BLO), we introduce a feasibility restoration scheme (FRG) that drives iterates toward the feasible region. Both (TMG) and (FRG) only require the first-order derivatives of the upper-level and lower-level objective functions, ensuring efficient computations in practice. We then develop a novel hybrid method that alternates between (TMG) and (FRG) and adaptively estimates its hyperparameters. Under mild conditions, we establish the global convergence properties of our proposed algorithm. Preliminary numerical experiments demonstrate the high efficiency and promising potential of our proposed algorithm.

Recent advances in secure hardware technologies, such as Intel SGX or ARM TrustZone, offer an opportunity to substantially reduce the costs of Byzantine fault-tolerance by placing the program code and state within a secure enclave known as a Trusted Execution Environment (TEE). However, the protection offered by a TEE only applies during program execution. Once power is switched off, the non-volatile portion of the program state becomes vulnerable to rollback attacks wherein it is undetectably reverted to an older version. In this paper we consider the problem of implementing reliable read/write registers out of failure-prone replicas subject to state rollbacks. To this end, we introduce a new unified model that captures multiple failure types that can affect a TEE-based system and establish tight bounds on the fault-tolerance of register constructions in this model. We consider both the static case, where failure thresholds hold throughout the entire execution, and the dynamic case, where any number of replicas can roll back, provided these failures do not occur too often. Our dynamic register emulation algorithm, TEE-Rex, provides the first correct implementation of a distributed state recovery procedure that requires neither durable storage nor specialized hardware, such as trusted monotonic counters.

One of the premier utilities of present day noisy quantum computers is simulation of many-body quantum systems. We study how long in time is such a discrete-time simulation representative of a continuous time Hamiltonian evolution, namely, a finite time-step introduces so-called Trotterization errors. We demonstrate that the truncated operator propagator (Ruelle-Pollicott resonances) is a powerful tool to that end, as well as to study prethermalization and discrete time crystals, including finding those phenomena at large gate duration. We show that the effective energy is more stable than suggested by Trotter errors -- a manifestation of prethermalization -- while all other observables are not. Even the most stable observable though deteriorates in the thermodynamic limit. Different than in classical systems with the strongest chaos, where the faithfulness time (the shadowing time) can be infinite, in quantum many-body chaotic systems it is finite. A corollary of our results is also that, opposite to previous claims, there is no Trotterization transition in non-integrable many-body quantum systems. We demonstrate our results on a one-dimensional (1d) kicked Ising model, as well as on 1d kicked XX model and 2d kicked Ising model. The truncated propagator is also used to calculate the energy diffusion constant in the tilted-field Ising model with high accuracy.

We present a new approach to verify the Elementary Type Conjecture for abstract Witt rings with small number of square classes. To do so, we make use of an abstract analogue of the 2-torsion part of the Brauer group, also verifying a certain case of the Arason-Pfister Hauptsatz in this setting. We develop a description of the entire structure of an abstract Witt ring with $2^n$ square classes in terms of a unique $n\times n$ matrix. Via computational search, we find all these matrices for $n$ up to $7$. All obtained results affirm the Elementary Type Conjecture.

We consider scattered data approximation on product regions of equal and different dimensionality. On each of these regions, we assume quasi-uniform but unstructured data sites and construct optimal sparse grids for scattered data interpolation on the product region. For this, we derive new improved error estimates for the respective kernel interpolation error by invoking duality arguments. An efficient algorithm to solve the underlying linear system of equations is proposed. The algorithm is based on the sparse grid combination technique, where a sparse direct solver is used for the elementary anisotropic tensor product kernel interpolation problems. The application of the sparse direct solver is facilitated by applying a samplet matrix compression to each univariate kernel matrix, resulting in an essentially sparse representation of the latter. In this way, we obtain a method that is able to deal with large problems up to billions of interpolation points, especially in case of reproducing kernels of nonlocal nature. Numerical results are presented to qualify and quantify the approach.

