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arXiv2026

A new lower bound for the sixth Schur number: $S(6)\ge 537$

We give an explicit partition of $\{1,\ldots,537\}$ into six sum-free sets, establishing $S(6)\ge 537$. This improves the lower bound 536 of Fredricksen and Sweet (2000). Known recursive constructions then give the improved lower bounds $S(8)\ge 5\,372$, $S(11)\ge 204\,208$, $S(13)\ge 2\,042\,082$, and $R_6(3)\ge 539$. We also report 131 pairwise inequivalent partitions found by the search. We describe their structure and provide the complete partitions in the supplementary material. These partitions provide starting points for future searches and structural studies of Schur partitions.

math.CO↗
arXiv2026

A Sharp Analysis of Fixed Offset Hash Tables

We consider the class of open addressing hash tables that uses a fixed offset sequence $r_0,r_1,\ldots,r_{n-1}$ to find an open slot in the table. As special cases this subsumes linear probing ($r_i=i$) and quadratic probing ($r_i=i^2$). We give a simple analysis that the expected insertion time when the load factor is $1-ε$ is $O(ε^{-2})$. This bound is tight for linear probing (but not for quadratic probing), so the analysis cannot be improved at this level of generality.

cs.DS↗
arXiv2026

A second-order linear decoupled scheme with a relaxed zero-energy variable for Rayleigh-Bénard convection

We construct a second-order, linear, and fully decoupled scheme for Rayleigh--Bénard convection between Navier-slip walls. The nonlinear terms are treated explicitly and multiplied by a scalar with exact value $1$ and no energy contributed to the continuous problem. As a result, each BDF2 step, with a rotational pressure correction, solves only constant-coefficient linear problems. For the time discretization, we prove that, for any time-step, the energy decays when the walls have equal temperatures, under a condition on the thermal weight, and the energy is bounded on finite time intervals when heated from below. Relaxing the scalar towards $1$ preserves both estimates. Energy stability does not control the accuracy of the scalar; the velocity and temperature fields are computed with its unrelaxed value, whose deviation from $1$ scales like $(E/C)(δt/τ)^3$ for kinetic energy $E$, time-scale $τ$ of the flow, and weight $C$ of the scalar equation. In our tests, choosing $C$ comparable to $E$ keeps this deviation small and doubles the largest time-step for which steady rolls are reproduced to a relative difference of $10^{-8}$. With Taylor--Hood elements, we observe second-order convergence in time, recover the critical Rayleigh numbers, and match published Nusselt numbers to within $0.5\%$. A three-dimensional implementation passes corresponding validation tests.

physics.flu-dyn↗
arXiv2026

FRB 20210601A: Extreme Temporal Clustering in a Repeating Fast Radio Burst Source as Evidence for a Hidden Class of Episodic Repeaters

Population studies of fast radio burst (FRB) repetition typically model sources using a single distribution of steady, time-averaged burst rates, yet individual sources show significant departures from this assumption. Here we present FRB 20210601A, a source discovered by the Canadian Hydrogen Intensity Mapping Experiment FRB project (CHIME/FRB) that represents the most extreme such departure yet identified. The source produced at least nine bursts within a single 106.5 s interval -- the highest instantaneous burst rate yet reported from a repeating FRB source at $\gtrsim 2$ Jy ms fluences -- yet remained otherwise undetected despite $\sim 320$ hr of exposure spanning 7 yr. The active interval occurred while the source was in a side-lobe of the CHIME/FRB digitized beams, implying an instantaneous burst rate $\gtrsim2.3\times$ higher ($\gtrsim$ 21 bursts) at main-lobe sensitivity. The source's intense, brief burst storm flanked by apparent quiescence despite substantial exposure is unique among the CHIME/FRB repeater sample, although its burst morphology, excess dispersion measure ($\rm{DM_{excess}} \sim 360$ pc cm$^{-3}$), Faraday rotation measure ($|\rm{RM_{excess}}| \sim 150-170$ rad m$^{-2}$), and temporal scattering ($τ\lesssim 0.37$ ms at 600 MHz) are otherwise typical. Modeling a population of episodically active repeaters (i.e., short burst storms separated by long recurrence times) and accounting for declination-dependent exposure, we forecast $\sim1-4$ additional source discoveries similar to FRB 20210601A over the next decade of CHIME/FRB operations, corresponding to a CHIME/FRB-detectable population of $9^{+200}_{-8}$ sources (68% credible interval). These results demonstrate that highly episodic repeaters could constitute a selection-biased but detectable component of the FRB population, possibly reflecting distinct magnetospheric states or triggering mechanisms.

