Papers

We prove optimal dual Furstenberg estimates for Frostman and Katz–Tao configurations for which the shadings are not locally concentrated near pencils. The starting point is the sticky estimate of Orponen and Shmerkin in a companion paper. We deduce restricted projection theorems in the plane and in higher dimensions. In particular, Ahlfors-regular measures on the circle satisfy a Marstrand-type projection theorem. Applications include restricted projections to surfaces in the $\mathbb{G}(4,2)$ Grassmannian, pinned distances, discretized sum-product, and a new proof of the dimension formula for planar self-affine sets.

Recent advances in physical 3D generation have enabled visually faithful assets with simulatable mechanical behavior. However, existing methods provide limited control over material behavior beyond appearance-based inference and struggle to estimate physical properties from complex interactions between objects and semantic parts. Therefore, we present PhysForge, a Video Diffusion Model (VDM)-based framework for addressing these challenges, comprising two complementary modules for material-controllable physical fitting and componentlevel interaction modeling. The first, PhysFit, leverages material-conditioned VDM dynamics as references for physical-property fitting, introducing windowed fitting to focus on material-revealing interactions and Hallucinated Forcing to disentangle nonphysical motion hallucinated by the VDM from material responses. The second, PhysComp decomposes complex interactions into individual objects and semantic components, enabling their physical parameters to be optimized independently and flexibly composed for joint simulation. Experiments on PhysAssets and Google Scanned Objects demonstrate that, compared with existing methods, PhysForge provides stronger control over material-dependent physical responses while supporting component-level physical modeling for complex interactions. It achieves the best CLIP and physical controllability scores on both common and uncommon materials, with qualitative comparisons, component-level ablations, and a user study further supporting these improvements. Our project page is available at https://masterwanna.github.io/PhysForge.html .

We consider the Helmholtz equation in a Dirichlet angular sector (which may be locally perturbed) covered on one side by a thin homogeneous layer. First, we propose an asymptotic expansion of the solution at any order and for any angle when the thickness of the layer tends to 0, using a matched asymptotic expansion. This consists in introducing different asymptotic expansions in three subdomains – the vicinity of the corner, the layer and the rest of the domain – that are linked through matching conditions. The presence of the corner makes these matching conditions delicate to derive because the fields have complex singular behaviors. We overcome this difficulty using a novel system of algebraic tools, including formal series and operators on these series. Moreover, we justify the asymptotic expansion with error estimates and provide details on the form of the expansion depending on the angle. We numerically validate this expansion up to order 5. This requires computing the behavior at infinity of some corner profiles. To do this, we introduce a Galerkin-type method enriched by complements at infinity. These complements are known functions that appear in the behavior at infinity of the corner profiles with coefficients that become unknowns of the system. Finally, we propose effective models of order 3, resp. 5 where, far from the corner, the layer is replaced with an effective Robin, resp. Ventcel, boundary condition and, near the corner, an artificial boundary is introduced and a Dirichlet-to-Neumann (DtN) type condition is imposed. The DtN map relies on a modal decomposition of the truncated asymptotic expansion at order 3, resp. 5. The modes take into account the singular behavior of the expansion terms, the corner profiles and some algebraic matching coefficients. We prove the well-posedness of our effective models, as well as error estimates, and provide numerical results.

The layout algebra behind NVIDIA's CuTe library describes how tensor tiles map onto threads and memory. A layout is a shape with a stride for each entry, so a coordinate's address is the sum of its indices times their strides. As formalized by Shah and by Cecka, its central operations (e.g., composition, complement, division, product) are guarded by at least six interacting admissibility conditions. We show that this complexity is an artifact of a representational choice, not of the domain. We present a layout algebra in which every layout names the space its inputs and outputs live in: logical, physical, or thread-value indices. A layout into logical or thread-value space must be a bijection from its $N$ coordinates onto $[0,N)$. A layout into physical space must be alias-free: distinct coordinates land on distinct addresses, except under broadcast, where many threads read one value on purpose and the layout must mark where. Nothing maps out of a physical index. Composing $f$ with $g$ requires only that $f$'s range be exactly $g$'s flattened coordinate set. The index arithmetic can always be emitted naively: compute $f$, split the result by division and remainder into $g$'s coordinates, and apply $g$. Often the composite simplifies to a layout; we show that when it does, there is a coarsest such form, whose digits every other one refines, and we emit that. A warp's or thread's slice is a restriction. A restricted layout is no longer composable, which forces the user to settle the whole view of a tile before taking slices. We validate the algebra by reproducing within it CuTe's operations, such as composition, complement, and division. We implement it in OCaml and use the implementation to rederive CuTe's atoms and usage patterns. The decision procedure is proved correct in Rocq.

