Short mid-infrared laser pulses efficiently facilitate ultrafast manipulation of ferroic order parameters, including full reversal of magnetization or ferroelectric polarization, with the invoked mechanisms relating to the properties of polar phonons in ionic crystals. Much less is known, however, about the behaviour of such order parameters in response to an excitation in the far-infrared range, where phonons are more collective and less polar. Here we investigate transient crystallographic strains and polarization switching in ferroelectric barium titanate (BaTiO3) driven by an excitation in the frequency range of 5-8 THz, or wavelengths of 35-60 um. We find that switching persists in a large part of this range, but is governed primarily by optical absorption rather than by the longitudinal optical phonons or epsilon-near-zero conditions that dominate in the mid-infrared regime.

We study the asymptotic behavior of a nonlattice random walk in a general cone of $R^d$ . Following the approach initiated by D. Denisov and V. Wachtel in [8], we use a strong approximation of random walks by the Brownian motion and prove local limit theorems, combining integral theorems for random walks in cones with classical theorems for unrestricted random walks.

Multiplex networks are a powerful framework for representing systems with multiple types of interactions among a common set of entities. Understanding their structure requires statistical tools capturing higher-order cross-layer correlations. We develop a comprehensive framework for estimating and testing dependence in exchangeable multiplex networks through motif counts. We first propose a moment-based estimation methodology that extends the multi-layer stochastic block model network histogram to arbitrary motif counts. This allows us to estimate the $2^d-1$ graphons defining a $d$-layer multiplex network. We then derive the joint asymptotic distribution of cross-layer motif counts, that is aligned motifs shared across layers. Extending existing results from the unilayer setting, we show that the limiting distribution in the motif-regular case exhibits a covariance structure involving minimum-based distances between graphons. Finally, we construct hypothesis tests to detect inter-layer similarity and dependence. This work provides a rigorous extension of motif-count asymptotics and inference procedures to the multiplex setting, providing new tools to study high-order dependencies in complex networks.

The vigorous development of the Internet has spurred exponential data growth, yet data is predominantly stored in isolated user entities, hampering its full value realization. In large-scale deployment of ``AI+industries'' such as smart medical care, intelligent transportation and smart homes, the gap between data supply and demand continues to widen, and establishing an effective data sharing mechanism is the core of promoting high-quality industrial development. However, data sharing faces significant challenges in security, performance, and functional adaptability. Privacy-enhancing encryption technologies, including Attribute-Based Encryption (ABE), Proxy Re-encryption (PRE), and Searchable Encryption (SE), offer promising solutions with distinct advantages in enhancing security, improving flexibility, and enabling efficient sharing. Statistical analysis of relevant literature from 2020 to 2025 reveals a rising research trend in ABE, PRE and SE, focusing on their data sharing applications. Firstly, this work proposes a data sharing process framework and identifies 20 potential attacks across its stages. Secondly, this work integrates ABE, SE, PRE with 12 enhancement technologies and examines their multi-dimensional impacts on the security, performance, and functional adaptability of data sharing schemes. Lastly, this work outlines key application scenarios, challenges, and future research directions, providing valuable insights for advancing data sharing mechanisms based on privacy-enhancing encryption technologies.

Applying the $q$-Zeilberger algorithm, we establish a unified $q$-analogue of the (C.2) and (G.2) supercongruences of Van Hamme, which can be viewed as a refinement of several previously known results. As consequences, we obtain a $q$-analogue of supercongruence involving Bernoulli numbers, as well as a refinement of (G.2) supercongruence.

We establish a long exact sequence for the homotopy K-theory groups of the algebraic Cuntz-Pimsner rings introduced by Carlsen and Ortega [CO11] by adapting Pimsner's original proof [Pim97] to Cuntz's formalism.

