Recent studies revealed discrepancies between observations and the predictions of the standard magnetic braking (MB). Although alternative models have been broadly discussed in neutron star binaries, they have not been systematically tested in cataclysmic variables (CVs). In this work, we investigate the performance of four MB models in CVs: the standard MB, the Convection And Rotation Boosted (CARB) model, the $τ$-boosted model, and the saturated, boosted, and disrupted (SBD) model. We find that both the CARB and $τ$-boosted models appear too strong so that it fails to reproduce the location of the period gap in CVs, indicating that they are not appropriate for CVs. Furthermore, we present a comparison between the standard MB and the SBD models. Compared with the standard model, although the SBD model can better reproduce some observational features, it also exacerbates certain discrepancies between theory and observations. We also find that different prescriptions for the convective turnover timescale have a significant impact on the results in the non-standard MBs. Finally, we discuss the impact of the SBD model on the formation and evolution of AM CVn.

The recent measurement of the branching ratio of $B^+ \to K^+ + \mathrm{inv}$ (where "inv" denotes invisible states) by the Belle~II collaboration is enhanced relative to the standard model expectation by 2.7$σ$. An older puzzle persists in measurements of the branching ratios and CP asymmetries of $B \to πK $ decays. We address these two anomalies in flavor-changing neutral current $B$ decays, with a short-lived axion-like particle (ALP) with mass close to that of the $π^0$. In the minimal model, the ALP has couplings to the photon, top quark and a heavy sterile neutrino $ν_N$. The ALP contributes to the $B \to π^0 K$ decays by mixing with the $π^0$. It contributes to $B^+ \to K^+ + \mathrm{inv}$ by its off-shell coupling to sterile neutrino pairs. We make predictions for all $B \to K^{(*)} + \mathrm{inv}$ modes and for the rare kaon decays $K^+ \to π^+ + \mathrm{inv}$ and $K_L \to π^0 + \mathrm{inv}$. We find an appreciable contribution to the magnetic moment of the muon, and negligible contributions to the magnetic moment of the electron and $b \to s e^+ e^-$.

Quadratic regularization has emerged as a potential alternative to the popular entropic regularization in computational optimal transport, offering the theoretical advantage of producing sparse couplings through its hinge density structure. Despite recent progress in one-dimensional settings and general upper bounds, fundamental questions about the localization rate of QOT optimizers around the Monge coupling have remained open. In this work, we establish a general lower bound showing that the support of the QOT optimizer cannot concentrate around the Monge graph faster than order $\varepsilon^{\frac{1}{d+2}}$ in the directed Hausdorff distance, matching the conjectured optimal exponent under standard regularity assumptions in \citet{wiesel2025sparsity}. We also show that the QOT value gap controls the mean-squared deviation $\mathbb E_{π_\varepsilon}\|y-T(x)\|^2$ by the scale of $\varepsilon^{\frac{2}{d+2}}$. As a corollary, in the affine Brenier regime, which includes Gaussian-to-Gaussian transport, we derive a sharp pointwise tube bound of order $\varepsilon^{\frac{1}{d+2}}$ by reducing the problem to self-transport and applying recent self-transport sparsity results. Finally, we validate our theoretical bound with a synthetic experiment in high-dimensional settings.

Quantum contextuality is a concept used to describe the property of hidden-variable theory that measurement outcomes predetermined by the hidden variables depend on the measurement context. The term measurement context can have different meanings, giving rise to different flavours of quantum contextuality. The first discovered flavour is Kochen-Specker (KS) contextuality where measurement outcomes will depend on what compatible measurements are jointly performed with the selected measurement. Another flavour, here to be compared with KS contextuality, is that referred to in Bohmian mechanics where outcomes of some specific measurements are not completely specified by the model state, but depend also on specifics of the measurement device used. It has been claimed that this type of Bohmian contextuality is necessary to enable KS contextuality in a hidden variable model. In this paper we show that this is not the case. The recently proposed Contextual Ontological Model (COM) [Hindlycke and Larsson, Phys. Rev. Lett. 2022] produces KS contextual predictions but does not have the Bohmian contextuality; the outcome of every measurement allowed by COM can be predicted from the model state itself. This distinguishes Bohmian contextuality from KS contextuality, and enables individual study of the two concepts.