Prior work suggests fraudulent scientists write their articles with higher rates of linguistic obfuscation than genuine scientists, with a possible mechanism suggesting obfuscation makes research more costly for a reader to analyze. This result has not undergone additional empirical scrutiny in the open science era, nor have other psychological mechanisms been evaluated. The current replication and extension addressed this opportunity by comparing 800 fraudulent and 800 genuine control papers (2013-2025). The results supported the linguistic obfuscation hypothesis, suggesting fraudulent papers were more obfuscated than genuine ones. A parallel mediation model evaluated three competing mechanisms, finding that reality monitoring, as measured through decreased rates of adjectives, best explained the link between fraudulent science and linguistic obfuscation. Cross-validated detection accuracy using obfuscation features (60.7%) and an elastic net of over 100 language dimensions (71.6%) were significantly greater than chance. Implications for scientific fraud and psychology of language research are discussed.

We introduce spread-out measures, a notion of local non-concentration near pencils for measures on Grassmannians. We establish an optimal $\delta$-discretized dual Furstenberg estimate for “sticky spread-out” shadings. A simple-to-state corollary is the following. Assume that $\nu$ is an Ahlfors $s$-regular measure on $\mathbb{G}(2,1)$ with $s > 0$. Let $E \subset \mathbb{R}^{2}$ be a Borel set with equal Hausdorff and packing dimension. Then, $\dim_{\mathrm{H}} \Pi_{V}(E) = \min\{\dim_{\mathrm{H}} E,1\} \quad \text{for $\nu$-a.e. } V \in S^1$. The general $\delta$-discretized estimate for “sticky spread out” configurations here is the starting point for a more general theorem in the companion paper of Cohen, Shmerkin, Wang and Zakharov, where the “stickiness” hypothesis is removed. For instance, the assumption of equal Hausdorff and packing dimension in the result stated above will be dispensed with in the companion paper.

Time series foundation models (TSFMs) increasingly offer covariate support, with calendar information being the most common type of categorical covariate in practice, as it is available for every point in time. However, how these features are encoded is largely treated as an implementation detail rather than as an object of study, despite encoding being one of the few design choices left for pretrained models at inference time. With the present study, we demonstrate that the way information is propagated into the model substantially affects forecast accuracy of TSFMs. Therefore, we systematically compare seven calendar encodings, including three conventional target-agnostic representations (ordinal, one-hot, and cyclic) and four target encodings, across four state-of-the-art TSFMs on nine datasets from multiple domains in a zero-shot rolling evaluation. The nine datasets are influenced by social activities, such as weekends and holidays, making calendar information particularly relevant for forecasting. We find that encoding choice affects forecast accuracy across all models, with MASE differences of up to 9.9 % on non-holidays and 83 % on holidays within a single model. Effectiveness and utilization vary substantially across models. Improvements are concentrated around holidays, where calendar information provides signals unavailable from target history, while some model encoding combinations also improve non-holiday forecasts. A SHAP-based attribution analysis locates the contribution of the covariates relative to the autoregressive component and explains why some model-encoding combinations degrade performance. Our findings suggest that current benchmarks for covariate-informed TSFMs can distort model comparisons by imposing fixed encoding strategies.

