Papers

Onsager solved the two-dimensional Ising model in zero field, but no solution is known in a magnetic field. We prove that in any dimension $d\ge2$, at every fixed dimensionless field outside a countable set, the free energy obeys no algebraic differential equation in the coupling, even locally. The obstruction arises from droplet geometry: the maximal droplet volume grows as the $d/(d-1)$ power of its boundary size, while an integer band in the pressure coefficients protects the extremal contributions against cancellation. Our result excludes Onsager-type solutions and closed forms in elliptic, hypergeometric, or Painlevé functions of the coupling at these fields.

Neural emulation need not require learning a substitute for known physics. We investigate PolyNets that couple a continuous-time neural representation of polynomial dynamics with a simple recurrent network for discrete-time integration. When the governing coefficients are known, the resulting tensor graph requires no training and preserves the prescribed numerical scheme while enabling efficient hardware execution and parallel simulation. Using the one-dimensional Korteweg-de Vries equation, we compare PolyNet with CPU and GPU finite-difference Runge-Kutta solvers and three trained neural models: a state-map emulator, a physics-informed predictor, and a learned-tendency integrator. Analytical soliton solutions provide a reference for amplitude and phase errors across grid resolutions, 10^6 dispersively scaled time steps, ensembles of up to 10^4 trajectories, and varying initial amplitudes. PolyNet achieves accuracy comparable to the numerical solvers, including at amplitudes outside the learned models' training conditions. It combines this fidelity with high GPU throughput, approximately constant over the tested resolution range, and favorable ensemble scaling.

We establish topological recursion for the spin enumeration of hypermaps, or, in the dual language, spin dessins d'enfants. This problem has remarkable features distinguishing it from other applications of topological recursion to enumerative combinatorics. First, the spectral curve is elliptic rather than rational, as in most cases studied before. Second, it contains degenerate ramification points, and one has to apply a version of degenerate topological recursion. Finally, the meromorphic correlator differentials have "moving" poles on the antidiagonals between pairs of distinct arguments, and one has to adjust the loop equations to this kind of degeneracy as well. The computations involve concepts of different kinds, such as BKP integrability, Gaussian integrals, the x-y swap, and symplectic duality.

A recent graph-theoretic criterion for a Hamiltonian to exhibit a free-parafermion spectrum is that its frustration graph be an indifference graph. We show that an arbitrary finite indifference graph can be realized as a frustration graph of the quantized six-vertex model at odd roots of unity, by a suitable choice of the defining wiring diagram and parameters. This provides a concrete realization of the criterion through the quantum Frobenius property of the quantized six-vertex model, recently established from its three-dimensional integrable structure.

Echo State Networks (ESNs) are fundamentally important from the conceptual perspective of AI, because they provide simple and tractable frameworks for understanding temporal information processing, memory, and learning in dynamical systems. They are composed by an interconnected reservoir whose connectivity matrix remains constant over time and only the output weights are updated during the training period. In particular, ESNs employing online training allow the incremental adaptation of the output weights as data is received, making them particularly suitable for applications in which new input data becomes available over time (non-stationary environments). A key objective in the study of such learning structures is to determine which statistical properties of the reservoir enhance computational efficiency and maximize prediction performance. To this end, we investigate the effects of reservoir topology on learning performance, considering weight matrices generated from Erdos-Renyi random networks and Watts & Strogatz small-world networks. As test data we use the Lorenz and Rossler models in the chaotic regime and the FitzHugh-Nagumo model in the limit cycle regime. In the case of random Erdos-Renyi reservoirs, our simulations show that best performances are obtained for sparse reservoirs and especially when the link-density is near and immediately above the critical point of mean-field bond percolation. Watts & Strogatz reservoirs are shown to demonstrate best performance also for small values of link-density, compatible with the mean-field bond percolation when taking into account the specific nonlocal ring structure of the Watts & Strogatz reservoirs. These studies on the influence of the reservoir architecture and hyperparameters on enhancing the accuracy of ESN predictions provide an efficient framework for real-time forecasting in dynamically evolving systems.