This work addresses a central challenge in the numerical analysis of the cutoff spatially homogeneous Boltzmann equation: the development of rigorously justified, accurate numerical schemes. We present (i) a novel Fourier spectral method for the equation with Maxwellian and hard potentials, (ii) the derivation of the first rigorous error estimates for the proposed schemes. Comprehensive numerical experiments validate the theory, confirming the predicted accuracy and illustrating the method's capability to capture solution dynamics, including the approach to equilibrium. The study thus provides a complete framework--from theoretical analysis to practical implementation--for the reliable computation of solutions to this foundational kinetic model.

With the explosive growth of graph-structured data, graph databases have become a critical infrastructure for supporting large-scale and complex data analysis. Among various graph operations, shortest distance queries play a fundamental role in numerous applications, such as path planning, recommendation systems, and knowledge graphs. However, existing encrypted graph query methods still suffer from limitations in computational efficiency and system scalability, making it challenging to support efficient query processing over large-scale encrypted graph data. To address these challenges, this paper proposes a tensor-based shortest distance query scheme for encrypted graph databases. The proposed method integrates an encrypted 2-hop cover indexing framework with the Pruned Landmark Labeling (PLL) technique, thereby constructing an efficient and privacy-preserving indexing mechanism. Furthermore, a tensorized representation is introduced to uniformly model graph structures, which effectively reduces computational complexity while ensuring data privacy, and significantly improves the scalability of the system. Extensive experimental evaluations on large-scale graph datasets demonstrate that the proposed approach achieves superior scalability and lower computational costs compared with existing encrypted graph query methods. Moreover, it provides strong privacy protection guarantees, making it well suited for privacy-preserving graph query applications in cloud computing and distributed environments.

We develop and analyze a class of structure-preserving discontinuous Galerkin schemes for the nonlinear Vlasov-Poisson-Fokker-Planck model, reformulated as a hyperbolic system through a Hermite expansion in the velocity variable. We discretize the Vlasov-Fokker-Planck equation with the discontinuous Galerkin method, while the Poisson equation is approximated with either a discontinuous Galerkin method or a Raviart-Thomas mixed finite element method. We prove the exponential relaxation to equilibrium for suitable initial data, uniformly with respect to the discretization parameters thanks to discrete hypocoercivity arguments. Moreover, we check that the resulting semi-discrete schemes preserve the physical invariants along with the L 2 variational structure of the linearized model. Numerical simulations verify the accuracy and the long-time behavior of the scheme.

Fluid antenna (FA) systems offer novel spatial degrees of freedom (DoFs) with the potential for significant performance gains. Compared to existing works focusing solely on optimizing FA positions at discrete time instants, we introduce the concept of continuous-trajectory fluid antenna (CTFA), which explicitly considers the antenna element's movement trajectory across continuous time intervals and incorporates the inherent kinematic constraints present in practical FA implementations. Accordingly, we formulate the total throughput maximization problem in CTFA-aided wireless communication systems, addressing the joint optimization of continuous antenna trajectories in conjunction with the transmit covariance matrices under kinematic constraints. To effectively solve this non-convex problem with highly coupled optimization variables, we develop an iterative algorithm based on block coordinate descent (BCD) and majorization-minimization (MM) principles with the aid of the weighted minimum mean square error (WMMSE) method. Finally, numerical results are presented to validate the efficacy of the proposed algorithms and to quantify the substantial total throughput advantages afforded by the conceived CTFA-aided system compared to conventional fixed-position antenna (FPA) benchmarks and alternative approaches employing simplified trajectories.

In a colouring of a graph, a vertex is b-chromatic if it is adjacent to a vertex of every other colour. We consider four well-studied colouring problems: b-Chromatic Number, Tight b-Chromatic Number, Fall Chromatic Number and Fall Achromatic Number, which fit into a framework based on whether every colour class has (i) at least one b-chromatic vertex, (ii) exactly one b-chromatic vertex, or (iii) all of its vertices being b-chromatic. By combining known and new results, we fully classify the computational complexity of b-Chromatic Number, Fall Chromatic Number and Fall Achromatic Number in $H$-free graphs. For Tight b-Chromatic Number in $H$-free graphs, we develop a general technique to determine new graphs $H$, for which the problem is polynomial-time solvable, and we also determine new graphs $H$, for which the problem is still NP-complete. We show, for the first time, the existence of a graph $H$ such that in $H$-free graphs, b-Chromatic Number is NP-hard, while Tight b-Chromatic Number is polynomial-time solvable.