Rust is a strong contender for a memory-safe alternative to C as a "systems" language, but porting the vast amount of existing C code to Rust remains daunting. In this paper, we evaluate the potential of large language models (LLMs) to automate the transpilation of C code to idiomatic Rust. We present SafeTrans, a generic framework that leverages LLMs to i) transpile C code into Rust, and ii) iteratively repair compilation and runtime errors. A key novelty of our approach is a few-shot guided repair technique for translation errors, which provides contextual information and example code snippets for specific error types, guiding the LLM toward the correct solution. Another novel aspect of our work is the evaluation of the security implications of the transpilation process, showing how some vulnerability classes in C persist in the translated Rust code. SafeTrans was evaluated with six leading LLMs on 2,653 C programs and two real-world C projects. Our results show that iterative repair improves the rate of successful translations from 54% to 80% for the best-performing LLM (gpt-4o).

Social media is nearly ubiquitous in modern life, raising concerns about its societal impacts -- from mental health and polarization to violence and democratic disruption. Yet research on its causal effects is still inconclusive: Various methods, spanning observational to experimental, can yield seemingly conflicting results. Considering the complexity of such socio-technical systems, with coupled networks, feedback loops and collective phenomena, this may not be surprising. Here, we enumerate and examine the features of social media as a complex system that challenge our ability to infer causality at societal scales. Attempts to ascertain and summarize causal effects have tended to prioritize findings from randomized controlled trials (RCTs). However, like observational studies, RCTs rely on assumptions that may frequently be violated in the context of social media, especially regarding societal outcomes at scale. Drawing on insight from disciplines that have faced similar challenges, like climate-science or epidemiology, we propose a path forward that combines the strengths of observational and experimental approaches while acknowledging the limitations of each. Progress, we argue, requires moving beyond isolated, linear effects to mechanistic explanations of how social media platforms generate collective outcomes.

We construct new sets of log-canonical coordinates on the $SL(2, \mathbb{C})$ character variety of compact Riemann surfaces. These are labelled by families of $1\leq m\leq 3g-3$ non-intersecting simple loops on the Riemann surface and are obtained by combining the complexified shear-type with length/twist-type coordinates. In the case $m=3g-3$ the loops define a trinion decomposition of the Riemann surface, and our coordinates are closely related to the (complexified) Fenchel-Nielsen ones.

The Yang-Baxter equation (YBE) and the reflection equation (RE) both come from mathematical physics, and they can be defined in any monoidal category. For cartesian monoidal categories, we prove that every solution to the RE provides a Drinfeld twist for a solution of the YBE. As we observe, Drinfeld twists of solutions are relevant for the following reason: two solutions to the YBE (in any strict monoidal category) are related by a Drinfeld twist, if and only if they induce equivalent representations of the braid group. In the category of sets, it is known that every solution is associated with a structure group, which is a braided group in the sense of Lu, Yan, and Zhu (2000). Using De Commer's notion of a braided action, we then define group reflections for a braided group. We prove that group reflections provide group Drinfeld twists in the sense of Ghobadi. Finally, we characterise when a reflection on a solution (X,r) can be extended to a group reflection on its structure group G(X,r).

A new thermodynamic state function is introduced to describe the thermodynamics relevant for the mean transverse momentum fluctuations of charged particles in heavy-ion collisions, which allows us to compute the temperature fluctuations of different orders in hot quantum chromodynamics (QCD) matter for the first time. Consequently, it is found that the temperature fluctuations are suppressed remarkably as the system transitions from the hadron resonance gas (HRG) to the quark-gluon plasma (QGP) with increasing temperature or baryon chemical potential, alongside a negative skewness. This is attributed to the general fact that the heat capacity of QCD matter increases significantly in QGP in comparison to that in HRG. These predictions provide a unique signature to discover the thermodynamical temperature fluctuations in upcoming heavy-ion collision experiments, which also paves a novel way to study QCD thermodynamics and QCD phase diagram through measurements of the mean transverse momentum fluctuations of charged particles.

