Papers

On 6 October 2026 OpenAI published an initial 722 AI-written mathematical manuscripts, organized into 372 problem families, in the public repository github.com/openai/math. The release includes claimed solutions to many prominent open problems. We approach this collection as a treasure mine of mathematical ideas. Our aim is to extract new lemmas, constructions, inequalities and proof mechanisms introduced in the source arguments, and to distinguish them from established tools and new combinations of those tools. We call these mathematical instruments. Extraction comes first; their validity, novelty and possible reuse then become subjects for further study. The resulting descriptions also help readers understand the selected papers without having to reconstruct every argument at the outset. We examine 38 families in two groups of 17 and 21, spanning number theory, geometry, analysis, combinatorics, complexity, algebra, operator algebras, topology and mathematical physics. Each entry records the claim, extracted instruments, antecedents and caveats. Eleven cross-cutting patterns connect the entries, and a scoreboard records formalization status and dependencies. Novelty assessments are provisional and sometimes mixed. Selected source comparisons correct mathematical statements, hypotheses and counts; they do not independently establish priority or certify the underlying proofs. No Lean development was compiled or kernel-checked for this survey. We also briefly describe a subsequent application of the signed restriction lemma from family 112 to circuit lower bounds at every fixed depth.

Existing embodied navigation tasks commonly assume that traversability and object states remain fixed during an episode. Yet reaching a goal may require opening a door to access a room and then opening a container to reveal a hidden target. We formulate interactive navigation beyond reachability, unifying channel-induced changes in spatial connectivity, container-induced changes in target visibility, and their sequential dependencies. We introduce MultINav-Bench, a 3K-episode ObjectGoal benchmark in MolmoSpaces spanning channel, container, and mixed tasks, with interaction-necessity annotations covering required and unnecessary episodes. Its data generation pipeline automates task construction across multiple task types. The evaluation protocol separately measures navigation success, interaction effectiveness, and cost. We further propose MultINav-Agent, a three-stage agentic framework built on a persistent, state-conditioned interaction graph. Semantic interactive perception grounds entities and interaction states; a vision-language model selects task-relevant actions based on expected changes in connectivity and target observability; and post-interaction verification updates the graph to guide subsequent navigation, exploration, or interaction. Experiments on MultINav-Bench show that MultINav-Agent consistently outperforms adapted baselines across channel, container, and mixed tasks.

We study privacy-preserving algorithms that redact information from a correlated dataset. The goal is to reduce the information leakage about sensitive parts. We analytically characterize the optimal tradeoff between privacy (leakage through correlation) and utility (number of redactions) for local mechanisms and binary Markovian correlations. Our result shows how to decompose the mechanism design problem into a sequence of local mechanism design problems and reveals their sequential relation. The optimal mechanism is the solution to a Finite-Horizon Deterministic Markov Decision Process.

We compute the index and nullity of any harmonic morphism from $S^3$ to $S^2$.

We construct an explicit countable group \(G_*=N(R)\rtimes A\) whose generated variety is a limit variety of groups: it has no finite identity basis, whereas every proper subvariety is finitely based. Here \(A\) is relatively free in the variety of groups of nilpotency class at most two and exponent dividing four, and \(R\) is an explicitly defined quotient of the group algebra \(\mathbb F_2[A]\). The proof establishes an ascending chain condition for coefficient spaces indexed by six ordered commutator entries and reduces every proper subvariety to a hereditarily finitely based approximation. Non-finite basability is obtained by transferring independent identities from the constructions of Gupta and Krasilnikov. The resulting variety is contained in \(\mathfrak B_4\mathfrak B_2\). This construction provides the first example of such a variety in which both the generators and a basis are explicitly given.

On many estimates, artificial general intelligence (AGI) is likely to emerge within the next few decades or even sooner. Once it does, it may quickly surpass human intelligence, possibly by a huge margin. Some believe that the country that develops AGI first could attain global dominance. This prospect risks setting off intense competition between the United States and China. Indeed, a narrative has already developed that the superpowers are locked in an AI race, with the Trump administration proclaiming its determination to 'dismantle unnecessary regulatory barriers' to winning it. We argue that this approach presents three profound dangers. The first two are familiar from past arms races: being overtaken by a rival or triggering a war. The third is new: losing control of human beings' future to agents far more intelligent than we are, which could endanger our very survival. Striving to get to AGI first risks a race to the bottom on safety practices and regulation concerning them. China does not yet appear bent on racing to superintelligence. The United States should prioritise both unilateral and co-operative efforts to improve AI safety and avoid provoking a race that all humanity could lose.

