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We generalize the universal approximation theorem for functional input neural networks (FNN) to differentiable maps by including the approximation of the derivatives. A FNN maps the input from a possibly infinite-dimensional weighted manifold to the real-valued hidden layer, on which a non-linear scalar activation function is applied, and then returns the output into a Banach space via some linear readouts. By proving a weighted Nachbin theorem, we establish a universal approximation theorem for differentiable maps, which goes beyond the usual formulation on compact sets and also includes the approximation of the derivatives. This leads us to approximation results for non-anticipative functionals including the horizontal and vertical derivatives. As a further application, we show that linear functions of the signature are able to approximate path space functionals including their directional derivatives.

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We introduce CLEVER1, a high-quality, curated benchmark of 161 problems for end-to-end verified code generation in Lean. Each problem consists of (1) the task of generating a specification that matches a held-out ground-truth specification, and (2) the task of generating a Lean implementation that provably satisfies this specification. Unlike prior benchmarks, CLEVER avoids test-case supervision, LLM-generated annotations, and specifications that leak implementation logic or allow vacuous solutions. All outputs are verified post-hoc using Lean's type checker to ensure machine-checkable correctness. We use CLEVER to evaluate several few-shot and agentic approaches based on state-of-the-art language models. These methods all struggle to achieve full verification, establishing it as a challenging frontier benchmark for program synthesis and formal reasoning. Our benchmark can be found on GitHub as well as HuggingFace. All our evaluation code is also available online.

A painting bought for less than $100 (£75) in a US charity shop in the 1960s has sold for almost £190,000 at auction. Art teacher Helene Plotkin bought the work by Scottish Colourist FCB Cadell in White Plains, New York in 1966, unaware of its true value. The painting, Interior: The Lady in Black, hung in her living room for 60 years - but the artist's signature was illegible and was only recently identified. It sold for £189,200, including buyer's premium, in Edinburgh as part of Lyon & Turnbull's Scottish painting and sculpture auction. The background to the painting only became clear when Helene's son Barry began his own research into it and took it for a valuation last year.

Multi-agent systems built on large language models (LLMs) require many coordination choices that are difficult to fix a priori: which skill protocol to invoke, which agent role should perform a subtask, which model to bind to each role, how roles should interact, when to use retrieval or verification, and when to omit a step entirely. These choices interact with task regime and operational constraints, so static pipelines and one-off model comparisons provide only a limited view of the design space. This paper introduces AgensFlow, an open-source framework that treats multi-agent coordination as an online policy-learning problem under partial observability. The framework makes coordination decisions observable and learnable from repeated trajectories, rather than treating skill, role, model, topology, and evaluation choices as fixed pipeline design. AgensFlow is evaluated on two corpora: distributed-systems incident tasks and security-advisory tasks. The evaluation shows three main results: learned routing reaches a higher-quality operating point than a fixed pipeline baseline on coordination-heavy classes; skip:X isolates topology compression as a meaningful part of the substrate; and warm-started policy graphs can reduce exploration cost while preserving plateau quality. Overall, the results support that learned, auditable routing can improve coordination-heavy multi-agent workflows over static wiring.

In the era of precision medicine, genome-wide epigenetic modifications offer rich data that could inform risk prediction. However, these data are high-dimensional and exhibit complex dependence structures, which makes it difficult to jointly model them with low-dimensional covariates when the goal is to obtain interpretable effect estimates for covariate adjustment. Standard Bayesian additive regression trees (BART) provide strong predictive performance but treat all predictors uniformly within the tree ensemble, obscuring the contributions of significant covariates and complicating variable selection in high-dimensional settings. We propose a semi-parametric BART model (spBART) that addresses this limitation by modeling low-dimensional covariates through a parametric component with interpretable coefficients, while capturing complex nonlinear associations among high-dimensional predictors through the tree ensemble. To perform stable variable selection, we develop a cross-validation-based procedure that aggregates posterior inclusion probabilities across folds and applies Bayesian false discovery rate control. We apply the proposed method to a pooled case--control analysis of high-dimensional genome-wide 5-hydroxymethylcytosine profiles derived from circulating cell-free DNA in two multiple myeloma studies ($N = 869$). The approach identifies a parsimonious set of candidate loci and achieves strong out-of-sample discrimination (AUC $= 0.96$) in a held-out validation set. Overall, spBART provides a unified framework for combining interpretable covariate inference with flexible modeling and variable selection in high-dimensional biomedical studies.

We study the grokking phenomenon through the lens of topology. Using persistent homology on point clouds derived from the embedding matrices of a range of models trained on modular arithmetic with varying primes, we identify a clear and consistent topological signature of grokking: a sharp increase in both the maximum and total persistence of first homology ($H_1$). Persistence diagrams reveal the emergence of a dominant long-lived topological feature together with increasingly structured secondary features, reflecting the underlying cyclic structure of the task. Compared to existing spectral and geometric diagnostics -- specifically, Fourier analysis and local intrinsic dimension -- persistent homology provides a unified geometric and topological characterization of representation learning, capturing both local and global multi-scale structure. Ablations across data regimes and control settings show that these topological transitions are tied to generalization rather than memorization. Our results suggest that persistent homology offers a principled and interpretable framework for analyzing how neural networks internalize latent structure during training.

Magnification shift is a major obstacle to robust histopathology classification, because models trained on one imaging scale often generalize poorly to another. Here, we evaluated this problem on the BreaKHis dataset using a strict patient-disjoint leave-one-magnification-out protocol, comparing supervised baseline, baseline augmented with DCGAN-generated patches, and a gradient-reversal domain-general model designed to preserve discriminative information while suppressing magnification-specific variation. Across held-out magnifications, the domain-general model achieved the strongest overall discrimination and its clearest gain was observed when 200X was held out. By contrast, GAN augmentation produced inconsistent effects, improving some folds but degrading others, particularly at 400X. The domain-general model also yielded the lowest Brier score at 0.063 vs 0.089 at baseline. Sparse embedding analysis further revealed that domain-general training reduced average signature size more than three-fold (306 versus 1,074 dimensions) while preserving equivalent predictive performance (AUC: 0.967 vs 0.965; F1: 0.930 vs 0.931). It also increased cross-fold signature reproducibility from near-zero Jaccard overlap in the baseline to 0.99 between the 100X and 200X folds. These findings show that calibrated, compact, and transferable representations can be learned without added architectural complexity, with clear implications for the reliable deployment of computational pathology models across heterogeneous acquisition settings.