In this paper, we solve the direct monodromy problem for the Borel-summed $0$-parameter solution of Painlevé~I. Using the auxiliary time equation, we first construct formal WKB wave matrices whose amplitudes are regular at the double turning point. Rigorous Borel summation then yields exact WKB matrices in each Stokes region. Uniform estimates on moving action strips provide a common exact wave matrix in a full neighbourhood of the double point, proving that all four incident jumps are identities for consistent column normalisations. We formulate the resulting normalised Riemann–Hilbert problem and prove its existence and uniqueness on the Borel domains of the construction. Finally, after Painlevé scaling, the five Borel-summed solutions coincide with the five rotated tritronquée solutions of Painlevé~I.

Sensitive dependence on initial conditions limits long-horizon prediction of chaotic systems even for deterministic dynamics, yet some dynamical information may remain recoverable after precise state information is lost. Recursive multistep methods are fragile because they reuse each intermediate prediction as input, allowing local errors to propagate through the rollout. We investigate whether partial multistep prediction can recover predictive information beyond the predictability horizon by omitting intermediate positions instead of assigning them unreliable values. The method uses a lightweight pattern-based local predictor. We analyze partial trajectories generated by different omission patterns and quantify the trade-off between accuracy on retained positions and the fraction classified as non-predictable. Experiments on the Lorenz system show that admissible masks contain trajectories that remain accurate far beyond the horizon of predictability, and an agreement-based ensemble recovers part of this information without future observations at the cost of a growing rejected fraction. At short horizons a broad accepted subset retains accuracy at moderate rejection, whereas at long horizons positive geometric skill requires increasingly strong selection. In this regime retained predictions no longer localize the state within a Lorenz scroll but still identify the occupied scroll, so topological structure outlasts geometric accuracy. A noise-based control reproduces this survival only after tuning of its perturbation scale, unlike masking. Across a heterogeneous collection of chaotic systems the ensemble strictly extends the positive-$R^2$ horizon of the recursive baseline for every retained system.

We study the stability of the inverse cascade in the 3D Nonlinear Schrodinger Wave Kinetic Equation (WKE) in the presence and absence of a condensate. Using theoretical arguments supported by numerical simulation, we show that, the Kolmogorov-Zakharov solution is stable with respect to small isotropic perturbations in the absence of a condensate. Conversely, introducing a condensate component leads to a growth of the condensate occupation and ultimately destabilizes the inverse cascade. The system then evolves towards a new steady-state whose properties depend on the condensate dissipation rate.

Resonantly driven nonlinear systems, from superconducting circuits to nanomechanical resonators, are often described in a rotating frame, where amplitude and phase evolve slowly within a fixed landscape of stable and unstable states. How such systems behave when this landscape moves on their intrinsic timescale remains to be understood. Here, we exploit the strong tension sensitivity of a graphene drum's resonance frequency to reshape the rotating-frame phase space of a bistable mechanical mode in real time, combining an MHz-frequency resonant drive with kHz-frequency tension modulation. When the modulation rate becomes comparable to the mechanical relaxation rate, adiabatic tracking of the fixed points breaks down. Instead, parametric instabilities emerge in the slow dynamics at frequencies set by libration around these points rather than by the mechanical resonance; stronger modulation drives period doubling and chaos. A slow-variable theory predicts where the instabilities occur, and Melnikov analysis captures the onset of chaos. These results show that the rotating frame itself can be parametrically driven, offering a general route to controlling nonadiabatic dynamics in systems ranging from Josephson bifurcation amplifiers to cold atoms.