In this short paper we provide a new proof of the geometric Forward-Reverse Brascamp-Lieb inequality, using the approach of the heat semigroup, or the heat flow. Furthermore, we characterize all the Forward-Reverse Brascamp-Lieb data such that the initial relation can be preserved by some heat flow.

Radio phoenixes are filamentary sources in the intracluster medium (ICM) of galaxy clusters, often extending over $>100$ kpc, arising from fossil radio lobes. Their soft, curved spectrum is widely attributed to aged relativistic electrons recently accelerated or compressed, but at high frequencies is shown to approach a power-law. Moreover, the full, curved spectrum is naturally reproduced by secondary $e^{\pm}$ from a pure power-law spectrum of relativistic ions, radiating in highly-magnetized filaments; this model provides a better fit to all phoenixes, with only three free parameters. Weaker magnetization shifts the curvature to low frequencies, explaining pure power-law phoenixes. Hadronic high-curvature phoenixes require $e^{\pm}$ heating, by a factor $\gtrsim 15$ if at ICM pressure. The $\sim$keV Compton- and $\sim$GeV $π^0$-decay-peaked counterparts of hadronic phoenixes may be detectable as non-thermal X-rays and $γ$-rays.

Quantum Extreme Learning Machine (QELM) is an emerging hybrid quantum machine learning framework that leverages quantum system dynamics to enhance classical models. However, QELM can suffer from the exponential concentration problem, where excessive entanglement reduces model expressivity. In this work, we gain insight into this challenge and demonstrate how tensor network methods specifically, the Time Dependent Variational Principle (TDVP) with Matrix Product States (MPS) can efficiently simulate quantum systems while controlling entanglement and mitigating exponential concentration. Using numerical experiments on the Modified National Institute of Standards and Technology (MNIST) dataset, we show that time-evolving an MPS system modeled as a chain of Rydberg atoms produces high-quality data embeddings with low classical computational overhead. Our findings indicate that exact simulation of quantum dynamics is not necessary for strong machine learning performance; even approximate quantum embeddings can yield competitive results. Furthermore, we observe that both increased disorder in the quantum state achieved by tuning Hamiltonian parameters and careful control of entanglement directly correlate with improved model accuracy, highlighting the importance of these factors in optimizing QELM performance.

Introduced in 2001 by Lecomte and Rigo, abstract numeration systems provide a way of expressing natural numbers with words from a language $L$ accepted by a finite automaton. As it turns out, these numeration systems are not necessarily positional, i.e., we cannot always find a sequence $U=(U_i)_{i\ge 0}$ of integers such that the value of every word in the language $L$ is determined by the position of its letters and the first few values of $U$. Finding the conditions under which an abstract numeration system is positional seems difficult in general. In this paper, we thus consider this question for a particular sub-family of abstract numeration systems called Dumont--Thomas numeration systems. They are derived from substitutions and were introduced in 1989 by Dumont and Thomas. We exhibit conditions on the underlying substitution so that the corresponding Dumont--Thomas numeration is positional. We first work in the most general setting, then particularize our results to some practical cases. Finally, we link our numeration systems to existing literature, notably properties studied by Rényi in 1957, Parry in 1960, Bertrand-Mathis in 1989, and Fabre in 1995

Deep Learning (DL) methods can reconstruct highly accelerated magnetic resonance imaging (MRI) scans, but they rely on application-specific large training datasets and often generalize poorly to out-of-distribution data. Self-supervised deep learning algorithms perform scan-specific reconstructions, but still require complicated hyperparameter tuning based on the acquisition and often offer limited acceleration. This work develops a bilevel-optimized implicit neural representation (INR) approach for scan-specific MRI reconstruction. The method automatically optimizes the hyperparameters for a given acquisition protocol, enabling a tailored reconstruction without training data. The proposed algorithm uses Gaussian process regression to optimize INR hyperparameters, accommodating various acquisitions. The INR includes a trainable positional encoder for high-dimensional feature embedding and a small multilayer perceptron for decoding. The bilevel optimization is computationally efficient, requiring only a few minutes per typical 2D Cartesian scan. On scanner hardware, the subsequent scan-specific reconstruction-using offline-optimized hyperparameters-is completed within seconds and achieves improved image quality compared to previous model-based and self-supervised learning methods.