astro-ph.HE↗
arXiv2026

AfterDeleteBench: Auditing Post-Deletion Information Persistence in LLM Agents

Many LLM agents maintain external persistent memory and derive additional state from past interactions, such as summaries. Deleting an original memory record therefore need not remove information carried by its descendants. We introduce AfterDeleteBench, a controlled behavioral audit for this failure mode in fixed-weight LLM agents. For each synthetic secret, we compare an exposed-then-deleted agent with an otherwise matched agent that was never exposed to the secret, allowing post-deletion recovery to be attributed to the original exposure. In our primary setting, a single query recovers the deleted secret from Qwen2.5-7B-Instruct in roughly two thirds of cases, while matched never-exposed controls have zero recovery. Crucially, recovery does not require a surviving verbatim copy: when the complete secret string is absent and only separately stored components remain under a public composition rule, exact reconstruction succeeds in 62-99% of cases across Qwen, Mistral, and Phi-3. We also show that the response interface is part of the measurement system: target-bearing replies can be scored as failures when the requested format and parser are misaligned. Finally, we separate reconstruction from write-back and fresh-session recall. Provenance-aware closure eliminates observed recovery in the tested positive-baseline settings, whereas an exact-literal write filter blocks reinsertion but cannot erase information already present. Together, these results show that source deletion should be evaluated behaviorally at the level of the agent and its memory system, not only as a storage operation.

cs.CR↗
arXiv2026

Synthetic 2D magnetic turbulence with controlled island structuring

Magnetic turbulence remains up to this day not fully understood, despite its important role in the behavior of plasmas. A recurring ingredient in plasma turbulence is the spontaneous formation and evolution of plasmoids. These represent magnetic islands on current sheets in two-dimensional turbulent magnetic fields that are consistently found in high-resolution direct numerical simulations (DNSs) using a magnetohydrodynamic description. However, such simulations require considerable amounts of computational power. Synthetic turbulence models provide an alternative, cheap approach to the generation of fully-developed turbulent fields. In this work, we first present an extension of our open-source, general synthetic turbulence BxC toolkit that is specifically tailored to generate 2D turbulent magnetic fields. We then use a novel physically motivated, but purely geometric construction that generates current sheet island chains, which results in fields that visually match the ones obtained through DNSs. We show how this construction affects the magnetic field lines and find a correlation with the formation of magnetic O-points and X-points. The magnetic flux and size of the islands in the field are found to match well with theoretical scaling laws. Lastly, we show BxC's capabilities by generating a 2D $N\times N$ turbulent magnetic field of resolution $N = 100000$, going beyond the state-of-the-art reached by simulations, on a single CPU with modest computational resources. Our construction maintains direct control over the inertial range, and allows to make morphological changes to the plasmoids and fields, demonstrating plasmoid appearance, disappearance and merging.

physics.plasm-ph↗
arXiv2026

Integral representations of high-order derivatives of Bessel functions with respect to the order

We derive integral representations for the $n$-th derivative with respect to the order of Bessel functions of the first, second and third kinds, as well as the modified Bessel function and the Macdonald function. Based on these integral representations, we have developed a highly-efficient algorithm for the numerical evaluation of these derivatives with respect to the order, applicable to any complex argument and any real order.

math.CA↗
arXiv2026

Sharp stability for the Talenti inequality

We prove a sharp quantitative version of the Talenti rearrangement inequality. Specifically, we show that, for any $p\in [1;+\infty]$, for any $f\in L^\infty(Ω),\, f\geq 0,\, f\not\equiv 0$, for any open set $Ω\subset \mathbb{R}^d$ with finite volume, the solution $u_{f,Ω}$ to $-Δu_{f,Ω}=f,\, u_{f,Ω} \in W^{1,2}_0(Ω)$, satisfies, in dimension $d\geq 1$, $\Vert u_{f^\sharp,Ω^\sharp}\Vert_{L^p(Ω^\sharp)}-\Vert u_{f,Ω}\Vert_{L^p(Ω)}\gtrsim α(Ω)^2+\inf_{z\in \mathbb{R}^d}\mathscr A(f,z)^2$ and, in dimension 1, $ \Vert u_{f^\sharp,Ω^\sharp}\Vert_{L^p(Ω^\sharp)}-\Vert u_{f,Ω}\Vert_{L^p(Ω)}\gtrsim α(Ω)+\inf_{z\in \mathbb{R}}\mathscr A(f,z)^2$ where $α(Ω)$ denotes the Fraenkel asymmetry of $Ω$, $\mathscr A(f,z)$ a weighted asymmetry of $f$ with respect to some center $z$ and $Ω^\sharp$ (resp. $f^\sharp$) is the Schwarz rearrangement of $Ω$ (resp. of $f$). In dimension $d\geq 2$, this estimate implies $ \Vert u_{f^\sharp,Ω^\sharp}\Vert_{L^p(Ω^\sharp)}-\Vert u_{f,Ω}\Vert_{L^p(Ω)}\gtrsim α(Ω)^2+\inf_{z\in \mathbb{R}^d}\Vert f-f^\sharp(\cdot-z)\Vert_{L^1}^2.$ Our result sharpens and extends recent quantitative versions, and does not rely on shape differentiation arguments, but on a generalisation of the quantitative Saint-Venant inequality.