We revisit the non-thermal leptogenesis mechanism during preheating using the quantum Boltzmann equations. This framework allows us to study the production and decay of heavy right-handed neutrinos in a consistent manner. Due to their fermionic nature, right-handed neutrino production can saturate after a few inflaton oscillations when their occupation numbers approach unity. Right-handed neutrino decays can deplete their occupation numbers, allowing more particles to be produced and generating a larger lepton asymmetry. Compared with the case where right-handed neutrinos decay only after production has finished, we find that the final baryon asymmetry can be enhanced by two orders of magnitude.

Large Language Models (LLMs) are predominantly trained on imperative programming languages, and existing work on LLM-based robot programming reflects this: it focuses on goal-directed tasks expressed as plans. However, many robotic applications require not tasks but behaviors–open-ended, continuous patterns of sensory-motor engagement with no termination condition (e.g., greeting visitors, patrolling an area). Such behaviors are better served by declarative and reactive programming, on which LLMs have virtually no training. We investigate whether this gap can be bridged. Using Playful, a reactive scripting language, we show that an LLM-based coding agent can generate correct declarative and reactive programs for robotic behaviors. These programs are produced from under-specified natural language descriptions, given suitable context and a set of reusable programming guidelines. The guidelines encode design principles that redirect the agent from imperative defaults toward proper reactive patterns. Developed once by an expert, they generalize across robots and behaviors. Starting from only the generic robot operating system API, the agent jointly designs both the imperative component (Python modules for sensing, acting, and evaluating) and the declarative component (the reactive script that composes them into behavior logic). The declarative nature of reactive programming supports both incremental refinement through conversational turns and one-shot generation of complete behaviors.

We present DeboRheo v2.0, an unstructured, geometrically unsplit Volume-of-Fluid framework implemented in OpenFOAM for viscoelastic two-phase flows. Version 2.0 improves numerical robustness when large viscosity contrasts coincide with strongly deformed interfaces and thin viscoelastic lamellae or filaments. The revised interfacial viscosity treatment retains a sharp geometric interface while regularizing the viscous contribution to the momentum equation. The solver combines mass-momentum-consistent phase advection, optional SIMPLEC pressure-velocity coupling, and a modular rheological framework with conformation-tensor stabilization. We assess the method using two-dimensional, axisymmetric, and three-dimensional benchmarks governed by the Oldroyd-B and exponential Phan-Thien-Tanner models. The results agree well with published numerical and experimental data and demonstrate improved robustness for demanding viscoelastic free-surface configurations on unstructured meshes.

The Transition Matrix formalism (T-matrix) offers an optimal solution to the problem of radiation scattering and absorption by particles characterized by a high degree of chemical and structural complexity. However, its application to particle models composed by large numbers of constituents has been classically limited by the substantial computational burden of the method. As a consequence, the investigation of light scattering and absorption processes, in cases where particle asymmetry and non-homogeneity play an important role, typically need to rely on simplifying assumptions which affect the correct interpretation of transmission spectra and extinction curves. This problem has a huge impact on many fields of scientific application. In particular, it has a critical role in astrophysics, because the ability to model highly non homogeneous particles is a fundamental requirement to understand the role of extinction in different phases of the interstellar medium (ISM), as well as to probe the physics of light transmission in planetary atmospheres within the Solar System and beyond. In this work we describe the application of a new implementation of the T-matrix formalism, based on the framework originally developed by Borghese, Denti & Saija and named Nano-particle Transition Matrix code, which leverages the power of modern hardware architectures to reduce the computational costs related to the solution of relevant particle models for the investigation of dust in the ISM and of hot-Jupyter class planets atmospheric aerosols. This new implementation of the T-matrix formalism can handle calculations of increasing complexity with a direct impact in the interpretation of the transmission spectra of transiting exoplanets produced by the James Webb Space Telescope (JWST) and of those expected from the Atmospheric Remote-sensing Infrared Exoplanet Large survey (ARIEL).