We present a rigorous formulation of the spin-1 Affleck--Kennedy--Lieb--Tasaki (AKLT) state within the framework of hidden quantum Markov models (HQMMs). We show that the AKLT ground state admits a natural representation as the observable output of a causal HQMM, thereby endowing it with an underlying hidden quantum memory that is fully consistent with its standard finitely correlated (matrix product state) description. This viewpoint yields a compact and structurally transparent characterization of the AKLT chain as a quantum spin system equipped with intrinsic quantum memory. Our results further indicate that the HQMM framework provides a promising setting for analyzing measurement-based quantum computation (MBQC) and related information-processing tasks.

We consider an initial boundary value problem of the space fractional Allen-Cahn equation with logarithmic Flory-Huggins potential. As an approximation technique, first-order weighted and shifted Grunwald difference formulae of the left and right fractional derivatives are used. The main focus of the present work is to study the spectral features of the underlying matrices and matrix sequences and to design proper preconditioners based on the spectral information. Then a computational and spectral analysis of the resulting preconditioned matrix sequences is performed. Numerical evidence and a short list of open problems complete the current study.

We study impurity-induced particle growth and scrambling in clean one-dimensional free-fermion systems. We show that a single local impurity can act as a branching source: particle or operator weight propagates coherently into the free bulk, returns to the impurity, and is locally converted into additional degrees of freedom. We develop this branching picture in three complementary settings: a monitored free-fermion model with feedback, a fully unitary interacting particle model, and Heisenberg operator dynamics with an interacting impurity. In the monitored model, we find a feedback-driven transition for both boundary and bulk impurities. In the unitary particle and operator models, a boundary impurity gives rise to an analogous transition from saturation to sustained growth and scrambling. These results reveal how a single impurity can generate complex many-body dynamics in an otherwise clean and free quantum system.

Polarization skyrmions are usually characterized through a unit Stokes vector field, which defines a map from the transverse plane to the Poincaré (Bloch) sphere and allows one to assign a skyrmion number under appropriate boundary conditions. This description assumes complete polarization. In many optical situations, however, the local polarization state is partially polarized and is therefore described by a spectral polarization matrix rather than by a single Jones vector. We formulate a density-matrix-based effective extension of the skyrmion-number construction for statistically stationary, quasi-monochromatic paraxial fields in terms of the normalized spectral polarization matrix. Its eigenvectors define two orthogonal polarization eigenstates, corresponding to antipodal points on the sphere, while the eigenvalue imbalance is fixed by the local degree of polarization \(P({\bf r})\). Averaging the opposite skyrmion-density contributions of these two eigenpolarizations leads to an effective skyrmion number in which the usual pure-state density is weighted by \(P({\bf r})\). The resulting quantity reduces to the conventional skyrmion number for fully polarized fields and vanishes in locally unpolarized regions. In general it is not an integer topological invariant; rather, it is an integrated effective measure, constructed from local Stokes parameters, of the skyrmionic content associated with the polarized part of a mixed optical field. We illustrate the construction with an analytic Gaussian-beam example in which a degree-one polarization texture is combined with a radially varying degree of polarization.

We introduce a halo solving and tracking procedure that intrinsically treats dark matter halos as non-spherical objects by leveraging the bound particle searching techniques used in Haskap Pie. The AGORA Collaboration's hydrodynamic simulation CosmoRun}project provides a useful laboratory to explore trends in dark matter halo morphology that are revealed by our new procedure in the context of any dispersions or similarities between the codes. We find that several morphological and shape measures were very responsive to high mass ratio mergers. The greatest difference in these measures between the simulation codes were related to timing discrepancies and the dynamical state of the halos prior to the mergers. Most other quantities were similar across codes, including several secular and redshift-dependent trends in various dynamical quantities that showed a departure from Virial Theorem (e.g., overdensity and halo mass). We find that halo spin and the ratio between the semi-major and the semi-minor axis peaked at 4>z>2 before declining at low redshift. Also, halo overdensity is both mass-dependent and redshift-dependent, diverging for low mass halos at low redshift. Our method contributes a new perspective on these trends that have not been fully replicated in other works due to our emphasis on fundamentally non-spherical halos and measures of morphology that correspondingly do not assume spherical symmetry.