We analyze the KdV–ghost system arising from the BRST formulation of BF-type topological field theory, its symmetries, Lax pair, spectral problem, zero-curvature formulation and Darboux formalism. The system is invariant under shifts of the ghost field, whose generator does not commute with the BRST symmetry of the KdV–ghost system. Their generators allow the introduction of a systematic procedure for generating new solutions from a given one, and the corresponding expressions in the Grassmann algebra are obtained explicitly. We introduce a Grassmann-valued Lax pair and prove an isospectral theorem relating the spectral problem of the spatial Lax operator for the KdV equation to that of its KdV–ghost extension. We also construct a triangular zero-curvature representation and show that the Grassmann-valued spatial Lax operator is related to the KdV spatial Lax operator by a nilpotent conjugation. The natural variable for a local Darboux formulation is the differentiated ghost $v=C_x$, for which the ghost equation coincides with the Fréchet linearization of the KdV flow. We then construct a nilpotent lift of the scalar KdV Bäcklund–Darboux transformation. Using Riccati variables and off-shell factorization identities for the KdV residue and its Fréchet derivative, we prove that the transformation closes on the local KdV–ghost system. Successive Darboux transformations yield a recursive ghost extension of the bosonic Darboux chain, with a new odd homogeneous contribution introduced at each step. For the solitonic examples, the resulting local ghost fields are naturally identified with tangent directions associated with the soliton phase parameters. The construction provides a Darboux-theoretic interpretation of the ghost sector while remaining distinct from the standard supersymmetric KdV hierarchies.

We apply large $N$ expansion techniques to derive kinetic equations for strongly coupled, far-from-equilibrium field theories with $O(N)$ symmetry. For large $N$ vector models, the quasi-classical approximation holds. Incorporating arbitrary momentum-dependent interactions, we show the dynamics obey the standard kinetic equation, with the scattering amplitude given by the retarded propagator of the auxiliary Hubbard-Stratonovich field. This has broad applications to systems like spin wave and Langmuir ion-acoustic turbulence. For large $N$ random coupling (SYK-like) models in $d$ dimensions, leading-order melon diagrams induce order-one propagator renormalization and finite spectral width. The corresponding kinetic equation, unlike those for standard weakly interacting systems, does not have a local-in-time collision integral. We find evidence for a stationary, scale-invariant infrared solution, revealing a potentially novel strong turbulence scaling exponent.

Let $G$ be a group and let $A$ be a set with at least two elements. A lazy cellular automaton over $A^G$ acts as the identity except at the cells where a fixed pattern occurs, where it replaces the central symbol by a fixed different symbol. Let $M(G;A)$ be the submonoid of $\mathrm{CA}(G;A)$ generated by the lazy and the invertible cellular automata. We study when $M(G;A) = \mathrm{CA}(G;A)$. Since no lazy cellular automaton is surjective, every surjective element of $M(G;A)$ is invertible; hence $M(G;A) \neq \mathrm{CA}(G;A)$ whenever $A$ is infinite or $G$ has an element of infinite order. Moreover, if $|A| = 2$ and $G$ has a quotient isomorphic to the Klein four-group $C_2 \times C_2$, then $M(G;A) \neq \mathrm{CA}(G;A)$. For locally finite $G$, and in particular for finite $G$, these are the only obstructions: we prove that $M(G;A) = \mathrm{CA}(G;A)$ if and only if $3 \leq |A| < \infty$, or $|A| = 2$ and $G$ has no quotient isomorphic to $C_2 \times C_2$.

We investigate whether the past of an auxiliary time series provides predictive information about a target beyond that contained in the target's own history. We compare matched out-of-sample target-only and target-plus-source forecasts and introduce the forecast-error distribution entropy (FEDE) predictive-causality framework, where predictive causality is understood in the Granger sense: auxiliary history is informative when it improves prediction beyond the target's own past. FEDE quantifies how the full signed forecast-error distribution changes across prediction horizons. Controlled nonlinear benchmarks hide source-target dependence behind nonlinear transformations not supplied to the predictors. LightGBM recovers these hidden channels, whereas Ridge does not. Across four equal-power nonlinearities and four noise levels, the mean response-sensitivity statistic for FEDE remains 29.00, 27.47, 23.22, and 28.67, compared with 14.09, 10.68, 12.67, and 12.71 for mean log-RMSE gain. Directional and source controls separate the true source-to-target channel from the reverse direction and from independent periodic, quasiperiodic, chaotic, and autoregressive auxiliaries. Chronological Granger-style comparisons show strong model-form dependence: a degree-2 polynomial model is exceptionally sensitive to the matched quadratic channel but becomes weak or oppositely aligned for other nonlinearities, while basic convergent cross mapping is weak or directionally inconsistent in this stochastic, partly non-invertible setting. FEDE remains positively aligned across all 108 tested combinations of window length, history length, and maximum forecast horizon. Together, these results establish FEDE as a sensitive and robust horizon-resolved tool that extracts predictive information beyond conventional error-magnitude measures when the nonlinear interaction form is unknown. Code available in Zenodo.