Multimodal decision models return probabilities over predefined alternatives, offering a structured interface for image classification. This study benchmarks inference with CLEF and CLEF-Flash on 2818 breast ultrasound images from Dataset B, BUSI, and BUS-BRA. Dataset B and BUS-BRA use binary benign-malignant classification, whereas BUSI includes normal, benign, and malignant categories. Model parameters remain unchanged, and original images are submitted without user-defined preprocessing. Three prompts provide minimal labels, an explicit classification task, or sonographic criteria. We report accuracy, macro F1, balanced accuracy, macro precision, throughput, and latency. CLEF-Flash with the clinical prompt achieves the highest accuracy on all three datasets: 74.85% on Dataset B, 66.28% on BUSI, and 51.63% on BUS-BRA. On BUSI, however, CLEF with the task-guided prompt achieves the highest macro F1 (64.51%) and balanced accuracy (68.72%). CLEF-Flash has lower latency in every matched comparison. Clinical instructions improve CLEF-Flash accuracy relative to the minimal prompt but reduce CLEF accuracy on all datasets. Confusion matrices reveal frequent benign-to-malignant errors, particularly on BUS-BRA, where all configurations fall below the 67.63% accuracy of a majority-class predictor. These findings demonstrate dataset-and prompt-dependent performance and motivate class-specific evaluation before applying multimodal decision models to breast ultrasound analysis.

Linearizing softmax attention offers a route to efficient Transformer inference, but preserving pretrained attention remains challenging. We identify an important source of approximation error overlooked by standard linearization methods: a small number of outlier dimensions contribute disproportionately to query-key dot products and account for a large fraction of the approximation error. Motivated by this heterogeneity, we propose Outlier-Aware Attention Linearization (OAL), a dimension-aware method for constructing quadratic attention approximations. Starting from a second-order Taylor approximation, we introduce a Hadamard-product formulation that makes individual query-key contributions explicit. This formulation relaxes globally shared coefficients to support dimension-specific linear terms and cross-dimensional quadratic interactions while preserving separable evaluation. To retain this flexibility with fewer parameters, OAL groups dimensions with similar contribution scales to share coefficients, while allowing distinct treatment of the few outliers. The resulting positive quadratic kernel is calibrated against pretrained attention outputs and admits linear-time evaluation with respect to sequence length for fixed head dimensions. On Qwen2.5-0.5B, OAL reduces frozen-attention output relative MSE by approximately 8.9% compared with DiJiang. Replacing middle-layer attention modules and jointly fine-tuning the kernel coefficients with LoRA further lowers WikiText-103 perplexity from 31.71 to 27.50 compared with a learned global quadratic, while improving accuracy on PIQA, HellaSwag, and ARC-E. These results demonstrate the value of outlier-aware dimension grouping for both attention approximation and pretrained-model conversion. The code is available at https://github.com/wuqianyue09/OAL

We consider the Burgers-type stochastic PDE \[ du = \theta\,\partial_x^2 u\,dt + f(u)\,dt + g(u)\,\partial_x u\,dt + \sigma\,dW(t), \] driven by space–time white noise on the one-dimensional torus, where \(\theta>0\) is the diffusivity parameter and \(\sigma>0\) the noise intensity. The transport term \(g(u)\,\partial_x u\) admits no classical interpretation, and we work with the mild solutions defined via rough paths following~\cite{Hairer2011}, for which the solution decomposes as \(u=\psi+R\) with \(\psi\) a stationary centred Gaussian process solving a damped stochastic heat equation and \(R\) a remainder. We first show that, away from the initial time, \(R\) is Lipschitz in space and Hölder in time of order strictly greater than \(1/2\), hence strictly smoother than \(\psi\) in both variables. We then compute exactly the spatial quadratic variation and the temporal quartic variation of \(\psi\), almost surely and not merely in \(L^{2}(\Omega)\), and prove that the corresponding variations of \(u\) are unaffected by the remainder. Combined with an exact time-scaling identity, this separation of regularity scales yields strongly consistent pathwise estimators of \(\theta\). To our knowledge this is the first inference procedure for an SPDE whose solution is defined only pathwise, as a measurable function of a rough path, rather than as an Itô process.

