Positive and negative flows of the Chen-Lee-Liu model and its various reductions, including Burgers hierarchy, are formulated within the framework of Riemann-Hilbert-Birkhoff decomposition with the constant grade two generator. Two classes of vacua, namely zero vacuum and constant non-zero vacuum can be realized within a centerless Heisenberg algebra. The tau functions for soliton solutions are obtained by a dressing method and vertex operators are constructed for both types of vacua. We are able to select and classify the soliton solutions in terms of the type of vertices involved. A judicious choice of vertices yields in a closed form a particular set of multi soliton solutions for the Burgers hierarchy. We develop and analyze a class of gauge-Bäcklund transformations that generate further multi soliton solutions from those obtained by dressing method by letting them interact with various integrable defects.

Spatial patterning and synchronization are pervasive features of plankton communities, yet the mechanisms that allow such patterns to persist coherently under environmental noise remain unresolved. In vertically structured aquatic ecosystems, plankton populations are often organized into distinct layers, raising the question of how interactions between layers shape both spatial self-organization and robustness. Here, we develop a spatiotemporal ecosystem model of a two-layer plankton community to examine the role of passive diffusive coupling under stochastic environmental fluctuations. We show that interlayer diffusion induces a sharp transition from independent, layer-specific Turing patterns to fully synchronized spatial patterns once the coupling strength exceeds a critical threshold. Importantly, the same coupling mechanism markedly enhances the stability of spatial patterns against environmental noise, extending their persistence far beyond that of non-coupled layers. Moreover, we uncover a trophic hierarchy in noise sensitivity, with zooplankton exhibiting substantially greater vulnerability than phytoplankton. Together, these results identify passive diffusive coupling as a unifying mechanism that simultaneously promotes spatial synchronization and robustness, providing a mechanistic explanation for the persistence of coherent plankton patterns in fluctuating aquatic environments.

It is shown that nonlocal coupling provides for controlling the collective noise-induced dynamics of non-excitable oscillators in the regime of coherence resonance. This effect is demonstrated by means of numerical simulation on an example of coupled generalized Van der Pol oscillators near the saddle-node bifurcation of limit cycles. In particular, it has been established that increasing the coupling radius allows to enhance the effect of coherence resonance which is reflected in the evolution of the dependence of the correlation time on the noise intensity and the power spectrum transformations. Nonlocal coupling is considered as an intermediate option between local and global coupling topologies which are also discussed as a tool for controlling coherence resonance.

Robustness of higher-order networks is often quantified by the instantaneous smallest positive eigenvalue of the Hodge $1$-Laplacian under simplex deletion. We show that this observable is generically ill-defined: along a deletion trajectory, eigenvalue branches can switch, so the quantity being monitored may correspond to different nonharmonic modes at different steps. The primary issue is therefore definitional rather than algorithmic. We resolve it by fixing the first nonharmonic branch of the intact complex and following that same branch throughout the damage process, which defines a branch-consistent functional robustness. Triangle sensitivities then follow directly from first-order perturbation theory, making the resulting mode-sensitive deletion protocol a consequence of the observable itself rather than an independent heuristic. Across synthetic and empirical clique complexes, removing only a small fraction of triangles is sufficient to drive the tracked mode to collapse, while graph-level observables remain unchanged because the $1$-skeleton is exactly preserved. The same framework also reveals bridge-like localization of functionally critical simplices and provides a compact predictor of dynamical timescales.

We show that PLDR-LLMs pretrained at self-organized criticality exhibit reasoning at inference time. The characteristics of PLDR-LLM deductive outputs at criticality is similar to second-order phase transitions. At criticality, the correlation length diverges, and the deductive outputs attain a metastable steady state. The steady state behaviour suggests that deductive outputs learn representations equivalent to scaling functions, universality classes and renormalization groups from the training dataset, leading to generalization and reasoning capabilities in the process. We can then define an order parameter from the global statistics of the model's deductive output parameters at inference. The reasoning capabilities of a PLDR-LLM is better when its order parameter is close to zero at criticality. This observation is supported by the benchmark scores of the models trained at near-criticality and sub-criticality. Our results provide a self-contained explanation on how reasoning manifests in large language models, and the ability to reason can be quantified solely from global model parameter values of the deductive outputs at steady state, without any need for evaluation of curated benchmark datasets through inductive output for reasoning and comprehension.