Chip-scale microresonator-based frequency combs, commonly referred to as microcombs, are promising sources for field-deployable metrology applications. However, intrinsic microresonator noise can make it difficult to generate microcombs with sufficiently low noise for applications. A widely used approach to overcome this limitation is to trap the DKS in the microresonator by phase-modulating the pump laser, although integrating a low-noise modulation reference in a fully on-chip platform remains challenging. Here, we propose an optoelectronic self-trapping scheme in which the impact of intrinsic microresonator noise on microcomb repetition-rate jitter is suppressed through feedback-controlled phase modulation of the driving laser so that the intrinsic noise of the laser sets the ultimate performance limit. This modification eliminates the need for an external low-phase-noise RF source, thereby enabling chip integration. We develop both analytical and numerical models for the proposed system and benchmark its performance against conventional externally referenced bulk implementations, showing that the proposed approach achieves comparable microcomb noise suppression while maintaining chip-scale compatibility.

Early warning signals are key tools for anticipating qualitative changes in the dynamics of complex systems. In this work, we investigate whether an information-theoretic measure designed to characterize directional causal dependencies between observables can provide early warning of the approach to bifurcations. The measure considered is the rate of information transfer, which has been successfully applied to characterize directional dependencies in a broad range of fields, from economics to climate science. Three fundamental local bifurcations are considered: saddle-node, pitchfork, and Hopf bifurcations. We first investigate idealized (quasi-)potential stochastic systems for which analytical expressions for the rate of information transfer can be derived. Distinct signatures are found for the three bifurcation types. In particular, the rate of information transfer displays a strong increase and divergence when approaching the saddle-node bifurcation, whereas smoother but systematic changes are found near the pitchfork and Hopf bifurcations. The approach is then tested using low-order climate models exhibiting the same types of bifurcations. The results confirm that changes in the rate of information transfer can precede the corresponding dynamical transitions, suggesting that directional information transfer provides a useful complementary early-warning indicator for detecting the approach to critical transitions in nonlinear stochastic systems.

Global synchronization has been largely investigated in static higher-order networks and little attention has been brought to directed, temporal higher-order networks. We hereby make a step in this direction by studying the emergence of global synchronization in periodically fast switching systems with directed higher-order interactions. By working in the framework of the Master Stability Function, we show that the linearized dynamics includes two time-dependent Jacobian matrices, both evolving on the timescale of the synchronous solution, together with a generalized Laplacian evolving on a fast switching timescale, resulting from the hypergraph time evolution. To address this two-timescale structure, we establish a fast-switching stability criterion for directed balanced higher-order structures. The latter condition allows to decouple the parallel and transverse perturbations to the synchronization manifold. By relying on the average theory for dynamical systems, we are able to prove exponential stability of global synchronization in the fast switching higher-order network. The proposed theory is then illustrated by using a higher-order Stuart–Landau model, which provides a natural benchmark for the theory. Directionality can move Laplacian eigenvalues into instability regions off the real axis, hence destabilizing synchronization; our results nevertheless reveal that temporal switching can counteract this destabilizing effect and restore global synchronization when the directed structures alternate sufficiently rapidly, even though the instantaneous directed higher-order structures are individually unstable.

Periodically driven quantum systems exhibit a wide range of complex dynamics, including chaos. Yet, at special parameter values, their evolution can recur exactly. Several quantum protocols have recently been proposed that exploit exact Floquet recurrences to demonstrate genuine quantum advantage. Such recurrences can be studied through the arithmetic structure of the Floquet unitary. In this work, we establish a necessary and sufficient arithmetic criterion for exact recurrence and determine the minimal recurrence period when one exists. Because recurrence is a global property of the unitary, it holds for every initial state. Applying our framework requires only knowledge of the number field associated with the Hamiltonian parameters. We demonstrate the method using the kicked Ising chain, establishing exact recurrences in both Clifford and non-Clifford regimes. We further show that, in the recurrent non-Clifford regime, magic–a resource for quantum computation–is generated at intermediate times and undergoes exact periodic revivals enforced by the recurrence of the full Floquet unitary. By connecting number theory, quantum many-body dynamics, and quantum information, our work shows that exact recurrence extends beyond Clifford circuits and opens new avenues for magic generation and quantum hardware benchmarking.