Large language models struggle to reliably identify and correct errors in their own reasoning. Unlike prior training-based approaches that fine-tune an LLM to generate revisions or feedback, we learn where to intervene in failed rollouts and determine how to redirect them. To choose where to intervene, we train a 130K-parameter MLP head on a frozen encoder using only final-answer correctness. For redirection, we find that under a fixed generation budget, the tested inline injections fail to reliably redirect reasoning, whereas presenting the same instruction in a new user turn changes the model's course. The resulting correction framework, UPTURN (User-Prompt TURN injection at a learned position), requires no LLM fine-tuning: it retains a selected prefix, injects an instruction as a new user turn, and regenerates the suffix. Across four reasoners and six math benchmarks, UPTURN improves average accuracy in every setting over a 32-rollout baseline. It outperforms self-correction and reflection-token injection baselines under matched correction budgets, transfers across reasoners and to GPQA-Diamond without retraining, and complements GRPO and self-correction methods.

We develop a field-redefinition-invariant description of hadronic amplitudes outside the WZW sector at $\mathcal{O}(1/f_a)$, including general flavour-conserving and flavour-changing quark couplings. Starting from the most general CP-conserving Lagrangian for ALP interactions with the QCD sector, we identify invariant Wilson coefficient combinations and match them onto the $U(3)$ chiral theory. Using LSZ reduction, we choose fields for which ALP–meson mixing vanishes at this order; amplitudes then depend only on direct invariant couplings, while the anomalous gluonic interaction remains encoded in the $U(1)_A$ sector. For the baryon octet, we derive the general vector and axial couplings and identify, to our knowledge for the first time, an independent invariant ALP–baryon interaction from a second topological singlet structure in the $U(3)$ baryon Lagrangian. We constrain flavour-changing couplings with rare hyperon decays and show in an explicit ultraviolet model that running and matching can generate them from flavour-diagonal high-scale couplings. Finally, we estimate baryon–antibaryon widths and find that, in a gluophilic benchmark, their sum can exceed individual exclusive mesonic widths for $2~\text{GeV} \le m_a \le 3~\text{GeV}$.

We establish a Thouless–Anderson–Palmer (TAP) variational formula for the free energy of spherical $2$-spin models whose interaction matrices have a compactly supported limiting spectral distribution. The formula holds at every inverse temperature and, in particular, extends the TAP representation previously obtained for the spherical Sherrington–Kirkpatrick model. Our main contribution is a new microcanonical proof designed to handle marginality. We partition the sphere according to the smallest collection of top eigendirections carrying a prescribed fraction of the norm. On every element of this partition, the remaining degrees of freedom are effectively at high temperature and their contribution can be computed through spherical-integral asymptotics. The resulting auxiliary variational problem separates the macroscopic low-temperature contribution from the high-temperature fluctuation term; its maximization recovers the usual TAP functional and Plefka condition. This decomposition also shows why neighborhoods of the TAP maximizers alone do not capture the Gibbs mass: all eigendirections associated with large eigenvalues contribute on the same exponential scale, a manifestation of the marginality of spherical $2$-spin states.

Curriculum learning typically organizes training samples from easy to hard. In GRPO, medium-difficulty samples are attractive for online selection because they are more likely to yield informative group-relative learning signals, and prioritizing them can improve training efficiency. However, it remains unclear whether persistently selecting such samples also leads to the strongest final model performance, or whether training can benefit from progressing toward harder samples. Meanwhile, sample difficulty changes as the policy evolves, so fixed difficulty estimates can become misaligned with the current policy. We therefore re-estimate sample difficulty under the current policy and, inspired by the easy-to-hard principle of curriculum learning, design and evaluate a staged curriculum that progressively shifts the active difficulty band toward harder samples. Experiments on GSM8K show that persistent medium-difficulty selection does not achieve the strongest observed performance. Across three random seeds, progressing to a moderately harder difficulty band improves test accuracy to $0.6591 \pm 0.0077$, compared with $0.6449 \pm 0.0024$ for the fixed-medium curriculum. However, further progression to a harder difficulty band reduces accuracy to $0.6405 \pm 0.0053$. These results show that staged difficulty progression can outperform persistent medium-difficulty selection, but its benefit is non-monotonic. Effective GRPO curricula can therefore benefit from progressing beyond medium difficulty, while avoiding excessive progression toward overly difficult samples.}