We study growth quenches, which are local quenches that may gradually destabilize a false vacuum in certain kinetic constrained quantum lattice models, such as the East-West model. We point out a formal analogy with the dynamics of a local operator in the Heisenberg picture. Exploiting this analogy, we obtain several results on growth quenches by adapting operator-dynamics concepts and methods. First, applying the Krylov approach (recursion method), we conjecture the linear growth of Lanzcos coefficients in generic quenches, $a_m \sim νm$ (diagonal), and $b_m \sim αm$ (off-diagonal), extending an operator growth hypothesis. We show that the growth quench dynamics is localized in both Krylov and Fock spaces when $|ν| > 2 α$, and derive a bound for the growth quench analogue of Lyapunov exponent $λ_L \le \sqrt{4 α^2 - ν^2}$ when $|ν| < 2 α$. Second, we realize the Fock localization in large $N$ solvable growth quenches inspired by Sachdev-Ye-Kitaev (SYK) models. The bound on Lyapunov exponent is saturated in large-$q$ SYK grow quench. By contrast, the growth quench is almost always Fock localized in a nonrandom all-to-all growth quench amenable to semiclassics. Finally, in the 1D East-West model, we interpret Fock space cage states as the existence of a conserved charge. We show that the latter has ballistic transport due to current conservation. Moreover, adding hopping with a fine-tuned amplitude induces a partial localization due to a flat band. Our work suggest growth quenches as a promising approach to realize non-equilibrium coherent phenomena in many-body systems.

Stray radiation is a considerable challenge for radio telescopes, requiring careful assessment due to its effects. This is crucial when the strong background flux from side lobes significantly affects the total flux, especially for extended sources. In this study, we introduced the beam pattern of the L-band receiver on the Five-hundred-meter Aperture Spherical Telescope (FAST), covering various frequencies based on recent observations. We discovered that the main beam efficiency of all beams exceeds 90\% throughout the L band frequencies, with efficiency decreasing slowly as frequency increases. Subsequently, we developed a module to mitigate stray radiation effects, incorporating it into FAST's standard \HI data reduction process, referred to as \texttt{HiFAST}. Our analysis shows that side lobe flux's influence, particularly for extended sources with significant surface density gradients, necessitates detailed evaluation. Corrections for the extended M33 galaxy can reach up to 20\%. Moreover, the pattern data presented here is vital for studying HI intensity maps at high redshift. The module, along with HiFAST and beam pattern data across 15 frequency bins, can be accessed at \textrm{https://hifast.readthedocs.io}. The datasets of beam pattern presented in this paper, are openly available at \textrm{https://doi.org/10.57760/sciencedb.j00113.00266} (https://www.scidb.cn/s/bqQRNv).

Recently, the Lobster Eye Imager for Astronomy (LEIA) detected the longest-lasting and most energetic stellar X-ray flare event from HD 251108. In this work, we re-determined the atmospheric parameters of HD 251108 using three spectroscopic observations obtained with the 2.4 m Lijiang Telescope. Combined with the stellar radius derived from spectral energy distribution fitting, we found that HD 251108 contains a K-type giant with a mass of approximately 1.3 $M_{\odot}$. Long-term photometric monitoring over 12 years reveals a modulation suggestive of a stellar activity cycle, but inconclusive given the limited time span to date. Light curve fitting indicates that the variations in both amplitude and shape are primarily driven by the evolution of a large spot. The fitting further indicates that the spot migrated from low latitudes toward the pole between 2014 and 2020, and began to recede from the pole after 2022. Using spot parameters from light curve fitting, we found that the observed radial velocity variations arise from both the spot-induced distortions and the Keplerian orbital motion of the giant star. Additionally, we detect a possible M-dwarf companion with a mass of approximately 0.25 $M_{\odot}$. Our finding suggests a notable effect on the radial velocity caused by stellar magnetic activity.