math.AP↗
arXiv2026

RIDDLE: Revealing Irregularities through Distributional Discrepancies in Latent-space via Stein-Witness Estimation

RIDDLE is a new method for resonant anomaly detection that uses a scalar Stein-witness function to identify discrepancies between unlabeled signal-region events and a background distribution inferred from dijet-mass sidebands. The witness is optimized in a latent representation of auxiliary jet observables against a differentiable background reference, without requiring an explicit signal model. The method is evaluated using simulated LHC Olympics dijet events and compared with idealized and supervised benchmarks that share the same statistical construction but have access to additional signal or background information. RIDDLE demonstrates strong signal--background discrimination comparable to the idealized benchmark, while largely preserving the background dijet-mass spectrum under anomaly-score selection. Tests incorporating additional angular information and deliberately mass-correlated features further demonstrate the method's robustness under the studied conditions. Together, these results highlight the potential of RIDDLE as a robust, data-driven approach for enhanced resonance searches that combines effective anomaly discrimination with background-shape preservation.

hep-ph↗
arXiv2026

Probabilistic Energy Modeling Unifies Versatile Swarm Control

Robotic swarms must coordinate across diverse missions, yet existing controllers often require redesign, retuning, or retraining when objectives change. Here we show that probabilistic energy modeling provides a unifying representation for versatile swarm control, in which diverse missions emerge from compositions of shared interaction principles rather than task-specific control formulations. By exploiting the correspondence between Gibbs random field energy minimization and structured reward maximization, we encode fundamental swarm behaviors as reusable energy terms within a unified probabilistic objective. Decentralized policies trained within this representation are composed into a single controller through lightweight policy gating, without retraining. An interpretable action-attention mechanism captures non-uniform local interactions, while dimensionless normalization supports transfer across swarm scales and platforms. Demonstrated on aerial swarms, the unified policy generalizes zero-shot across six mission classes, 10-100 agents, two hardware platforms, and indoor cluttered and GPS-denied woodland environments. Large-scale simulations achieve 0.96-1.00 task success rates with 1-10 ms per-agent inference latency, while physical deployments confirm zero-shot transfer to real platforms and unstructured environments. These results establish probabilistic energy modeling as a scalable and transferable framework for decentralized swarm control.

cs.RO↗
arXiv2026

RIDAR: Safeguarding Research Integrity in Autonomous Research Agents via Automated Detection and Runtime Intervention

Autonomous research agents are increasingly used to automate large portions of scientific research, yet they are prone to research integrity violations (RIVs): behaviors that compromise the validity, reproducibility, or faithfulness of the research process. We present RIDAR, a framework for automated RIV detection and runtime intervention. First, we introduce RIVT, a comprehensive taxonomy of 14 RIV types spanning the full research lifecycle. Building on RIVT, we present RIDAR-post-hoc, a framework to detect RIVs within agent trajectories. RIDAR-post-hoc achieves 85.2% F1 performance on RIV-Gold, a benchmark for RIV detection with 60 expert-annotated trajectories. Applying RIDAR-post-hoc to RIV-Wild, a dataset of 300 agent trajectories across 50 distinct research tasks, we detect 3,964 RIV instances in 99.7% of trajectories, revealing the prevalence of RIVs in current agents. Finally, we present RIDAR-runtime, a framework for dynamically detecting and correcting RIVs during agent runtime. RIDAR-runtime effectively reduces agent RIVs remaining at task completion by 36.8%. We will release our code and data to facilitate future research.

cs.AI↗
arXiv2026

Stochastic Geometry Analysis and Optimization for Multi-LEO Satellite-Assisted Communications