With the advances of coding agents, software developers are spending more time reviewing code than writing code. As human review velocity falls behind agentic code generation, it is tempting to build automated review agents towards a future of "lights-off" software factory with little to no human supervision. To assess the practicality of such a fully autonomous software factory, we construct CRBugBench, a new benchmark that evaluates whether review agents detect human-verified bugs. We build a data pipeline that mines bugs from repository history, and pair each bug with its introducing PR to form a review task. CRBugBench consists of 1,000 review tasks across 98 open-source GitHub repositories, and 800 additional tasks from proprietary industry repositories at one large company. We then use the benchmark to study agent review quality: We found that SOTA review agents do well: the best (claude-opus-5 with copilot-cli) catches 71.9% of bugs within its top-5 comments, up from 47.7% (opus-4.6) in five months of model evolution. Yet agents remain less effective on complex changes and bugs requiring non-local reasoning, with sharp performance drops on proprietary industry repositories where project context is more often needed. To improve bug detection, we found scaling compute helps most through prompt changes: exploration-nudging prompts gain up to 5.8% performance at 10% higher cost. From these findings, we derive concrete guidance for developers to adopt review agents - scaling review effort to PR complexity, steering agents by project priorities, supplying repository knowledge, dispatching models by PR characteristics, and monitoring quality as models evolve.

We introduce a discrete spectral curve from $\mathscr{N}=2$ double-scaled SYK whose topological recursion defines quasi-polynomial volumes in integer boundary lengths. The curve has two edge theories: a local $\mathscr{N}=2$ theory at the lower edge and a bosonic theory at the upper edge. Focusing on the local $\mathscr{N}=2$ theory, we compute its finite-$q$ volumes and independently check the first few using the string equation. In the $q\rightarrow 1$ continuum limit, these recover the super Weil-Petersson volumes of Turiaci and Witten. Finally, we give an all-genus intersection-theoretic interpretation of these local volumes on the moduli space of stable marked curves, in terms of a deformed spin class with a gap-dependent $\kappa$-class dressing.

We develop an analytical approach to the Kondo problem based on an effective partition function that generalizes the sophomore's dream integral. The Kondo crossover is represented as a passage through an approximately equidistant ladder of its Lee-Yang zeros in the complex plane, rather than as a singularity on the real axis. Summation over this ladder yields the Gamma function $\Gamma(\alpha + T_K/(2 \pi T))$ as the universal crossover backbone, where the offset $\alpha =1/2$ is fixed by the Friedel sum rule. The Euler-Maclaurin expansion connects the discrete zero structure to the Nozieres Fermi-liquid theory through Bernoulli numbers. A two-cutoff formula provides a compact, essentially one-parameter description of the full crossover in terms of a temperature-space Fano lineshape which is a thermal average of the energy-space Fano resonance observed in STM. Drift corrections are treated. The approach extends to $\mathrm{SU}(N)$ Coqblin-Schrieffer, underscreened and Anderson impurity models, and Kondo lattices.

This paper presents a dual-quadrotor hook- suspended transport system for cooperative aerial manipulation of a payload that is not rigidly attached to a fixed point along the cable. Two quadrotors are connected by a single inextensible cable that passes through a frictionless hook mounted on the payload, allowing the cable to slide between its two segments. This introduces a cable-distribution variable that couples the quadrotor motion, cable geometry, and payload dynamics. A feasible reference-generation framework is formulated using the desired payload pose, cable distribution, and system shape while satisfying the cable-length constraint and elliptical workspace. The quadrotors track the geometrically generated trajectories while maintaining positive cable tension and respecting actuator limits. The payload pose is estimated from the system geometry, and its measured position is used only to validate the desired trajectory. The approach is evaluated through experimental demonstrations.