Large collections of tabular data from data lakes, web tables and open data portals often originate from heterogeneous sources, leading to representational inconsistencies. Understanding and organizing such repositories therefore remains a major challenge. While prior work has primarily focused on dataset discovery and exploration, this paper addresses the complementary problem of conceptual schema inference: automatically deriving a conceptual schema that captures entity types, attributes and inter-type relationships directly from raw tables. We propose two large language model (LLM)-based approaches that use only column headers and cell values: GeSI uses generative LLMs to infer hierarchical types and their attributes from table- and column-level semantics, and to integrate them into a global schema that also captures relationships across types; EmSI employs LLM-based table embeddings to group tables by column-level semantics, infer attributes within each group, and construct hierarchical structures from shared attribute patterns. Finally, we report an experimental analysis demonstrating the effectiveness of our approaches in terms of the conciseness and structural quality of the inferred schema components, their scalability to large repositories, and a case study illustrating end-to-end schema inference.

Precision measurements in storage rings are increasingly limited by the ability to monitor collective spin dynamics coherently over long time scales. Existing polarimetry techniques rely on destructive scattering processes that preclude continuous, non-intercepting tracking of spin evolution and constrain both statistical sensitivity and systematic control. Here we introduce a non-destructive, phase-coherent polarimetry method in which the stored beam polarization is treated as a continuous dynamical observable rather than a quantity inferred from scattering events. Spin-dependent electromagnetic fields generated by a polarized relativistic beam establish a symmetry-selected differential signal on pickup electrodes. This signal is transduced into a narrowband phase modulation of a high-Q resonator interrogated with a coherent probe, while dominant charge-induced backgrounds are rejected through geometric symmetry, helicity reversal, and synchronous demodulation. Controlled spin precession (spin-wheel operation) provides a stable phase reference enabling phase-coherent detection of slow spin evolution. Combined with optimized lattice symmetry and beam cooling, this approach can substantially extend the usable spin coherence time, with values approaching 10^5 s appearing realistic within existing accelerator technology. The resulting readout supports optimal slope-based estimation with T^{-3/2} statistical scaling while eliminating the efficiency penalties inherent to scattering-based polarimetry. For storage-ring EDM experiments, this combination enables sensitivity approaching the level expected within the Standard Model. More broadly, the method establishes a general phase-coherent architecture for collective spin measurements in storage rings, adapting resonant sensing concepts from axion dark-matter searches to charged-particle precision experiments.

We study the dynamics of a binary orbiting a supermassive black hole (SMBH), focusing on both binary scattering in unbound orbits and eccentric von Zeipel-Lidov-Kozai (vZLK) oscillations in bound orbits. The motion is described in a local inertial frame in Kerr spacetime, where tidal effects are encoded in the Riemann curvature. For unbound (parabolic and hyperbolic) orbits, we identify four scattering regimes-adiabatic, tidally affected, chaotic, and disruptive-depending on the binary semi-major axis. As the binary becomes softer, tidal interactions near periapsis lead to strong eccentricity excitation, large changes in the orbital parameters, and eventually chaotic behavior or tidal disruption, with a sensitive dependence on the argument of periapsis. For eccentric bound (elliptic) orbits, the vZLK mechanism differs qualitatively from the standard one, although the $z$-component of the angular momentum in the local inertial frame remains approximately conserved. The evolution proceeds on a dynamical timescale and exhibits step-like changes driven by repeated periapsis passages, which can be interpreted as a sequence of scattering events. We refer to this behavior as scattering-type vZLK oscillations. The rotation of the SMBH also modifies the oscillation profiles, although its effect is less significant than the dependence on the initial orbital parameters. These results suggest a unified picture of periapsis-driven tidal dynamics in galactic nuclei.