Low Earth orbit (LEO) satellites are vital in providing seamless connectivities to ground users (GUs). However, complex satellite-to-ground propagation and non-negligible satellite failure probabilities make single-satellite transmission unreliable. To address this issue, we consider a multi-LEO-satellite communication network (mLEO-CN), where a typical GU is served by multiple LEO satellites. The satellite locations are modeled as a binomial point process on a sphere and ordered according to their distances to the GU. To improve communication reliability, a serial communication scheme is adopted, in which the subsequent satellite is activated only when all preceding satellites fail to satisfy the target signal-to-noise ratio threshold. By employing the tool of stochastic geometry, we derive key performance metrics, including overall transmission success probability, average transmission rate, and end-to-end delay under the serial communication scheme by incorporating both shadowed Rician (SR) fading and satellite failure probabilities. To improve analytical tractability, we derive asymptotic expressions for the end-to-end transmission success probability in two special cases by leveraging the asymptotic expression of the SR fading model and a Gamma-distribution, and further propose a cluster-based approximation approach for the overall transmission success probability. Building on these analyses, we formulate an optimization problem to maximize the ratio of the overall transmission success probability to the end-to-end delay, and obtain its solution using a one-dimensional search approach. Numerical results validate the analysis, reveal the impact of key parameters, and identify the optimal number of serving LEO satellites under different constellation altitudes.

eess.SP↗
arXiv2026

Symmetric and antisymmetric cumulant hierarchies and Wick polynomials of operators

There has been much progress in understanding mathematical foundations of kinetic theory in recent years, in particular, for the original model of a rarefied gas of classical particles with repulsive interactions, as well as for weakly nonlinear wave evolutions related to wave turbulence. These results are mainly based on refined analysis of evolution hierarchies of moments of the relevant random variables, or of related quantities such as of cumulants or of correlation errors. In this work, we extend two of such basic tools, cumulants and Wick polynomials, to a setting where the random variables no longer need to commute, with the goal of handling expectations of creation and annihilation operators on a Fock space, both in the bosonic and fermionic case. We call these constructions symmetric and antisymmetric cumulants and Wick polynomials, respectively. As an application, we derive a hierarchy of evolution equations satisfied by such cumulants, usually called truncated correlation functions in this context, for a gas of bosons or fermions hopping on a finite lattice and interacting via a two-body potential. With sufficient control over solutions to these hierarchies, one could control the magnitude of the error terms related to an approximation of the state of the quantum system by a finite lattice kinetic equation studied in our earlier work, closely connected to standard bosonic and fermionic Boltzmann--Nordheim equations.

math-ph↗
arXiv2026

Adaptive Local Proper Orthogonal Decomposition via Graph Partitioning and Provable Error-Bounded Lossy Compression for Parametric Series in Finite-Dimensional Function Spaces

We introduce a spatially adaptive proper orthogonal decomposition (POD) method for error-bounded lossy compression of snapshot data in finite-dimensional function spaces. The method combines hierarchical graph partitioning with independent low-rank approximations on spatial subdomains, allowing the local approximation ranks to adapt to the spatial distribution of information in the data. For partitions that are orthogonal in the discrete inner product, we establish an exact decomposition of the global squared reconstruction error into local contributions and characterize the optimality of the resulting local POD approximations for fixed partitions and ranks. These results provide a rigorous mechanism for allocating a prescribed global error tolerance among subdomains and selecting the smallest admissible local ranks. We further develop a sufficient error bound for the coefficient-space normalized root-mean-square error (NRMSE). A greedy hierarchical refinement algorithm accepts a spatial subdivision only when it reduces the storage required by the local low-rank factors while preserving the prescribed reconstruction-error bound. The method is assessed using synthetic data and snapshot sequences from two- and three-dimensional compressible-flow simulations. The numerical results illustrate how spatially adaptive ranks and partitioning can improve compression relative to global POD and fixed spatial partitions, particularly when the data exhibit spatially heterogeneous dynamics.

math.NA↗
arXiv2026

Certified Execution Harness for Agentic AI: Safe and Least-Intrusive Petri-Net Supervisory Control Between Reasoning and Action

This paper studies the problem of safe tool execution for large language model (LLM) agents under prompt injection. Using Petri nets, we formalize the execution harness between the LLM and its tools as a supervisory control problem, in which marks represent capabilities, approvals and budgets. Since a transition fires only if the marks it needs are present, an injected request whose preconditions are unmet or already spent is refused regardless of its wording. Treating tool outcomes as uncontrollable, we enforce safety by the conventional maximally permissive supervisor, and this safety transfers to the running system under four assumptions. Moreover, to capture an agent that may propose again after a refusal, we introduce the notion of a replanning closed loop, in which each retry costs an inference call. We show that, in this loop, the conventional supervisor can be arbitrarily more intrusive than a stricter safe one. Furthermore, for a given proposer model, the maximally permissive least-intrusive supervisor is characterized as a value-threshold refinement of the conventional one. Under precise conditions, this refinement orders two concurrent actions by at most one monitor place. On an autonomous release agent, we find by synthesis that safe deployment requires a step that the supervisor can refuse after a rollout starts.