Rare-earth-ion based molecular crystals are emerging as a promising platform for quantum information applications, combining long optical coherence times with the chemical tunability of coordination chemistry. Here we present a comprehensive optical characterisation of twelve Eu$^{3+}$-based molecular complexes, with a focus on the coherent $^{5}\text{D}_{0} \longrightarrow \ ^{7}\text{F}_{0}$ transition at $\sim$580 nm. Judd-Ofelt analyses are performed across all complexes, providing insights into ligand-field environments, local point-group symmetries, and the resulting branching ratios of the coherent transition. We find that lower point-group symmetry (C$_1$, C$_2$) yields higher branching ratios of the $^{5}\text{D}_{0} \longrightarrow \ ^{7}\text{F}_{0}$ transition, exceeding 1% in several complexes. Optical inhomogeneous linewidths measured at 4.2 K reveal narrow linewidths for some complexes of below ten GHz, indicative of a highly ordered crystalline environment. Optical coherence measurements of one of the complexes, 5-Eu, yields an optical coherence time of $T_{\text{2}}^{\text{opt}} = 2.3~\mu\mathrm{s}$ at 4.2 K, comparable to the best-performing molecular systems reported to date. Systematic characterisation and quantitative comparison will be key to find structure-property relations guiding rational design of optimised complexes for quantum information applications.

Many machine learning approaches are two-level: an expert selects a class of methods, within this class the choice of key components is critical and usually manual. This paper addresses the automatic tuning of these components and proposes a framework transferable across models of different nature. The multi-agent large language model (LLM) framework has three roles—analyst, coder, and recorder~—with an experiment log and a sandboxed execution environment. Unlike systems that optimize the entire ML script or search architectures in a discrete space, the framework operates over an open but semantically narrow space: the agent generates code only for the specified components, leaving the overall network structure unchanged. The approach is validated in two directions: tuning a classification spiking neural network with correlation learning, and solving high-order differential equations with Physics-Informed Neural Networks (PINN). Across five tasks—spiking classification on MNIST and Digits, and boundary-value problems for the generalized nonlinear Schrödinger equation (GNLSE) and Kuramoto–Sivashinsky (KS) equations—the agent outperforms the baselines: classification accuracy is 13 percentage points (MNIST) and 6 percentage points (Digits) higher; the error against the analytical single-soliton solution is reduced approximately 2491-fold; the invariant-preservation error drops approximately 325-fold in the selected GNLSE run with a Gaussian perturbation and by more than 100-fold on average over repeated initializations; and the mass drift drops 14-fold for the KS problem with a Gaussian perturbation. The framework thus tunes key components on tasks of different nature without changing its internal structure.

Given the challenges associated with obtaining large volumes of real-world audio, the advancement of Automatic Speech Recognition (ASR) systems increasingly depends on synthetic data. This study focuses on the call center-banking domain, providing a targeted analysis of ASR models in this context. We evaluate real and synthetic datasets, using Word Error Rate (WER) and Character Error Rate (CER) as metrics, and employ a speech quality model to assess the impact on ASR performance. Our research compares Text-to-Speech (TTS) and advanced voice conversion methods, including KNNVC, Seed-VC, and Vec2Wav2, revealing significant improvements in speech quality and ASR accuracy, particularly with Seed-VC. Additionally, our domain-specific experimentation provides insight into the unique challenges and opportunities that arise when applying ASR technologies to industry-relevant settings. This highlights the potential of voice conversion technologies to enhance ASR systems, guiding future research in diverse linguistic scenarios, and paving the way for the broader application of ASR innovations across various fields.