Unified multimodal models (UMMs) provide an interface for interleaved text-image generation, where generated visual states can directly inform subsequent reasoning and synthesis. However, existing systems are still primarily demonstrated on short trajectories or simplified visual content. We develop a large UMM natively trained on interleaved multimodal data, with strong capabilities in image synthesis, text rendering, complex composition, and multimodal planning, and study how to extend these native capabilities to substantially longer autoregressive trajectories. The main challenge is that generated images continuously accumulate in context: irrelevant or imperfect visual states interfere with future synthesis, while previously established characters and scenes must still be recovered after long intervals. We introduce a visual-memory framework that maintains persistent visual references, selectively retrieves scene-relevant historical states while preserving the complete semantic history, and uses few-step self-rollout to adapt the model to imperfect generated histories. We further introduce LongMIG-Bench for evaluating long-horizon text-to-multi-image and image-to-multi-image generation. Our model achieves high-fidelity long-horizon interleaved generation with strong character and scene consistency, demonstrating how visual memory can scale native interleaved generation to substantially longer trajectories.

Knowledge-based Visual Question Answering (VQA) requires reasoning that extends beyond visible image content, demanding external commonsense and factual knowledge. While recent work leverages Large Language Models (LLMs) as knowledge sources, the question of how deeply such guidance should be integrated into a vision language architecture remains underexplored. In this work, we generate structured reasoning guidance, visual focus concepts, distractor elements and external knowledge using Llama~3.3~70B and systematically compare two integration strategies within a unified frozen ViT + FLAN-T5 framework on the OK-VQA benchmark: decision level late fusion and attention-matrix injection. Contrary to the prevailing assumption that deeper integration is superior, we find that shallow late fusion achieves the strongest results (39.4% exact match, +10.0 over the no guidance baseline), outperforming deep attention modification (31.8%) under data scarce fine-tuning. We further contribute an LLM based augmentation pipeline that expands training data more than tenfold via question paraphrasing. Our findings indicate that integration depth must be matched to the data regime.

Let $X$ be a connected graph and let $\lambda_1(X)$ denote the spectral radius of its adjacency matrix. An edge $e$ of $X$ is sustainable if it can be deleted and replaced by a non-edge so that the resulting graph is connected and its spectral radius does not decrease; $X$ is sustainable if every edge is. We study this edgewise notion of spectral robustness for trees, where each replacement is an elementary exchange between trees on the same vertex set. We establish basic properties of sustainability and exhibit infinite families of sustainable and non-sustainable trees. Our main results concern trees with a vertex of large degree: if $T$ is a tree on $n$ vertices with $\Delta(T)\geq n/2$, then every vertex of maximum degree is a principal vertex, that is, a vertex at which the Perron vector is maximal. Building on this, we prove that every tree with $\Delta(T)\geq n/2+1$ is non-sustainable, and we classify the sustainable trees in the boundary case $\Delta(T)=n/2$: for $n\geq 4$, the only such trees are the double broom $B(2,n/2,n/2-1)$ and one exceptional tree on eight vertices. We conclude with open problems on the generalized tree-shift order and the asymptotic proportion of sustainable trees.

Online discussions on platforms like Reddit are a productive testbed for fallacy detection: short texts, broad topical range, and the full inventory of informal fallacies. The Touche 2026 lab at CLEF formalises this as a shared task on a Reddit-derived corpus that combines fallacy detection with argument-scheme classification. This paper describes our participation in all three subtasks (ST1 binary fallacy detection, ST2 eight-way fallacy classification, ST3 four-way argument-scheme classification) and both evaluation tracks. Our submitted systems are fine-tuned RoBERTa-large encoders with task-specific heads, augmented with a small batch of LLM-generated synthetic contrastive pairs; for ST2 and ST3 on the enhanced track we additionally added confidence-filtered pseudo-labelled examples from the unlabelled pool. A wider backbone comparison spanning additional encoders (DeBERTa-v3, ModernBERT) and an LLM track (four Qwen variants from 3B to 32B, zero-shot through CoT-LoRA) showed no validated benefit over RoBERTa-large at this corpus scale, with 5-fold variance wide enough that we call the comparison underpowered rather than settled. We also characterise how much of the enhanced track's lift reflects organiser-rewritten inputs carrying label information rather than genuine argument clarification. On the official Touche 2026 test set our systems ranked first on both base-track classification cells (ST2 and ST3) and second on both binary-detection (ST1) tracks, with the two enhanced-track classification cells topped by cascaded and multitask systems. We close with the rare-class limitation on ST3 and the most useful follow-ups. Code: https://github.com/psychias/fallacy_detection_clef_2026