Large language models (LLMs) are increasingly deployed as optimization modules in agentic systems, yet the fundamental limits of such LLM-mediated improvement remain poorly understood. Here we propose a theory of LLM information susceptibility, centred on the hypothesis that when computational resources are sufficiently large, the intervention of a fixed LLM does not increase the performance susceptibility of a strategy set with respect to budget. We develop a multi-variable utility-function framework that generalizes this hypothesis to architectures with multiple co-varying budget channels, and discuss the conditions under which co-scaling can exceed the susceptibility bound. We validate the theory empirically across structurally diverse domains and model scales spanning an order of magnitude, and show that nested, co-scaling architectures open response channels unavailable to fixed configurations. These results clarify when LLM intervention helps and when it does not, demonstrating that tools from statistical physics can provide predictive constraints for the design of AI systems. If the susceptibility hypothesis holds generally, the theory suggests that nested architectures may be a necessary structural condition for open-ended agentic self-improvement.

We model the spatiotemporal dynamics of cellular protein concentrations near membranes composed of different lipids using a three-variable continuum model for membrane-bound protein, cytosolic protein, and the local composition of a binary lipid membrane. The model contains two globally conserved quantities: the total protein content and the average fractions of the two lipid species. It combines a conserved reaction-diffusion model for protein dynamics with a Cahn-Hilliard equation for lipid demixing. Linear stability analysis of the homogeneous steady state and direct numerical simulations show that the lipid dynamics undergoes classical phase separation, whereas the protein dynamics exhibits oscillatory phase separation for intermediate total protein contents, associated with a long-wavelength instability and traveling domains. In parameter regions where both instabilities are present, we find multiscale patterns with larger-scale traveling and rotating protein domains coexisting with smaller-scale stationary lipid domains. In this regime, traveling protein domains coexist with arrested coarsening of stationary lipid domains above a critical coupling. We further show that the main instabilities and phase diagram are well captured by an extension of a recently proposed conserved FitzHugh-Nagumo model for non-reciprocal pattern formation. The extended model consists of two non-reciprocally coupled Cahn-Hilliard equations with different interface tensions, reflecting the distinct physical properties of lipids and proteins. This also explains the observed asymmetry between static lipid patterns and traveling protein patterns.

The adoption of generative AI across commercial and legal professions offers dramatic efficiency gains -- yet for law in particular, it introduces a perilous failure mode in which the AI fabricates fictitious case law, statutes, and judicial holdings that appear entirely authentic. Attorneys who unknowingly file such fabrications face professional sanctions, malpractice exposure, and reputational harm, while courts confront a novel threat to the integrity of the adversarial process. This failure mode is commonly dismissed as random `hallucination', but recent physics-based analysis of the Transformer's core mechanism reveals a deterministic component: the AI's internal state can cross a calculable threshold, causing its output to flip from reliable legal reasoning to authoritative-sounding fabrication. Here we present this science in a legal-industry setting, walking through a simulated brief-drafting scenario. Our analysis suggests that fabrication risk is not an anomalous glitch but a foreseeable consequence of the technology's design, with direct implications for the evolving duty of technological competence. We propose that legal professionals, courts, and regulators replace the outdated `black box' mental model with verification protocols based on how these systems actually fail.

In several natural and engineering systems, changes in control parameters can trigger bifurcations that lead to sustained or growing periodic oscillations, indicating the onset of oscillatory instabilities. Such emergent behaviour often results from positive feedback between interacting subsystems, resulting in large-amplitude oscillations that can be detrimental. Several precursors are available to provide early warning of an impending oscillatory instability. In reality, practical systems may exhibit different sequences of bifurcations, including a primary bifurcation to an oscillatory state that may be either continuous or abrupt, followed by an abrupt secondary bifurcation, and further transitions beyond the secondary bifurcation. Existing precursors for oscillatory instabilities typically forewarn the onset of the primary bifurcation to an oscillatory state and tend to saturate once the system enters the oscillatory regime. Notably, primary bifurcations often involve lower amplitudes compared to the more severe states after secondary bifurcation. In this study, we propose a methodology based on spectral visibility graphs to get forewarning for both primary and secondary bifurcations. The method inherently captures the evolution of the harmonic content of the signal relative to other frequency components. The approach employs tuning of a single sensitivity parameter to detect different sequences of bifurcation. We demonstrate the usefulness of our method for thermo-acoustic and aero-acoustic instabilities in multiple engineering systems involving turbulent flow and reactions. Our methodology can help systems prepare in advance or even avoid undesirable transitions. Tuning the sensitivity parameter allows adaptive, risk-based warnings, ensuring high performance without tipping into undesirable regimes.