In [1, Proposition 8.1], Alexakis and Mazzeo showed that the renormalized area of minimal surfaces in the hyperbolic space $\mathbb{H}^3$ approaching the ideal boundary $\partial_\infty\mathbb{H}^3$ is the negative of the Willmore energy of their Euclidean conformal compactifications. In this paper, we investigate whether there are other geometric surface equations for which these two quantities satisfy a universal proportionality relation. We answer this question in the affirmative by extending the Alexakis–Mazzeo result to a broader class of geometric equations containing, as two distinguished cases, the equations defining generalized singular minimal surfaces in $\mathbb{R}^3$ and linear Weingarten surfaces in $\mathbb{H}^3$. Moreover, for rotational linear Weingarten surfaces, we express both energies in terms of generalized hyperelliptic integrals and show that their doubles form a family of rotational tori in $\mathbb{R}^3$ whose Willmore energies realize all values in $(2\pi^2,\infty)$, with a particular subfamily converging to the Clifford torus in $\mathcal{W}^{2,p}$ for every $1\leq p<\infty$, as the optimal value $2\pi^2$ is approached.

The Abstraction and Reasoning Corpus (ARC) is increasingly used to evaluate whether neural networks can infer task-specific transformations from few demonstrations. Recent ARC models differ in architecture, but also in broader training and evaluation protocols, including pretraining data, data augmentation, test-time setup, and optimization budgets, making conclusions about architecture benefits debatable. In this work, we study how the performance of the vision-inspired VARC and recursion-based TRM models is affected by pretraining and test-time training (TTT) budgets under a common inductive protocol on ARC-AGI-1. We cross five pretraining budgets with four TTT budgets for each test task. Each budget specifies a task-variant count and total sample presentations, defining the controlled training budgets. We observe that increasing TTT sample presentations improves the standard ARC pass@2 metric in all settings for both models. Increasing the TTT task-variant count consistently helps only TRM, whereas VARC often performs better with fewer variants. VARC also remains competitive without pretraining, while TRM benefits much more from it. These results show that pretraining and TTT budgets affect ARC evaluation and motivate reporting at comparable budgets.

We study sampling of a diffusion on $[0,T]$ conditioned on its terminal position lying on a prescribed ray, i.e. on an exactly observed bearing. For a linear drift we derive closed-form expressions for the Doob $h$-function and its gradient. For a broad class of tracking models admitting a Kronecker block structure, the guiding term is obtained from a single row of a matrix exponential, and the conditioned process decomposes into independent radial and angular components. For a nonlinear drift, we construct a guided process using the $h$-function of the linear model and prove absolute continuity of the law of the true conditioned process with respect to the law of the guided process, with an explicit likelihood ratio. This result is used to devise an auxiliary particle filter for bearings observed sequentially in time without noise. Numerical experiments show that the proposed filter approximates the filtering distribution and estimates the log-likelihood considerably more accurately than the shifted Rayleigh filter, at the cost of a higher computing time.

We apply the generalized monoid of words and logarithmic-prefix functionals of \cite{C1} to the regularization of the divergent product \[ P_{\omega}=\prod_{n=1}^{\infty} n^{(-1)^{\omega(n)}}, \] where \(\omega(n)\) is the number of distinct prime divisors of \(n\). The associated Dirichlet series \(D_{\omega}(s)=\sum_{n\ge1}(-1)^{\omega(n)}n^{-s}\) has a natural boundary at \(\Re(s)=0\), as established by Meretov (1987) and recovered here as a special case of the Estermann–Dahlquist theory of Euler products, which makes zeta regularization impossible: the derivative \(D_{\omega}'(0)\) does not exist. This product therefore lies completely beyond the reach of classical regularization methods. We construct the canonical net encoding the sign sequence, define the logarithmic-prefix functional with exponent \(\alpha=2\), and extract a finite value through the weighted evaluation functional of the monoid framework. The evaluation functional is shown to converge absolutely, and this convergence is unconditional: it follows from the classical zero-free region of \(\zeta(s)\) (de la Vallée Poussin), which yields the effective bound \(T_{\omega}(x)=O(x\,e^{-c\sqrt{\log x}})\) for some \(c>0\). Numerical computation up to \(N=10^7\) gives the regularized value \[ P_{\omega,\mathrm{ren}}\approx 0.7731, \] a new constant of arithmetic origin that cannot be obtained by any classical summation method. We also compute the second derivative of the Dirichlet series at \(s=1\), \[ D_{\omega}''(1)=8\,C_2\ln 2, \] where \(C_2\) is the Hardy–Littlewood twin prime constant, which provides an unconditional analytical link to the arithmetic structure of the sequence \((-1)^{\omega(n)}\). A self-contained proof of the natural boundary, based on the Estermann–Dahlquist theorem, is given in Appendix~A.