Real-time 3D volumetric streaming is a transformative technology that enables the seamless transmission and rendering of high-fidelity 3D models, enhancing applications in virtual reality (VR), augmented reality (AR), gaming, telepresence, and remote collaboration. However, latency remains a major challenge, affecting immersion, causing motion sickness, and disrupting real-time interactions. Addressing these latency issues is essential for improving user experience and ensuring system efficiency. This study conducts a comprehensive latency measurement and analysis within a real-time volumetric streaming environment. We systematically break down the streaming process into three key layers: the application layer, the transport protocol layer, and the network layer. By evaluating each layer in a real-world system, we identify latency bottlenecks, quantify their impact, and uncover the underlying causes of delay. Based on these findings, we propose targeted optimization strategies to mitigate latency and enhance system responsiveness. Through this research, we establish best practices and innovative solutions to improve the efficiency, scalability, and overall user experience of real-time 3D volumetric streaming. Our insights contribute to advancing the field, paving the way for more immersive and responsive digital environments.

We present a review on quantum metrology and sensing, from its foundations to current applications. Highlights of the review include consideration of both frequentist and Bayesian approaches to parameter estimation; single as well as multiparameter estimation; estimation for different encoding processes comprising unitary as well as noisy channels, quantum thermometry, and channels involving indefinite causal order; different estimation strategies incorporating also recent advances like quantum error correction-aided methods and reservoir engineering; usefulness of quantum Fisher information to detect resources; applications of quantum metrology in diverse arenas covering quantum many-body sensors, sensing protocols in atomic ensembles, atom-photon systems, and continuous-variable systems, quantum imaging, quantum illumination, atomic clocks and atom interferometry, etc; and experimental realizations of quantum sensors in different physical platforms.

We initiate the study of gravitational-wave lensing in the wave-optics regime within modified gravity. We consider a phenomenological setup in which the gravitational-wave amplitude obeys a curvature-coupled propagation equation. This framework reproduces the standard GR behaviour in the geometric-optics regime, while leading to qualitatively different infrared dynamics. In particular, the usual argument implying that the amplification factor approaches unity in the zero-frequency limit no longer applies. This is due to the persistence of curvature-induced interactions in the infrared, which modify the natural propagation basis itself. As a result, the standard Fresnel treatment ceases to be valid at sufficiently low frequency. The correct infrared regime is instead controlled by an interacting static Green function, with a finite-frequency completion provided by a partial-wave formulation. We show that this structure admits an equivalent distorted-wave interpretation, in which the curvature interaction is absorbed into a dressed reference propagation basis, while the residual lensing effect is encoded in finite-frequency phase shifts. We further demonstrate that these phenomena admit a natural interpretation in the language of scattering amplitudes. Wave-optics lensing can therefore probe propagation-level departures from GR that remain entirely invisible in geometric optics.

We introduce the Compression-Directional Entropic Stress (CoDeS) method inspired by information geometric regularization. CoDeS replaces scalar multidimensional entropic pressure with a tensor stress aligned with the principal directions of compression. The stress has the form $\boldsymbolΠ_Σ=σ\boldsymbol{M}$, where $σ$ is obtained from a modified-Helmholtz equation and $\boldsymbol{M}$ is constructed from the compressive eigenspace of the symmetric velocity-gradient tensor. The source is gated by volumetric and principal-strain compression, so the regularization vanishes in smooth expansion, rigid-body rotation, and ideal contacts, while recovering the compressive one-dimensional IGR mechanism at planar shocks. The same tensor stress is used in the conservative momentum flux and the stress-work energy flux. CoDeS is tested on one-, two-, and three-dimensional problems including smooth expansion, double rarefaction, the Sod shock tube, multidimensional Riemann flow, a viscous shock tube, a two-fluid triple point, a Mach-3 slot jet, and a supersonic Taylor--Green vortex. The results show that CoDeS remains inactive in expansive and contact regions, supplies localized stress at shocks, and concentrates regularization along compressive wave structures while remaining weak in shear- and vorticity-dominated regions. At matched resolutions, the three-dimensional Taylor--Green results are comparable to or more energetic than seventh-order WENO/TENO references. These results indicate that CoDeS provides a compression-selective shock regularization compatible with high-order finite-volume resolution of contacts, interfaces, shear layers, and vortical structures. All the code, case settings, and code for plotting figures of this paper are available at https://github.com/xubonan/code\_for\_CoDeS.