We study the mean-field limit of a generic class of dynamic co-evolving latent space networks motivated by the social and opinion dynamics literature. Such models include $n$ agents, whose opinions are given by latent stochastic processes, and a dynamic network process describing agent interactions. Models in this class incorporate (a) bi-directional feedback between the latent processes and the network process, (b) persistence effects, meaning that the network structure at the current time depends on the value of the latent processes at the current time but also on the network structure at the previous time instance and (c) localized interactions, meaning that individual agents do not have global information. We characterize the distributional limit of a random sample taken from the latent space network as the number of nodes in the network diverges. We describe the rich conditional probabilistic structure of the resulting limiting model which we use to establish the limiting behavior of the following quantities: (i) the empirical measure of the latent process, (ii) a conditional empirical measure relating the latent process to the network process and (iii) the network process graphon. In proving our main results, we derive a general conditional propagation of chaos result, which is of independent interest. Our novel approach to studying the limiting behavior of random samples proves to be a very useful methodology for fully grasping the asymptotic behavior of co-evolving particle systems. Numerical results are included to illustrate the theoretical findings.

Weak mixing in lattice models is informally the property that ``information does not propagate inside a system''. Strong mixing is the property that ``information does not propagate inside and on the boundary of a system''. In dimension two, the boundary of reasonable systems is one dimensional, so information should not be able to propagate there. This led to the conjecture that in 2D, weak mixing implies strong mixing. The question was investigated in several previous works, and proof of this conjecture is available in the case of finite range Gibbsian specifications, and in the case of nearest-neighbour FK percolation (under some restrictions). The present work gives a new proof of these results, extends the family of models for which the implication holds, and, most interestingly, provides a ``percolative picture'' of the information propagation.

Gravitational-wave (GW) ringdown signals from black holes (BHs) encode crucial information about the gravitational dynamics in the strong-field regime, which offers unique insights into BH properties. In the future, the improving sensitivity of GW detectors is to enable the extraction of multiple quasi-normal modes (QNMs) from ringdown signals. However, incorporating multiple modes drastically enlarges the parameter space, posing computational challenges to data analysis. Inspired by the $F$-statistic method in the continuous GW searches, we develope an algorithm, dubbed as FIREFLY, for accelerating the ringdown signal analysis. FIREFLY analytically marginalizes the amplitude and phase parameters of QNMs to reduce the computational cost and speed up the full-parameter inference from hours to minutes, while achieving consistent posterior and evidence. The acceleration becomes more significant when more QNMs are considered. Rigorously based on the principle of Bayesian inference and importance sampling, our method is statistically interpretable, flexible in prior choice, and compatible with various advanced sampling techniques, providing a new perspective for accelerating future GW data analysis.

We give a complete classification to when a finite group of outer automorphisms preserves a bi-order on a non-abelian free group and bi-orderable surface groups. We also give another new criterion for an outer automorphism of $F_n$ induced by action of an $n$-strand braid to preserve a bi-order on $F_n.$ Using the new criterion, we produce examples of order-preserving whose underlying permutation is a full cycle which answers in affirmative a question of Kin and Rolfsen.

The self-catalytic branching Brownian motions (SBBM) are extensions of the classical one-dimensional branching Brownian motions by incorporating pairwise branchings catalyzed by the intersection local times of the particle pairs. These processes naturally arise as the moment duals of certain reaction-diffusion equations perturbed by multiplicative space-time white noise. For the subcritical case of the catalytic branching mechanism, we construct the SBBM allowing an infinite number of initial particles. Additionally, we establish the coming down from infinity (CDI) property for these systems and characterize their CDI rates.