The collapse of massive stars drives explosions that power relativistic fireballs. If only a small amount of matter is entrained, such clean fireballs can expand with Lorentz factors $Γ> 100$, accounting for gamma-ray bursts (GRBs). It has been hypothesized that energetic explosions with more baryon contamination, dubbed ``dirty fireballs'', may exist in nature, but they have not been observed. Here we report the observation of an extragalactic fast X-ray transient, EP241113a, detected by Einstein Probe. Compared to GRBs, it has a similar isotropic energy of $1.4\times 10^{51}$ erg, but significantly lower spectral peak energy. Theoretical modeling of its early X-ray afterglow suggests a relativistic jet with a low Lorentz factor of $Γ\sim 20$ aligned close to the line-of-sight, signifying the prototype of a dirty fireball.

A topological group $G$ is said to have no small subgroup (resp. no small normal subgroup) if it admits an open neighbourhood of the identity containing no non-trivial subgroup (resp. normal subgroup) of $G$. These properties are usually denoted by NSS (and respectively NSnS). The NSS property plays an important historical role in the solution to the fifth problem of Hilbert due to Gleason, Montgomery-Zippin and Yamabe for the characterization of Lie groups. In 2019, Shakhmatov and the author proved that a free group $F$ with countably infinitely many generators admits a metric Hausdorff group topology $\mathscr{T}$ which satisfies the so-called algebraic small subgroup generating property ASSGP: for each open neighbourhood $U$ of the identity of $F$, the family of subgroups contained in $U$ algebraically generates $F$. In particular $(F,\mathscr{T})$ admits no non-trivial continuous homomorphisms to either NSS or locally compact groups, making it minimally almost periodic. In this paper, we prove that $(F, \mathscr{T})$ can be made topologically simple; namely, $(F, \mathscr{T})$ contains no closed normal subgroups other than $\{e\}$ and $F$. In particular, this implies that $F$ satisfies the no small normal subgroup (NSnS) property.

We review recent progress on state-dependent Lieb-Robinson bounds for Bose-Hubbard Hamiltonians. In particular, Kuwahara, Vu, and Saito established that, for general bounded-density initial states, the Lieb-Robinson velocity is bounded by $t^{d-1}$ for large times, where $d$ denotes the lattice dimension. We present a shorter proof of the weaker, but still polynomial velocity bound $t^{d+ε}$.

We establish a Bayesian analysis framework with a general Walecka-type relativistic mean-field model to study dense nuclear matter under constraints from nuclear matter properties and neutron star observations. With experimental and observational data well described, we find that pure hadronic descriptions can generate a peak structure in sound velocity by $ω$, $ρ$, $σ$, and $a_0$ meson mixing, which is crucial for describing both medium and massive neutron stars. As the peak structure is frequently interpreted as a signature of phase transitions, our findings provide a new perspective on the microscopic origin of the sound velocity peak just with pure hadronic matter.

In this paper, we investigate the bulging of escaping or oscillating Fatou components on invariant fibers for general skew-products, with a focus on the dependence on the perturbation. We show that any orbitally unbounded component is non-bulging for an appropriate choice of perturbation, whereas sufficiently well-behaved perturbations can render it bulging when the fiber is attracting. Our results highlight that bulging is influenced by more than just the dynamics on the fiber and in the one-dimensional coordinate, contrasting sharply with established results for non-escaping Fatou components.