Direct and parametric driving together generate nonlinear response structures beyond conventional Duffing hysteresis, including multiple switching regions and isolated branches. Understanding how these structures are connected requires identifying the bifurcations that reorganize the response as the parametric excitation is varied. At a phase difference of 45 degrees between the direct and parametric excitations, we experimentally and analytically determine how the steady-state response changes as the parametric excitation is increased. A single resonance peak first splits into two while the response remains single valued. Two successive cusp bifurcations then create two saddle-node pairs and establish a double-hysteretic response. At the parametric-instability threshold, one saddle node of each pair is displaced towards unbounded amplitude and detuning, and an isolated-branch structure emerges above threshold. A parametric representation of the steady state distinguishes amplitude extrema from genuine saddle-node bifurcations and provides analytical conditions for these changes. Measurements on a dielectrically actuated pre-stressed SiN nanostring reproduce the predicted response and associated switching behavior. These results establish the bifurcation pathway connecting the different response structures at this phase configuration, providing a systematic basis for controlling multistability and switching in parametrically driven nonlinear resonators.

We study the amplification and concentration of wave energy in smooth, periodic space-time media through the dynamics of the characteristic curves of the associated wave equation. In an impedance-matched setting, we show that stable limit cycles attract neighbouring characteristics, focusing the wave energy density. The exponential attraction rate is estimated analytically from the system's Lyapunov exponent. We further show that the attraction of characteristics and energy accumulation persist under a range of nonperiodic perturbations. Finally, we present a Floquet-Fourier numerical method for computing stable and unstable limit cycles and their attraction rates. Throughout, we exemplify the results using the example of modulation given by a superposition of travelling waves, revealing saddle–node bifurcations where limit cycles may appear depending on the relative wave speeds. Full-wave simulations confirm that the characteristic mechanism continues to describe energy accumulation beyond the impedance-matched case.

Quantum aging transition has recently been reported, whereby the oscillatory dynamics of a network of globally coupled quantum self-sustained oscillators is degraded sharply when the fraction of inactive nodes exceeds a critical value. Here we explore the reverse process: the revival of rhythmic dynamics from an aged state in a quantum oscillator network by introducing a controlled incoherent single-boson pumping channel. Using a self-consistent quantum master equation in the Lindblad form for globally coupled active–inactive quantum Stuart–Landau oscillators, we show that increasing the pumping strength progressively suppresses the aging transition and eventually eliminates it, restoring rhythmicity even when all oscillators are inactive. Remarkably, complete compensation of the coupling-induced dissipative loss is not required; a substantially weaker pumping strength is sufficient to eliminate the aging transition. We show that the revival is governed by a redistribution of the Fock-state population of the inactive oscillators toward higher-occupation states. Our results establish controlled incoherent pumping as an effective mechanism for restoring dynamical robustness in quantum oscillator networks.

This article aims to introduce the bosonic asymmetric quantum conditional mutual information (QCMI), $I_{B}(A; B|C)$, as a rigorously defined information-theoretic quantity for continuous-variable (bosonic) systems. After defining $I_{B}(A; B|C)$ through quantum instruments on infinite-dimensional Fock spaces, we prove that Bose-Einstein condensation (BEC) induces a logarithmic divergence $ I_{B}(A; B|C) \geq \alpha \log_2 (N_{0}^C+1)$, which further establishes $I_{B}(A; B|C)$ as a quantum-information order parameter for BEC with power-law scaling near the critical point. We provide the complete five-step proof of this result, connecting measurement back-action, off-diagonal long-range order, and Jensen's inequality. In the condensed phase, the effective causal propagation velocity is set by the Bogoliubov sound speed $c_s$ in the long-wavelength limit, and vanishes continuously at the superfluid–Mott-insulator transition. Applications to composite bosons, the BEC–BCS crossover, and a quantum-gas-microscope measurement protocol are presented. Finally, we demonstrate how the reduced bosonic QCMI serves as a finite-size scaling probe analogous to the Binder cumulant.