cs.AI↗
arXiv2026

Measurements of the branching fractions of $Λ_c^+\toΛK^+ π^0$ and $Λ_c^+\toΛK^+ η$ decays at Belle and Belle II

We present a study of the three-body, singly Cabibbo-suppressed decays $Λ_c^+ \to ΛK^+ π^0$ and $Λ_c^+ \to ΛK^+ η$ using data recorded with the Belle and Belle II detectors at the KEKB and SuperKEKB $e^+ e^-$ colliders. The data samples correspond to integrated luminosities of 980$\,\mathrm{fb}^{-1}$ and 570$\,\mathrm{fb}^{-1}$, respectively. We measure their branching-fraction ratios relative to the Cabibbo-favored decay $Λ_c^+ \to Λπ^+ π^0$ as $\mathcal{B}(Λ_c^+ \to ΛK^+ π^0)/\mathcal{B}(Λ_c^+ \to Λπ^+ π^0)=(1.82 \pm 0.07 \pm 0.07)\%$, with significantly improved precision compared to the previous measurement, and $\mathcal{B}(Λ_c^+ \to ΛK^+ η)/\mathcal{B}(Λ_c^+ \to Λπ^+ π^0)=(8.73 \pm 0.35 \pm 0.43)\times10^{-3}$, corresponding to the first observation of the $Λ_c^+ \to ΛK^+ η$ decay with a statistical significance exceeding 10 standard deviations. The first and second uncertainties are statistical and systematic, respectively. Using the world-average value of the branching fraction for $Λ_c^+ \to Λπ^+ π^0$, we obtain the branching fractions $\mathcal{B}(Λ_c^+ \to ΛK^+ π^0)=(1.29\pm 0.05\pm 0.05\pm 0.06)\times10^{-3}$ and $\mathcal{B}(Λ_c^+ \to ΛK^+ η)=(6.20\pm 0.25\pm 0.31\pm 0.30)\times10^{-4}$, where the third uncertainty arises from the uncertainty in the world-average value of $\mathcal{B}(Λ_c^+ \to Λπ^+ π^0)$. The measured $\mathcal{B}(Λ_c^+\toΛK^+η)$ is significantly larger than predictions based on $SU(3)$ flavor symmetry.

hep-ex↗
arXiv2026

The classical two-loop amplitude at leading logarithmic accuracy for graviton emission in multi-Regge kinematics

Using the gravitational shockwave formalism, we compute the classical part of the $\mathcal{O}(G^{7/2})$ five-point amplitude at leading logarithmic accuracy describing the emission of a graviton in the collision of two massless scalars in multi-Regge kinematics (MRK). The result splits into a two-particle reducible term and a three-particle reducible one, the latter being the radiative counterpart of the $H$-diagram with the emission taken in MRK limit. The same amplitude can be equivalently calculated as the ultrarelativistic limit of the corresponding massive amplitude by applying the optical theorem according to the conjectured universality of the leading logarithmic amplitudes in classical gravity.

hep-th↗
arXiv2026

The Penrose Correspondence over a Non-Archimedean Field

Let $K$ be a non-Archimedean local field of characteristic different from $2$ and let $V$ be a four-dimensional symplectic $K$-vector space. We study the Penrose incidence correspondence $\mathbb{P}(V) \leftarrow \mathrm{Flag}(1,2;V) \rightarrow \mathrm{Gr}(2,V)$ after Berkovich analytification. The Klein quadric $\mathrm{Gr}(2,V)$ decomposes into the open orbit $\mathrm{Spin}(5)/\mathrm{Spin}(4)$ and the Lagrangian quadric, which parametrize respectively the projective lines transverse and tangent to the standard contact distribution on $\mathbb{P}(V)$. We construct explicit free discrete subgroups $Γ\subset\mathrm{PSp}(V)$ and analytic domains, cut out by Drinfeld upper half-spaces, in all three members of the incidence diagram. Both incidence projections are surjective on these domains, the three $Γ$-actions are free and properly discontinuous, and the Penrose double correspondence descends to the analytic quotients. The quotient of the twistor domain is a contact analytic threefold, and the Grassmannian domain is contained in the analytification of the $\mathrm{Spin}(5)/\mathrm{Spin}(4)$ open orbit.

math.AG↗