We study the nonlinear chaotic dynamics in a system of linear oscillators coupled by social network links with an additional stratification of oscillator energies, or frequencies, and supplementary nonlinear interactions. It is argued that this system can be viewed as a model of social stratification in a society with nonlinear interacting agents with energies playing a role of wealth states of society. The Hamiltonian evolution is characterized by two integrals of motion being energy and probability norm. Above a certain chaos border the chaotic dynamics leads to dynamical thermalization with the Rayleigh-Jeans (RJ) distribution over states with given energy or wealth. At low energies, this distribution has RJ condensation of norm at low energy modes. We point out a similarity of this condensation with the wealth inequality in the world countries where about a half of population owns only a couple of percent of the total wealth. In the presence of energy pumping and absorption, the system reveals features of the Kolmogorov-Zakharov turbulence of nonlinear waves.

Vaccination campaigns play a pivotal role in controlling infectious diseases. Their success, however, depends not only on vaccine efficacy and availability but also significantly on public opinion and the willingness of individuals to vaccinate. This paper investigates a coupled opinion-epidemic model on heterogeneous networks, where individual opinions influence vaccination probability, and opinions themselves evolve through a combination of peer interaction and local risk perception derived from observed infection rates. Embedding the coupled dynamics in scale-free networks, particularly barabasi-Albert structures, allows us to examine the role of network heterogeneity beyond homogeneous-mixing assumptions. Using Monte Carlo simulations and a semi-analytical microscopic Markov-chain approach, we derive and numerically validate analytical expressions for the critical infection threshold and stable vaccinated population where risk perception dominated peer influence. Our results show that stronger local risk perception enhances pro-vaccination opinions and suppresses infection, while dominant peer influence can increase long-term infection levels. These findings underscore the importance of accounting for social behavior and network structure when designing effective vaccination and epidemic control strategies.

We identify a geometric principle governing the location of Hopf and Bogdanov--Takens bifurcations in planar predator--prey systems. The prey coordinate of any coexistence equilibrium undergoing such a bifurcation lies between consecutive critical points of the prey nullcline. The mechanism is algebraic. At critical points of the nullcline, the vanishing of its derivative induces constraints on the Jacobian that prevent the spectral conditions required for bifurcation from being satisfied. We refer to this phenomenon as \emph{spectral rigidity}. The principle is established for three model families and one discrete counterpart with qualitatively different nullcline geometries: a quadratic case (Bazykin model), a cubic case (Holling type~IV with harvesting), and a rational case (Crowley--Martin functional response). In each case, the localization follows from explicit parametric characterizations and symbolic reduction. The analysis extends to discrete-time systems. For a map obtained by forward Euler discretization of the Crowley--Martin model, the Neimark--Sacker bifurcation occurs on the descending branch of the nullcline, providing a continuous--discrete duality governed by the same mechanism. We conjecture that this localization holds for general smooth prey nullclines, with critical points acting as spectral barriers that organise the bifurcation structure.

We establish sharp nonlinear stability results for fronts that describe the creation of a periodic pattern through the invasion of an unstable state. The fronts we consider are critical, in the sense that they are expected to mediate pattern selection from compactly supported or steep initial data. We focus on pulled fronts, that is, on fronts whose propagation speed is determined by the linearization about the unstable state in the leading edge, only. We present our analysis in the specific setting of the FitzHugh-Nagumo system, where pattern-forming uniformly translating fronts have recently been constructed rigorously, but our methods can be used to establish nonlinear stability of pulled pattern-forming fronts in general reaction-diffusion systems. This is the first stability result of critical pattern-selecting fronts and provides a rigorous foundation for heuristic, universal wave number selection laws in growth processes based on a marginal stability conjecture. The main technical challenge is to describe the interaction between two separate modes of marginal stability, one associated with the spreading process in the leading edge, and one associated with the pattern in the wake. We develop tools based on far-field/core decompositions to characterize, and eventually control, the interaction between these two different types of diffusive modes. Linear decay rates are insufficient to close a nonlinear stability argument and we therefore need a sharper description of the relaxation in the wake of the front using a phase modulation ansatz. We control regularity in the resulting quasilinear equation for the modulated perturbation using nonlinear damping estimates.