While the robustness of collective dynamics in static networks is well established, how agent mobility and network topology jointly shape dynamical robustness remains largely unexplored. In this letter, we investigate the robustness of mobile Stuart-Landau oscillators performing random walks on networks and interacting diffusively when co-located. In the rapid-mobility regime, we reduce the dynamics to a mean-field framework linking network degree statistics to an effective global coupling strength. We show analytically that increased degree heterogeneity in the network enhances the effective coupling strength, and strictly lowers the critical threshold of functional degradation for amplitude death. Numerical simulations confirm the predicted heterogeneity-dependent ordering of the transition thresholds, showing an earlier collective collapse in scale-free networks than in random or regular networks. Our results establish mobility and network heterogeneity as key determinants of robustness in mobile oscillator populations, with implications for resilient biological and engineered collectives.

The focusing coupled Kundu-Eckhaus (CKE) equation extends the coupled nonlinear Schrödinger (CNLS) equation by adding quintic and derivative nonlinearities. We study the Cauchy problem for the focusing CKE equation with Schwartz initial data under the assumption of no discrete spectrum and no spectral singularities on the real axis. Based on the gauge equivalence between the focusing CKE and CNLS equations, we analyze the associated $3\times3$ matrix Riemann-Hilbert problem by the Deift-Zhou nonlinear steepest descent method. By establishing joint error estimates for the CNLS solution and the gauge phase, we obtain a uniform long-time asymptotic formula for the dispersive radiation of the CKE equation with an $O(t^{-1}\log t)$ remainder in bounded velocity cones.

This paper investigates the geometric detection of latent dynamic transitions (where the latent space is defined as a finite-dimensional state space reconstructed via a delay-embedding Hankel matrix) in mutually coupled phase-locked loops (PLLs) exhibiting on-off intermittent chaos. We propose a data-driven approach to identify the transition of the effective dimension in latent dynamics as a precursor to physical bursting events. By applying Singular Value Decomposition (SVD) to a Hankel matrix constructed from high-resolution oversampled data, we extract the effective dimension that reflects the dynamical shift of intrinsic variables - from a synchronized two-dimensional manifold to a higher-dimensional chaotic state - prior to the actual burst. This geometric indicator enables a proactive, event-driven intervention using Pyragas-type delayed feedback control, which is activated only when the detected dimension exceeds a critical threshold. Simulation results demonstrate that this geometric precursor effectively compensates for control latencies and substantially reduces accumulated control energy compared to conventional continuous feedback. The findings establish a methodological strategy for stabilizing infinite-dimensional delayed systems by exploiting early warning signals embedded in dimensional transitions of latent dynamics.

Ensemble forecasts from the ECMWF's Integrated Forecasting System (IFS) and the Artificial Intelligence Forecasting System (AIFS-ENS) are compared using a multi-dimensional evaluation framework. Within this framework, the uncertainty is interpreted as the linear subspace spanned by the ensemble members in phase space, from which various geometric properties are extracted. As revealed by the analysis, the error does not entirely lie in this subspace, but is reasonably well calibrated in all directions. Moreover, the framework allows to quantify that this subspace critically depends on the number of ensemble members, suggesting that the current setups only partially represent the uncertainty subspace. This phase-space interpretation furthermore allows a direct comparison of the error-representation patterns of AIFS-ENS and IFS by computing the principal angles between the subspaces: IFS and AIFS ensemble subspaces do not entirely overlap in phase space, even though they were shown to have similar abilities to represent the amplitude and the structure of the error.

We investigate the computational capabilities of a one-dimensional lattice of negatively coupled circle maps in its traveling wave regime. We interpret propagating waves as information carriers and exploit their interactions to implement Boolean logic. The logical states are encoded by the propagation direction of the traveling waves, while the presence or absence of a wave at a designated output location represents the logical output. Using controlled wave collisions, we demonstrate the realization of NAND and NOR gates along with other basic gates. Since NAND and NOR operations are functionally complete, the results establish that the traveling wave regime of the negatively coupled circle-map lattice supports universal Boolean computation in principle. The demonstrated logical operations remain valid over a finite range of the coupling strength \epsilon and map parameter \omega, indicating robustness of the computational scheme to parameter variations. Our results therefore demonstrate that computational functionality can emerge from the collective nonlinear dynamics of a coupled map lattice, providing a con- nection between the dynamical properties of traveling wave states and collision based information processing.