We analyse the sizes and specific angular momentum content of galaxies in the Colibre cosmological hydrodynamical simulations spanning two orders of magnitude in mass resolution. We compare the predicted size-mass and angular momentum-mass relations to a broad range of observational measurements spanning redshifts $z=0$ to $4$. At $z=0$, Colibre reproduces observed size-mass relations over the sampled mass range $10^8 \lesssim M_\star/{\rm M_\odot}\lesssim 10^{11.5}$, and for multiple size definitions, including two- and three-dimensional stellar half-mass radii, half-light radii across several wavelengths, as well as alternative measures such as baryonic half-mass radii and characteristic radii defined by stellar surface density thresholds. The simulations also recover the observed segregation of galaxies in the size-mass plane by morphological type and star formation rate, and reproduce the distinct, approximately parallel sequences followed by star-forming discs and quenched spheroids in the stellar specific angular momentum-mass plane. The angular momentum content of star-forming Colibre galaxies match that of observed systems out to $z\approx 1.5$. At higher redshifts, massive galaxies ($ 10^{9.5}\lesssim M_\star/{\rm M_\odot}\lesssim 10^{11}$) in the simulations are somewhat smaller than observed, and the separation between star-forming and passive populations in the size-mass plane is reduced relative to observations, while at lower masses the agreement remains good. This apparent discrepancy may reflect the effects of dust attenuation, which is neglected in our analysis and may preferentially obscure the central regions of observed systems. Overall, our findings highlight the close connection between galaxy size, angular momentum, and morphology over cosmic time.

Information units are progressively approaching the fundamental physical limits of the integration density, including in terms of extremely small sizes, multistates and probabilistic traversal. However, simultaneously encompassing all of these characteristics in a unit remains elusive. Here, via real-time in situ electrical monitoring, we clearly observed stochastic alterations of multiple conductance states in Sc2C2@C88. The true random bit sequence generated exhibited an autocorrelation function whose confidence interval fell within \pm 0.02, demonstrating high-quality randomness. The alterations of multiple conductance states are controllable, that is, whose probability distributions could traverse from 0 to 1, enabling us to factorize 551 into its prime factors. Furthermore, we proposed a matrix-chain multiplication scheme and experimentally verified the multiplication of two 4 \times 4 state-transition matrices with a small maximum error < 0.05. Combined with theoretical calculations, the stochastic but controllable multistates are probably attributed to the rich energy landscape, which could be stepwise changed by the electric field. Our findings reveal extremely small multi-level probabilistic bit for matrix multiplication, which pave the way for ultracompact intelligent electronic devices.

Creative approaches to attract students to software engineering at an early age are emerging, yet their differential impact on gender remains unclear. This study investigates whether design thinking's empathy-driven approach addresses the documented gender gap in interest in software engineering. In a 10-week curriculum-integrated design thinking software development course with 55 secondary school students aged 13-15 from two schools in Canada, we examined gendered differences in perceived gains in knowledge and interest, as well as in social-emotional experiences. Our results show that both girls and boys gained perceived knowledge in software development. However, girls showed significant improvements in self-efficacy, interest, engagement with sustainability topics, and well-being, including optimism, sense of usefulness, and social connectedness. Positive emotions were strongest during creative, collaborative phases, while technical tasks led to some boredom, especially among boys, though they still benefited overall. This suggests that human-centred design thinking might be one effective way to address gender equity challenges, though we need more differentiated technical implementations.