This concise manuscript investigates the nonlocal nature of natural deduction, particularly the possibility of introducing assumptions at intermediate stages of a derivation after those stages have already been passed. By examining the structural correspondence between natural deduction and sequent calculus, we demonstrate how the structural rule of left-weakening (LW) permits the enlargement of an antecedent after its corresponding succedent has been derived, without modifying the original inference. Through a number of symmetrical derivations, we establish the proof-theoretic relationship between the introduction of assumptions in natural deduction and antecedent expansion in sequent calculus. On this basis, we formulate the principle of Nonlocal Assumption Introduction (NAI), according to which assumptions may be introduced at any intermediate stage prior to the final conclusion, provided that the corresponding structural expansion is proof-theoretically valid. We further specify the conditions governing this operation, including the preservation of inferential validity, assumption dependencies, and the scope and discharge requirements of the resulting derivation.

We propose a nonparametric approach that fuses deterministic initial forecasts and a partial realization into a probabilistic forecast using the framework of functional curve completion. Our motivation is to construct intra-day forecasts for renewable energy generation, which is a key operational task in modern power grids. The aim is to combine the day-ahead (DA) forecast and the morning realized generation profile to create trajectories for the rest of the day, quantifying operational uncertainty. Starting with a large database of complete forecast-actual pairs across many assets, we adopt a functional point of view and construct a weighted neighborhood based on similarity metrics for the observed partial curve, the DA forecast, the day, and the geographical location. The selected complete nearest-neighbor curves provide a full predictive distribution, including functional focal envelopes and predictive bands based on functional depth. The hyperparameters governing neighborhood size, distances, and fusion rates are tuned across forecast update times and directly target probabilistic calibration in terms of the predictive probability integral transform. Three case studies using synthetic wind and solar assets, as well as real-life regional wind generation, showcase that our algorithm is able to achieve excellent calibration.

Does randomization improve the worst-case first-order complexity of nonconvex optimization when the objective is very smooth? We prove that, for every fixed finite integer $p\ge3$, finding an $\eps$-stationary point requires $\Omega_p(\eps^{-5/3})$ exact value-and-gradient queries, even for $C^\infty$ objectives with initial gap at most one and simultaneous global Lipschitz budgets of one for derivatives of orders $1,\ldots,p$. Together with the known deterministic upper bound, this determines the accuracy exponent up to a single logarithmic factor. The lower bound holds for unrestricted adaptive randomized algorithms, unbounded query locations, and outputs that need not have been queried. One algorithm-independent hard distribution suffices in dimension $O_p(\eps^{-47/3})$. The construction combines a block-chain potential with an exact quartic core, repeated quadratic spectra, and two static Gaussian tensor perturbations. A scheduled Haar argument controls the entire adaptive query span. An innovation-only Gaussian comparison then bounds exact transcript information without assuming finite precision or a minimum query separation. This establishes the $5/3$ randomized accuracy barrier under every prescribed finite smoothness order, extending the previously matched third-order case.

Electrical control of orbital magnetism is a central goal of orbitronics. While current-induced orbital polarization has recently been demonstrated in inversion-asymmetric systems, nonvolatile electrical control of an equilibrium orbital moment has remained elusive. Here, using x-ray absorption spectroscopy with in situ electric-field control of the FE polarization, we demonstrate non-volatile switching of the equilibrium orbital magnetization of the oxide q2DEGs formed at the LaAlO$_3$/EuTiO$_3$/Ca-STO (001) interface. X-ray linear dichroism reveals a reversible modulation, driven by the FE-polarization direction, of the Ti-3$d$ crystal-field splitting and orbital occupancy, establishing a ferroelectric-like metallic state. X-ray magnetic circular dichroism shows a polarization-controlled switching of the Eu$^{2+}$ orbital magnetic moment induced by the orbital magnetization of the q2DEG projected onto Eu-5$d$ states and detected at the Eu-$M_{4,5}$ edge through Eu-5$d$/Eu-4$f$ hybridization in the inversion-symmetry-broken interface. These results provide spectroscopic evidence of ferroelectric control of orbital magnetism in an oxide q2DEG.