Many digital systems are designed as collections of asynchronous processes orchestrated by a domain-specific scheduler. The verification of such scheduler-restricted asynchronous systems (SRA) is challenging due to process-process and process-scheduler interactions. In this paper, we tackle the problem of verifying configurable SRA. A configurable SRA describes an unbounded family of possible SRA, each resulting from an instantiation satisfying given configuration constraints; our goal is proving at once that every legal instantiation of a configurable SRA is correct. We propose a contract-based, deductive verification approach that combines (i) compositional proof rules that abstract the scheduler to prove top-level invariant properties, (ii) automatic summarizations of the methods invoked by the scheduler, (iii) simplification with respect to the nature of the space of configurations. The approach is grounded in (object-oriented) first order logic, requires reasoning over quantified statements, and leverages the Dafny software verifier as a backend. An experimental evaluation on industrial case studies demonstrates that the framework scales effectively and enables practical reasoning about complex parameterized behaviors.

For a weighted tree, the Lin--Lu--Yau Ricci curvature admits an explicit formula in terms of the edge weights. Consequently, the constant-curvature equation is equivalent to an eigenvalue problem for an edge-indexed Ricci matrix $R_T$. Building on the spectral characterization of discrete Einstein metrics on trees, we classify all finite trees whose discrete Einstein metric has positive curvature, equivalently all trees satisfying $λ_{\max}(R_T)<0$. For caterpillars with spine order $m\ge 12$, this occurs precisely for the endpoint families $T_m(a,0,\ldots,0,b)$ with $1\le a,b\le 3$ and $(a,b)\ne(3,3)$. The remaining cases $3\le m\le 11$ are settled by an exact finite verification using rational characteristic polynomials and Sturm root counts. We also determine the zero level set $λ_{\max}(R_T)=0$: among caterpillars, it consists of the stable family $(3,0,\ldots,0,3)$ together with nine exceptional short-spine caterpillars, while $S_3^2$ is the unique non-caterpillar zero example.

Machine Learning Interatomic Potentials (MLIPs) achieve near ab initio accuracy at a fraction of the cost of quantum-mechanical simulations, yet they remain prone to silent failures on out-of-distribution configurations, making principled uncertainty quantification (UQ) essential for error-aware simulations and active learning. Existing non-ensemble UQ methods for MLIPs rely either on variational inference or on parametric distributional assumptions, both of which add architectural complexity and hyper-parameters that must be tuned per task. Inspired by recent advances in probabilistic weather forecasting, we propose a simpler alternative: turn a deterministic MLIP into a probabilistic one through learned functional perturbations and finetune it end-to-end with the Continuous Ranked Probability Score (CRPS), a proper scoring rule. We validate the approach with an equivariant GNN (P-EGNN) trained from scratch and by finetuning the foundation model the Orb-v3 for silica. On the N-body charged particle benchmark, P-EGNN improves CRPS over the state-of-the-art Bayesian MLIP method BLIP by 19-32% across all training sizes; on silica, P-Orb raises the Spearman correlation between predicted uncertainty and actual error from 0.75 (BLIP-Orb) to 0.84.

Contract theory typically assumes full commitment by the principal, but many contracts fix some payoff-relevant decisions while leaving others discretionary. We ask when imperfect commitment is equivalent to full commitment. For contracts in which a committed baseline is followed by a bounded discretionary adjustment, as in commercial-insurance schedule rating or civil penalties, bounded discretion is allocation-neutral. When contractible and non-contractible decisions are distinct instruments, the equivalence fails. We characterize optimal single-principal contracts and show that simple-offer equilibria are robust under competing principals. The methodological contribution is an extended taxation principle that makes these analyses more tractable.