Complex salt geometries and strong velocity contrasts pose significant challenges for velocity model building and subsalt imaging. Although full waveform inversion (FWI) provides high-resolution velocity models, its performance strongly depends on the accuracy of initial model. On the other hand, gravity focusing inversion (GFI) can recover compact density distributions and provide reliable long-wavelength structural information for seismic exploration, but it suffers from poor depth resolution and inherent non-uniqueness. To better invert salt structure by leveraging the complementary advantages of full waveform and gravity data, we propose a multi-physics alternating coupled inversion strategy for salt dome model. The proposed strategy mainly includes three parts. First, we perform FWI using a simple layered velocity model to obtain preliminary velocity updates and extract the salt top boundary. Second, this structural information is used as a constraint in GFI to recover a compact salt density distribution beneath the salt top. Third, the resulting salt geometry is used to construct an improved velocity model for the next stage of FWI. Through iterative alternation, FWI provides reliable structural constraints for GFI, while GFI supplies a more reasonable macroscopic salt model for FWI, effectively mitigating the strong dependence on the initial model. In addition, a depth-varying density contrast is introduced in GFI to better represent sediment compaction effects. Compared with unconstrained GFI and conventional FWI using a horizontally layered initial model, the proposed method effectively improves both velocity and density reconstruction in the modified BP salt model and SEG/EAGE salt model.

The general construction of lattice (co)homology assigns to a lattice $\mathbb{Z}^r$ and a weight function $w:\mathbb{Z}^r \to \mathbb{Z}$ a bigraded $\mathbb{Z}[U]$-module $\mathbb{H}_*$. The weight function $w$ is often obtained from some geometric data as the difference of two `height functions'. In this paper we consider the case when these height functions are Hilbert functions of valuative multifiltrations on a Noetherian $k$-algebra $\mathcal{O}$ and a finitely generated $\mathcal{O}$-module $M$. We introduce the notion of `realizable submodules' in $M$, the prime example of which are finite codimensional integrally closed submodules in the sense of Rees (or integrally closed ideals when $M=\mathcal{O}$). We prove, that whenever two sets of `extended' discrete valuations `realize' the same submodule $N \leq M$, then, although the corresponding lattices and weight functions might be different, the resulting lattice homology modules are isomorphic and have Euler characteristic $\dim_k(M/N)$. In this way, we associate a well-defined lattice homology to any quotient of type $M/N$, where $N$ is a realizable submodule of $M$. We also present some structural and computational results: e.g., we geometrically characterize the (lattice) homological dimension of integrally closed monomial ideals of $k[x,y]$. The main upshot of the paper, however, is the possibility of categorifying numerical invariants defined as codimensions of realizable submodules or integrally closed ideals. The geometric applications include: the delta invariant $δ(C, o)$ of a reduced curve singularity; the geometric genus $p_g(X, o)$, the irregularity $q(X, o)$ and the various plurigenera of higher dimensional isolated normal singularities. The corresponding categorifications generalize the analytic lattice homologies of Ágoston and the first author.

Assume $L$ is a $k$-assignment of a graph $G$. An $L$-packing $ϕ$ of $G$ is a sequence $ϕ=(ϕ_1, \ldots, ϕ_k)$ of $k$-mappings such that each $ϕ_i$ is an $L$-coloring of $G$, and for each vertex $v$ of $G$, $\{ϕ_1(v), \ldots, ϕ_k(v)\} = L(v)$ (and hence $ϕ_i(v) \ne ϕ_j(v)$ when $i \ne j$). We say $G$ is list $k$-packable if for any $k$-assignment $L$ of $G$, there is an $L$-packing of $G$. The list packing number $χ_l^{\star}(G)$ of $G$ is the minimum integer $k$ such that $G$ is $k$-packable. For a positive integer $d$, let $t(d)$ be the maximum packing number of graphs of tree-width at most $d$. It was known that $d+1 \le t(d) \le 2d$ for any $d$. In this paper, we prove that $t(d) \le 2d-1$ for $d \ge 3$, and $t(d) \ge d+2$ for $d \ge 2$. In particular, $t(2)=4$ and $t(3)=5$. Furthermore, we show that for constant positive integers $k, d$, the problem of determining $χ_l^{\star}(G)\leq k$ or not for a graph $G$ of tree-width at most $d$ is solvable in linear time.