Secure multi-party computation (MPC) lets parties evaluate a function over private inputs without revealing them. In garbled-circuit (GC) protocols, prior optimization has largely treated the number of non-linear gates, especially AND gates, as a proxy for cryptographic cost, and has focused on minimizing multiplicative complexity. We identify an orthogonal source of savings: a gate need not be garbled when one party knows or can locally compute its inputs; that party can evaluate it outside the secure protocol, even when the gate is non-linear. We introduce ownership-aware optimization for GC-based MPC, which minimizes non-linear gates requiring joint evaluation rather than total non-linear gates. We formalize this objective in a cost model and implement a compiler-integrated, gate-level framework that restructures Boolean circuits to increase party-local computation while preserving secure-computation semantics. The method is automated and general-purpose, and composes with existing multiplicative-complexity optimization. Experiments show that it reduces jointly evaluated computation and improves end-to-end GC performance on circuits with party-local structure, revealing protocol-cost savings missed by conventional GC cost models.

High-charge Weyl semimetals (WSMs), particularly those hosting charge-three Weyl point (C-3 WPs), have attracted considerable interest owing to their unconventional topological properties and enhanced boundary phenomena. However, ideal C-3 WSMs remain rare, and the symmetry-constrained coexistence of charge-three and charge-one Weyl point (C-1 WPs) in nonmagnetic crystals remains largely unexplored. Here, we investigate a representative eight-node Weyl configuration, consisting of two C-3 WPs and six C-1 WPs with opposite chirality, corresponding to two identical $\{3,-1,-1,-1\}$ irreducible Weyl molecules. By combining topological charge conservation with space-group symmetry analysis, we systematically classify all 230 space groups and identify 12 and 23 candidate groups without and with spin–orbit coupling (SOC), respectively. First-principles calculations without SOC reveal that the boron allotropes TBR-B$_{48}$ and chiral OBD-B$_{48}$ realize this configuration, hosting hourglass-type and non-hourglass-type C-3 WPs, respectively. Surface-state calculations reveal six Fermi-arc branches at the shared projection of the charge-three pair and three at each individual projection. Reversing the structural chirality of OBD-B$_{48}$ reverses the signs of all Weyl charges, establishing a direct correspondence between crystal and momentum-space chirality. Furthermore, breaking the protecting $C_6$ symmetry in TBR-B$_{48}$ splits each C-3 WP into three C-1 WPs and reconstructs the surface Fermi arcs. Our work establishes symmetry-based guidelines for designing and searching for nonmagnetic WSMs hosting charge-three and charge-one Weyl-point coexistence.

We study various symmetry conditions related to the Auslander-Reiten formula for finite-dimensional algebras. In particular, we define a new class of algebras which we call $E$-symmetric. A theorem by Derksen-Weyman-Zelevinksy implies that all Jacobian algebras are $E$-symmetric. We prove the converse for the class of monomial algebras.

In Nonlinear Model Predictive Control (NMPC), the control law is implicitly determined by the solution of an Optimal Control Problem (OCP). Consequently, OCP design is often tailored to specific platforms and tasks, demanding extensive expert knowledge and tuning effort. To address this, we propose a unified NMPC framework that directly plans and controls the robot joints subject to task-space motion and force requirements and is capable of seamlessly generalizing across multiple robotic platforms and diverse tasks. Furthermore, optimization-based methods often treat interactions as rigid contacts, making the torque solution sensitive to the modeling errors of the robot dynamics and the environment. We study the intrinsic task-space impedance of our proposed controller, its partial contraction for differential stability and differential passivity under unmodeled interactions. As a result, our framework enables closing the force-feedback loop in the high-level planning layer to achieve precise motion-force tracking performance, addressing the long-standing challenge of force-feedback NMPC. Finally, we validate our framework on physical hardware, demonstrating hybrid motion-force control on a redundant manipulator and compliant locomotion on a quadruped, while preserving core NMPC advantages such as hard constraint satisfaction, optimal redundancy exploitation, and real-time planning under unforeseen interactions.