Automated Market Makers face a geometric dilemma: expanding liquidity depth to reduce execution slippage increases Liquidity Providers' exposure to toxic arbitrage, quantified as Loss-Versus-Rebalancing (LVR). We study the Hybrid Liquidity-Collateral Pool (HLCP), a stylized architecture that aims to partially decouple execution quality from active risk exposure through an N-scaled virtual invariant and a collateral buffer. The analysis first characterizes the geometric divergence between execution slippage and marginal-price deviation, then uses this divergence to motivate a trigger-based collateral injection rule. In a stylized duopoly model, under hyper-saturated background liquidity and non-zero volatility or collateral yield, adopting the HLCP is a Nash equilibrium and Pareto-improving relative to a standard AMM benchmark. Empirically, we examine two settings. Under a stochastic-volatility-with-jumps stress scenario, the trigger policy avoids one-shot total buffer depletion under the imposed control law and simulated shock path. Using 2025 Uniswap V2 data with zero collateral yield, the HLCP exhibits lower realized LVR and higher net LP return than the standard CPMM benchmark in the sample considered.

Chemistry's rules carry exceptions: the octet rule, Hess's Law, detailed balance, orbital symmetry selection rules, all with disclaimers memorised separately. Their cause: a question from a richer structural level posed in the vocabulary of a simpler one, i.e. level incompleteness. This monograph makes the levels explicit, constructing a canonical tower of nine categorical levels from stoichiometry through thermochemistry, equilibrium, kinetics, electron-pushing mechanisms, stereochemistry, potential energy surfaces, and electronic structure to all-particle quantum mechanics. Each level emerges from pairs of reactions distinct yet indistinguishable at the previous level; the minimal extension resolving each ambiguity is provably unique, certified by a non-trivial cokernel in an automorphism exact sequence, and recovers Feinberg's deficiency theorems as homological corollaries. A perpendicular dimension: every ML model for chemistry (yield predictors, neural kinetic networks, equivariant force fields, learned wavefunctions) is a morphism in the Para-enrichment of one tower level, with equivariance and thermodynamic consistency as universal properties. Three incompleteness results (Eyring, Wegscheider, topological output gaps) apply to the current literature. The framework descends to code: an operational functor from a Para-enriched product of the first four levels into the Kleisli category of the probabilistic sub-monad of Haskell IO, instantiated as a simulator of the Briggs-Rauscher oscillating reaction: the first Kleisli semantics of Gillespie's next-reaction method and first Para application outside ML. The passage to all-particle quantum mechanics, Born-Oppenheimer as the classical limit of a continuous field of C*-algebras, remains the deepest open construction; four candidate conjectures including Woolley-Primas have obstructions the framework makes specific.

Large language model (LLM)-based agents that reason, plan, and act through tools, memory, and structured interaction are emerging as a promising paradigm for automating complex workflows. Recent systems such as OpenClaw and Claude Code exemplify a broader shift from passive response generation to action-oriented task execution. Yet as agents move toward open-ended, real-world deployment, relying on from-scratch reasoning and low-level tool calls for every task become increasingly inefficient, error-prone, and hard to maintain. This survey examines this challenge through the lens of \emph{agent skills}, which we define as reusable procedural artifacts that coordinate tools, memory, and runtime context under task-specific constraints. Under this view, agents and skills play complementary roles: agents handle high-level reasoning and planning, while skills form the operational layer that enables reliable, reusable, and composable execution. Skills are therefore central to the scalability, robustness, and maintainability of modern agent systems. We organize the literature around four stages of the agent skill lifecycle -- representation, acquisition, retrieval, and evolution -- and review representative methods, ecosystem resources, and application settings across each stage. We conclude by discussing open challenges in quality control, interoperability, safe updating, and long-term capability management. All related resources, including research papers, open-source data, and projects, are collected for the community in \textcolor{blue}{https://github.com/JayLZhou/Awesome-Agent-Skills}.

We introduce the Ceiling Continued Fractions (FCT) framework for constructing three-term Egyptian fraction representations in the Erdős-Straus conjecture. The approach exploits divisor structures of shifted integers p+i rather than congruence-based techniques. We derive a super-polynomial upper bound on the failure probability; its convergence, together with the Borel-Cantelli lemma, provides heuristic evidence that counterexamples, if any exist, form a finite set. Computational tests on 10^9 primes in ranges around 10^17, 10^52, and 10^131, show no counterexamples with very small search depth.