The non-Hermitian skin effect is an anomalous localization phenomenon induced by nonreciprocal dissipation and has attracted considerable attention in recent years both theoretically and experimentally. In this article, we review the multifractal aspects of the non-Hermitian skin effect. In particular, we discuss how the many-body skin effect exhibits multifractality in many-body Hilbert space, unlike the trivial Hilbert-space occupation of the single-particle skin effect on crystalline lattices. We further highlight that the many-body skin effect can coexist with random-matrix spectral statistics, in contrast to the multifractality associated with many-body localization, which typically accompanies the absence of ergodicity. We also introduce a solvable model on a Cayley tree as an effective description of the many-body Hilbert space, in which the multifractal dimensions can be obtained analytically. This review provides a unified perspective on multifractal structures in the non-Hermitian skin effect across single-particle, many-body, and tree models, and clarifies their distinctive relation to ergodicity in open quantum systems.

We consider smooth plane curves $\mathcal{X}$ of degree $d\geq4$, defined over an algebraically closed field of characteristic $0$, that possess a unique outer Galois point. This geometric condition forces the curve to be a cyclic covering of the projective line, and ensures that its automorphism group fits into a specific theoretical framework. For each possible non-cyclic reduced automorphism group $\operatorname{Aut}_{\operatorname{red}}(\mathcal{X})$, we fully characterize the defining equation of $\mathcal{X}$ and the precise structure of its full automorphism group $\operatorname{Aut}(\mathcal{X})$. This comprehensive analysis not only identifies the exact form of the equation for each automorphism type but also establishes the detailed criteria under which these scenarios can occur, thereby offering a complete classification of defining equations for smooth plane curves with a unique outer Galois point and a non-cyclic reduced automorphism group.

Non-Hermitian systems exhibit anomalous scaling, a striking departure from conventional bulk laws, rooted in the non-Hermitian skin effect (NHSE). Here, we experimentally uncover this scaling and demonstrate its active control in a temporal photonic lattice. By tracking the real-time evolution of all eigenstates as system size varies, we directly observe scaling-driven spectral reshaping and eigenstate localization, revealing phenomena absent in Hermitian or NHSE-free lattices. In a Su-Schrieffer-Heeger lattice, scaling alone can trigger a non-Hermitian topological phase transition, with edge modes remaining protected. Crucially, Kerr interactions open the frontier of nonlinear non-Hermitian physics: weak nonlinearity accelerates or decelerates anomalous scaling, while strong nonlinearity suppresses it entirely. These results establish the first experimental platform for linear and nonlinear anomalous scaling engineering, paving the way for compact non-Hermitian devices and exploration of nonlinear and many-body non-Hermitian phenomena.

We study ratios of eigenvalues of the Laplacian on compact metric graphs. Our goals are threefold: First, we prove a sharp Ashbaugh--Benguria-type bound for the ratio of the first two eigenvalues on compact trees with Dirichlet conditions at all leaves, concretely showing that the ratio is maximized when the graph is an interval or an equilateral star. This improves a previous Payne--Pólya--Weinberger-type result due to Nicaise [Bull. Sci. Math., II. Sér. 111 (1987), 401--413]. Second, we extend this bound to a set of inequalities for the ratio of any pair of eigenvalues of such compact Dirichlet trees which respect the Weyl asymptotics up to an absolute constant. Third, we show that on non-trees, on which we also allow any mix of Neumann and Dirichlet conditions at the leaves, it is possible to recover bounds on the eigenvalue ratios depending only on the number of independent cycles and the number of Neumann leaves, in addition to the eigenvalue indices. This complements previously known counterexamples to analogues of the Ashbaugh--Benguria bound for general quantum graphs, by showing that the only way the bound can fail is through cycles and Neumann leaves, and by explicitly quantifying the extent to which it can fail.