Misperceptions can run in two directions. Individuals can either believe false information or reject true information, with experimental work finding that debunking efforts are effective at correcting both forms of misperception. Yet despite the efficacy of true corrections, current debunking systems focus almost entirely on the correction of falsehoods, with professional fact-checkers publishing few true verdicts and crowdsourced systems like X's Community Notes program unable, by rule, to display notes verifying true claims. As a result, we do not have a clear understanding of the extent to which overall misperceptions are driven by doubting true information. To answer this question we first draw on a novel dataset of 47,899 queries to X's AI assistant which track users asking for verification on post content. Surprisingly, among questioned posts that received a true or false verdict, Grok judged nearly identical shares as either accurate or inaccurate. Of the questioned posts, more than 83% were not flagged by Community Notes and only 3.3% carried a displayed note. Among questioned posts judged accurate by Grok, nearly 90% received no note at all while fewer than 2% displayed a note flagging them as misleading. This discrepancy does not seem to have stemmed from a lack of effort, with contributors writing 2,981 notes verifying questioned posts despite there being no way for Community Notes to display them publicly. Alongside further exploration of these findings we discuss both the system- and code-level changes that would be required to effectively address this asymmetric correction paradigm.

Automating aerial wildlife surveys requires reliable animal detection, consistent delineation of observation strips, and correction for repeated coverage between successive images. We present a unified three-branch deep learning framework that integrates these tasks through WildSeg, WildMatch and WildDet. WildSeg combines rod segmentation with geometric refinement to delineate counting strips from aircraft-mounted markers. WildMatch uses teacher-student distillation to learn dense image correspondences and identify overlapping survey areas without relying on geolocation metadata. WildDet combines dual-scale detection with feature fusion, density supervision and empty-image gating to support animal detection and counting. We evaluated the framework using oblique aerial imagery of four antelope species from the Great Nile Migration Landscape in South Sudan. WildSeg achieved reliable rod extraction, while WildMatch approached teacher-level accuracy with a substantially lighter and faster model. Both modules maintained high accuracy on resized images, improving processing efficiency. WildDet improved detection performance across all four species and reduced detection-based count MAE, RMSE and mean bias by 21.0%, 4.6% and 27.8%, respectively, relative to YOLO11. By connecting spatial survey constraints with animal detection, the framework provides a practical and computationally efficient approach to processing large aerial image collections for wildlife monitoring and population surveys.

Virtual cell models should capture phenotypes induced by specific perturbations. However, image-based models are typically trained and evaluated to match real-image distributions, which alone does not explicitly prioritize perturbation-specific biological responses. Recent approaches add biological signals through reinforcement-learning post-training or inference-time steering, raising a central question: where should biological response information enter the generative pipeline? We introduce PerturbBridge, a stochastic data-to-data bridge model that directly supervises perturbation-specific phenotypes in predicted perturbed images during training. PerturbBridge jointly optimizes bridge regression and differentiable biological response losses, with biological gradients passing through a frozen image encoder. We further introduce perturbation-response evaluations based on downstream classification and retrieval. Across three microscopy benchmarks for perturbation profiling and subcellular imaging, PerturbBridge surpasses CellFlux in biological fidelity, image quality, and sampling efficiency. On BBBC021, mechanism-of-action classification improves by 5.66 and 7.5 percentage points with morphological and learned visual features, respectively. On JUMP and AllenCell, retrieval Hit@10 and Hit@1 improve by 1.77 and 7.15 percentage points, respectively. On BBBC021 and JUMP, PerturbBridge achieves lower FID with 80% fewer model evaluations than CellFlux. Overall, these results show that directly supervising perturbation responses during training improves phenotypic fidelity while enabling efficient cellular image generation. Code is available at https://github.com/yyc1215lll-creator/PerturbBridge.

Real-world tables are often heterogeneous, combining numerical features with free-form string columns. Table foundation models (TFMs) excel on numerical tables but require strings to be encoded numerically, while large language models (LLMs) hold semantic knowledge but are not designed for tabular predictions. Most existing bridges use LLMs as plug-in encoders for cell strings, yet how to extract representations effectively for tabular learning remains underexplored. We show that the interface between LLMs and tables matter as much as the LLM itself. Applied cell-wise, larger LLMs do not give better features. With our interface, the same LLMs improve steadily with size. TILE (Table-Interfaced LLM Embeddings) combines four ingredients: row serialization to provide cross-column context, intermediate-layer extraction to avoid layers specialized for next-token prediction, columnar pooling to remove mean pooling's length bias, and by-pass features to route numbers and categories past the LLM. Paired with TFMs, TILE sets a new state of the art for tables with strings. Across 100 datasets, it beats the strongest baseline on 89% of splits, with gains across 7 LLMs and 10 tabular learners, and ranks first on the 108-dataset STRABLE benchmark.