We present VLQBounds, a public, data-driven Python framework for testing Vector-Like Quark (VLQ) scenarios against Large Hadron Collider (LHC) exclusion limits from ATLAS and CMS. The framework incorporates public results on both pair and single VLQ production and supports the main parameterisations used in experimental interpretations, including mass-mixing, mass-coupling, and mass-width representations. For each parameter point, the predicted cross-section or effective coupling is compared channel by channel to the corresponding observed and expected experimental limits through interpolation over machine-readable grids. The most sensitive analysis is automatically identified and a 95\% Confidence-Level exclusion verdict is returned, together with the observed and expected sensitivity ratios and the metadata needed for reproducible reinterpretation. The modular structure of VLQBounds makes it suitable for fast phenomenological scans, validation of public limits, and future extensions to new collider searches and non-minimal VLQ decay patterns.

Recent occurrence-rate analyses have shown that the transition between the Neptunian desert and the savanna is not smooth but instead exhibits an overdensity of planets at $P_{\rm orb}\simeq3$-$6$ d, known as the Neptunian ridge. We confronted the high-eccentricity tidal migration (HEM) scenario with this updated desert-ridge-savanna landscape. We mapped the HEM tidal survival constraints onto the period-radius plane using empirically inferred mass-radius relations and provided an independent consistency check in the period-density plane. The HEM tidal survival formalism reproduces the slope of the desert boundary across the sub-Neptune to super-Neptune/sub-Saturn regime ($1.8\,\rm R_\oplus \lesssim R_{\rm p} \lesssim 6\,\rm R_\oplus$), with a single representative tidal encounter parameter setting the overall period offset. In the Jovian regime, the boundary remains broadly consistent with the survival limit, with residual deviations likely due to radius inflation or orbital decay. Incorporating the observed density dispersion transforms the disruption limit into a finite tidal survival band that traces the ridge. Because tidal dissipation rises steeply towards the disruption threshold, HEM survivors are expected to circularise just beyond this limit, clustering within the band and naturally producing the ridge overdensity. In the period-density plane, the population follows the predicted density-dependent survival and clustering pattern, with a persistent concentration of ridge planets near $ρ_{\rm p}\simeq1.7\,\mathrm{g\,cm^{-3}}$. High-eccentricity tidal migration thus provides a self-consistent explanation for the ridge and desert boundary geometry.

This paper lays out the foundations of graded $K$-theory for Leavitt algebras associated with higher-rank graphs, also known as Kumjian-Pask algebras, establishing it as a potential tool for their classification. For a row-finite $k$-graph $Λ$ without sources, we show that there exists a $\mathbb{Z}[\mathbb{Z}^k]$-module isomorphism between the graded zeroth (integral) homology $H_0^{gr}(\mathcal{G}_Λ)$ of the infinite path groupoid $\mathcal{G}_Λ$ and the graded Grothendieck group $K_0^{gr}(KP_\mathsf{k}(Λ))$ of the Kumjian-Pask algebra $KP_\mathsf{k}(Λ)$, which respects the positive cones (i.e., the talented monoids). We demonstrate that the $k$-graph moves of in-splitting and sink deletion defined by Eckhardt et al. (Canad. J. Math. 2022) preserve the graded $K$-theory of associated Kumjian-Pask algebras and produce algebras which are graded Morita equivalent, thus providing evidence that graded $K$-theory may be an effective invariant for classifying certain Kumjian-Pask algebras. We also determine a natural sufficient condition regarding the fullness of the graded Grothendieck group functor. More precisely, for two row-finite $k$-graphs $Λ$ and $Ω$ without sources and with finite object sets, we obtain a sufficient criterion for lifting a pointed order-preserving $\mathbb{Z}[\mathbb{Z}^k]$-module homomorphism between $K_0^{gr}(KP_\mathsf{k}(Λ))$ and $K_0^{gr}(KP_\mathsf{k}(Ω))$ to a unital graded ring homomorphism between $KP_\mathsf{k}(Λ)$ and $KP_\mathsf{k}(Ω)$. For this we adopt, in the setting of $k$-graphs, the bridging bimodule technique recently introduced by Abrams, Ruiz and Tomforde (Algebr. Represent. Theory 2024).