Spontaneous symmetry breaking (SSB) is conventionally understood through static vacuum selection in low-energy approximations. However, the nonperturbative dynamical pathways connecting symmetric configurations to asymmetric states remain an open question. In this paper, we propose an operator-theoretic framework that characterizes $Z_{2}$ symmetry breaking in scalar field theory through spectral branching. Specifically, we identify an effective invariant manifold of the nonlinear Klein-Gordon equation termed the "symmetric dual breather mode," whose phase-space geometry is traced using a $t$-dependent finite-dimensional representation. Our analysis demonstrates that static SSB represents merely a small-amplitude limit ($A \ll \phi_{v1}$). As the amplitude increases into the heteroclinic regime ($A > \phi_{v1}$), the field dynamically overflows the central barrier, yielding a vanishing time-averaged order parameter ($\langle \phi \rangle = 0$) and driving a geometric curvature cancellation ($\langle V''(\phi) \rangle_{t} \to 0$). Governed by a universal scaling function $\mathfrak{F}(y)$, the logarithmic period divergence near the heteroclinic boundary naturally generates sub-eV effective mass scales without fine-tuning bare parameters. Furthermore, this dynamic barrier overflow naturally favors Majorana-like self-conjugate states over Dirac-type states. Additionally, nonlinear phase interference along the heteroclinic channel induces asymmetric phase-space contraction, decoupling the right-handed mode and dynamically giving rise to the left-handed Weyl equation. These results bridge nonperturbative wave dynamics, topological mass suppression, chiral selection, and the dynamic origin of active neutrino masses.

We study the role of momentum fractionation in the development of chaos on smooth 3-charge supergravity solutions of D1-D5 theory, specifically momentum-fractionated superstrata. We probe the chaos through the dynamics of null geodesics and waves. On fractionated backgrounds, both geodesics and waves show uniformly strong chaos (large chaotic sea and near-perfect agreement with the Berry random wave hypothesis). In contrast, geodesics and waves on single-mode (non-fractionated) superstrata are closer to separability (although still chaotic). We discuss a possibility that such dichotomy can be related to the expectation that monotone BPS states are less chaotic than fortuitous ones. Finally, we explain why our chaos probes are meaningful for microstate geometries even though they become trivial in the black hole limit.

What makes natural languages diversify, in their syntactic structure, even when sharing a largely common origin? We refine our simple model, which focuses on the glassy dynamics induced by a network of interactions between syntactic parameters. Distinguishing between symmetric interaction weights that can be inferred from the observed statistics of syntax across many languages, and additional asymmetric weights assigned at random, we show that both components are needed, with similar overall strength, to account for plausible long-term syntactic dynamics. An approximate balance can combine the glassy metastability provided by the inferred weights with the chaotic diversification induced by the residual asymmetric weights.

We study geometry-controlled, initial-density-dependent stationary states in exclusion processes with branching-merging structures. We consider two boundary geometries for connecting the parallel branches to the reservoirs: the \(1\)-\(n\)-\(1\) model, in which a single lane branches into \(n\) parallel lanes and then merges back into a single lane, and the \(0\)-\(n\)-\(0\) model, in which particles directly enter and leave the \(n\) parallel lanes. For both models, we derive critical initial densities separating the low-density, shock-wave, and high-density regimes, and show that the theoretical predictions reproduce the phase diagrams and branch-averaged densities observed in simulations. In the \(1\)-\(n\)-\(1\) geometry, the critical initial densities depend on both the number of branches and the lengths of the upstream and downstream single-lane segments, whereas in the \(0\)-\(n\)-\(0\) geometry the corresponding estimates depend only on the number of branches. These results show that initial-condition dependence can arise from the global geometry of a branching-merging exclusion process, even without an explicit friction rule at the merging point.