People make strategic decisions many times a day - during negotiations, when coordinating actions with others, or when choosing partners for cooperation. The resulting dynamics can be studied with learning theory and evolutionary game theory. These frameworks explore how people adapt their decisions over time, in light of how effective their strategies have been. The outcomes of such learning processes depend on how sensitive individuals are to the performance of their strategies. When they are more sensitive, they systematically favor strategies they deem more successful. When they are less sensitive, their learning process is noisier and more erratic. Traditionally, most models treat this sensitivity as a fixed parameter - like the "selection strength" parameter in evolutionary models. Instead, we study how strategies and sensitivities co-evolve. We find that the co-evolutionary endpoints depend on both the type of strategic interaction and the learning rule employed. In prisoner's dilemmas, we often observe sensitivities to increase indefinitely. But in snowdrift and stag-hunt games, sensitivities often converge to a finite value, or we observe evolutionary branching altogether. These results shed light on how evolution might shape learning mechanisms for social behavior. They suggest that noisy learning does not need to be a by-product of cognitive constraints. Instead, it can serve as a means to gain strategic advantages.

Boolean networks are a widely used modeling framework in systems biology for studying gene regulation, signal transduction, and cellular decision-making. Empirical studies indicate that biological Boolean networks exhibit a high degree of canalization, a structural property of Boolean update rules that stabilizes dynamics and constrains state transitions. Despite its central role, existing software packages provide limited support for the systematic generation of Boolean functions and networks with prescribed canalization properties. We present BoolForge, a Python toolbox for the random generation and analysis of Boolean functions and networks, with a particular focus on canalization. BoolForge enables users to (i) generate random Boolean functions with specified canalizing depth, layer structure, and related constraints; (ii) construct Boolean networks with tunable topological and functional properties; and (iii) analyze structural and dynamical features including canalization measures, robustness, modularity, and attractor structure. By enabling controlled generation alongside analysis, BoolForge facilitates ensemble-based investigations of structure-dynamics relationships, benchmarking of theoretical predictions, and construction of biologically informed null models for Boolean network studies. Availability and Implementation: BoolForge is implemented in Python ($\geq$3.10) and can be installed via \texttt{pip install boolforge}. Source code and documentation are available at this https URL. A PDF tutorial compendium is provided as Supplementary Material.

Recently, Fendley et al. (2025) [arXiv:2511.04674] revealed a new simple way to demonstrate the integrability of XYZ Heisenberg model by constructing a one-parameter family of integrals of motion in the matrix product operator (MPO) form with bond dimension 4. In this work, I report on the discovery of two-parameter families of MPOs that commute with Heisenberg spin chain Hamiltonian in case of various anisotropies (XXX, XXZ, XX, XY and XYZ). These solutions are connected by taking appropriate limits. For all cases except XYZ, I also write down Floquet charges of two-step Floquet protocols corresponding to the Trotterization. I describe a symbolic algebra approach for finding such integrals of motion and speculate about possible generalizations and applications.

The quantum XXZ spin-1/2 chain features non-Gaussian spin current fluctuations in the regime of easy-axis anisotropy. Using ballistic macroscopic fluctuation theory, we derive the exact probability distribution of typical spin-current fluctuations in thermal equilibrium. The obtained nested Gaussian distribution is fully characterized by its variance which we analytically relate to the spin diffusion constant and static spin susceptibility, and compare with numerical simulations. By unveiling how the same mechanism which leads to anomalous charge current fluctuations in single-file systems manifests itself in the XXZ chain, our approach establishes the universal hydrodynamic origin of the observed anomalous fluctuations.