Using two observations obtained with the Five-hundred-meter Aperture Spherical radio Telescope (FAST), we present a detailed single-pulse analysis of the high-nulling pulsar PSR J1820-0509. We measure an exceptionally high nulling fraction of approximately 81.78%, significantly exceeding previous estimates from Parkes observations. The single-pulse energy distribution exhibits a clear bimodal structure, consistent with classical nulling behavior. However, stacking the identified null pulses reveals a statistically significant residual profile above the noise level, indicating that the nulls correspond to a very weak emission state rather than a complete cessation of radio emission. The pulsar shows clustered burst activities spanning several hundred rotation periods, with prominent quasi-periodicities at 1191 +/- 81 and 590 +/- 15 pulse periods in the two observations. Based on temporal clustering and integrated profile morphology, we identify three distinct emission modes (A, B, and C) and a pseudo-null state (D). These modes exhibit systematic differences in pulse morphology, polarization, and energy statistics. The pulse width-energy relations reveal clear transitions between low- and high-energy regimes. The energy distributions of Modes A and C are well described by lognormal functions, while Mode B follows a composite Gaussian-lognormal distribution. These results suggest that the radio emission of PSR J1820-0509 is governed by multiple quasi-stable magnetospheric states. The presence of weak emission during pseudo-nulls, together with systematic mode-dependent variations, supports the interpretation that pulsar nulling reflects transitions between different magnetospheric activity levels rather than a complete shutdown of emission.

We study the inverse problem for a semilinear wave equation on metric tree graphs. From the Dirichlet-to-Neumann map defined at all but one of the boundary vertices, we recover unknown connectivity of the graph, lengths of the edges, the time-independent potential and the time-dependent coefficient of the nonlinear term of the equation.

We propose a new family of inequality indices that bridges the Hoover index and the Gini coefficient. The measure is defined as the normalized expected absolute value of a convex combination of deviations from the mean and pairwise differences, providing a continuous interpolation between these two classical indices. We establish key theoretical properties, including scale invariance, boundedness, continuity, and compliance with the Pigou-Dalton transfer principle. Analytical representations are derived, allowing explicit evaluation under gamma distributions and leading to closed-form expressions involving incomplete gamma functions. From a statistical perspective, we study the plug-in estimator, obtaining a general expression for its expectation and explicit formulas for its bias under gamma populations. Simulation results indicate good finite-sample performance, with decreasing bias and mean squared error as the sample size increases. An empirical application to GDP per capita data illustrates the practical usefulness of the proposed index as a flexible tool for inequality analysis.

Predictability analysis, which focuses on perturbation growth dynamic, is a key problem in both weather and climate prediction. Among all perturbations, the conditional nonlinear optimal perturbation (CNOP) leads to maximum uncertainties in forecasts, which is fundamentally important for theoretical studies and applications. Traditionally, CNOPs are solved through iterative optimization of numerical weather prediction (NWP) systems. Their large computational demands pose significant challenges to long-term predictability analysis. In our study, using a fast and accurate Artificial intelligence (AI) model, i.e. FuXi, a low-cost optimization framework for solving 5-day tropical cyclone (TC) CNOP is developed. For the first time, CNOPs that achieve the optimal (i.e., fastest) nonlinear development of long-term TC forecast errors are solved, with their optimality and physical explainability verified. Results demonstrate that perturbations with specific spatial structures undergo significant development. In both AI and NWP models, AI-based CNOPs exhibit rapid and physically consistent error growth across diverse TC cases, faster compared to random and lagged forecast perturbations. Furthermore, sensitivity analysis reveals that far-environment systems and processes are more crucial for long-term TC forecasts. Structural analyses of the CNOP emphasizes the interactions between TC internal and external processes for rapid perturbation growth. The success derivation of AI-based CNOPs, with their rapid growth and physical explainability verified in both AI and NWP models, suggests that AI models can capture the most rapidly growing perturbation patterns and their subsequent nonlinear evolution. Thus, potential of AI models is highlighted for advancing atmospheric predictability researches, including theoretical analysis, targeting observations and ensemble forecasts.