Phase retrieval recovers complex-valued objects from intensity-only measurements. In its classical Fourier form, reconstruction suffers from twin-image stagnation. Coded diffraction imaging, which adds a random phase mask before the Fourier transform, has so far required several masks. We show that an image prior makes single-shot coded diffraction imaging possible, with a single random mask followed by a Fourier transform. We validate this claim in simulation and in an optical experiment, in which a spatial light modulator displays the phase object together with the random mask. Our experimental validation covers five test signals and 530 images from the public Brno quantitative phase microscopy dataset of five adherent cell lines. With no image prior or with Fourier-only measurements, reconstruction of complex-valued objects from a single diffraction pattern consistently fails. An out-of-the-box pretrained image denoiser enables accurate reconstruction, in simulation even with fewer measurements than unknowns. Five regularization frameworks built on the same denoiser and one learned weakly convex regularizer all reconstruct the experimental images with peak signal-to-noise ratios within one decibel of each other. A single random phase mask and image priors thus turn coded diffraction imaging into a practical single-shot modality for quantitative phase imaging.

Anomalous diffusion in heterogeneous media is ubiquitous across physical and biological systems, where spatially varying microstructures give rise to distinct transport regimes. Variable-order nonlocal operators provide a natural modeling framework for describing heterogeneity and transitions between diffusion regimes. However, the position-dependent long-range interactions inherent in these operators impose prohibitive memory costs and make numerical methods developed for constant-order operators computationally impractical. Consequently, accurate and computationally efficient numerical methods for variable-order nonlocal operators remain scarce. In this paper, we propose accurate finite difference discretizations for the variable-order fractional Laplacian $(-\Delta)^{\alpha(\mathbf{x})/2}$ and develop a fast Fourier transform (FFT)-accelerated scheme for their efficient implementation. The computational and memory costs are $O\!\big(MN\log(2N)\big)$ and $O(MN)$, respectively, where $N$ is the total number of spatial points and $M \ll N$. The method is also naturally parallelizable. We establish its consistency and accuracy through a rigorous error analysis that quantifies the local interplay between solution regularity and the spatially varying order $\alpha(\mathbf{x})$. To the best of our knowledge, this is the first work to quantify how these two factors jointly determine convergence rates, with numerical experiments demonstrating this dependence. Extensive numerical experiments confirm the accuracy and efficiency of the proposed method. Applications to anomalous diffusion in composite materials further demonstrate its ability to capture the effects of material heterogeneity.

We propose a mitigation to reward hacking in reinforcement learning. Our setup involves a misspecified reward function that gives rise to unintended behavior, and a monitor used to detect unintended behavior. As with frontier AI systems, the monitor makes systematic errors. Consequently, using the monitor for data filtering or to apply a reward penalty fails to prevent the model from learning unintended behavior ("reward hacking"). To address this problem, we introduce GRAFT: Gradient Routing Adapter Fine Tuning. Rather than penalizing or filtering flagged samples during training, GRAFT constrains gradient updates on those samples to a sacrificial set of new parameters. This has two benefits: First, because the monitor is not part of the reward function, the model does not learn to hide misbehavior. Second, ablating the sacrificial parameters reduces or entirely stops the unintended behavior while maintaining improved performance on target tasks. We demonstrate these benefits in RL environments where reward penalties cause models to learn to reward hack in undetected ways, training models ranging from 135M to 8B parameters. Additionally, we show a similar effect at a larger scale by distilling frontier LLM trajectories containing reward hacking into Qwen3-32B, without teaching that model to reward hack. Overall, our work suggests GRAFT may be a promising approach for leveraging imperfect monitors during post-training in settings where true performance is difficult to evaluate.

We study in-context binary classification in a simple attention model trained to minimise a logistic loss. Each prompt has a latent logistic teacher, and the learner must infer its direction from Bernoulli-labeled context examples. We derive an asymptotic prediction for the test logit in a high-dimensional regime with context length proportional to token dimension, and total training prompts proportional to dimension squared. Our formula cleanly separates calibration, test-time context noise, and non-isotropic parameter-estimation noise as interpretable effects on in-context performance. Our analysis also identifies the ridgeless separability regime and addresses the effects of context length, pretraining-set size, regularisation, and sampling temperature on in-context classification performance. The theory's predictions closely agree with finite-dimensional simulations and fully trained linear-attention models, while experiments with standard softmax attention show qualitatively related behaviour beyond the regime assumed by our analysis.