We prove that on a compact locally conformally Kähler manifold, the existence of a non-trivial parallel vector field with respect to a Gauduchon connection $\nabla^t$, for $t\ne \frac{1}{3}$, forces the metric to be either Vaisman (for $t=-1$), globally conformally Kähler (for $t=1$), or Kähler for all other values of $t$. As a consequence, we characterize compact Vaisman manifolds in terms of their Bismut holonomy group. Moreover, we show that in complex dimension two, the existence of a non-trivial $\nabla^\frac{1}{3}$-parallel vector field forces the metric to be globally conformally Kähler.

Latent field models are essential for spatial statistics, but the posterior distribution of the latent field, even with a Gaussian process (GP) prior, is analytically intractable whenever the likelihood is non-Gaussian. Existing approximations are either too slow or too restrictive for large non-Gaussian datasets. We present a scalable variational inference framework that mirrors the sparse nearest-neighbor autoregressive structure of the prior in the variational family, uses normalizing-flow conditionals for expressiveness, and amortizes all variational parameters via a single shared neural conditioner. Doubly stochastic optimization with reduced ancestor sets achieves polylogarithmic complexity per iteration.

We establish a general metastability principle showing that the local stability of a McKean-Vlasov equilibrium implies exponentially long confinement of the associated particle system. Consider interacting diffusions on $\mathbb{R}^d$ with a general smooth and bounded interaction kernel and a globally Lipschitz drift that is dissipative at large distances. Assume the McKean-Vlasov equation has at least one equilibrium which is locally exponentially stable. We prove that the probability that the empirical measure leaves a sufficiently small fixed Wasserstein-1 neighborhood of the selected equilibrium before time $T$ is bounded by $C(1+T)e^{-cN}$, where $N$ is the number of particles and $C,c>0$ are independent of $N$ and $T$. Consequently, confinement holds up to times $e^{\gamma N}$, for every $0<\gamma<c$, with exponentially small exit probability. The proof combines an explicit Volterra construction of the linearized Fokker-Planck semigroup, a mild representation of empirical fluctuations, and exponential maximal estimates for stochastic convolutions. Our approach does not require a gradient flow structure or reversibility of the equations.

Tabular foundation models (TFMs) are increasingly popular for their strong predictive performance. Many also provide posterior predictive distributions (PPDs), whose reliability remains less studied. We assess reliability through point prediction, distributional quality, marginal and subgroup calibration, and interval efficiency. We then introduce TabReliBench, a large classification and regression benchmark for these comparisons and dataset-level analyses. We find that TFMs' rankings change between TabArena and our benchmark and across reliability metrics. Regression TFMs usually achieve near-nominal marginal coverage and retain narrow intervals, but this favorable marginal behavior does not imply valid conditional coverage. Dataset-level analyses show that classification difficulty often accompanies poor calibration, whereas regression predictive performance is not a reliable indicator of calibration. Ensembling improves predictive performance but does not consistently improve calibration of PPDs; TFMs also benefit little from conformal post-processing, which does not reliably resolve subgroup gaps. Reliable TFM selection therefore requires multidimensional evaluation beyond aggregate rankings. Code is available at https://github.com/Tianqi-Zhao/TabReliBench.

Reinforcement learning with verifiable rewards (RLVR) has substantially improved LLM reasoning, yet growing evidence suggests it mainly performs distribution sharpening: concentrating probability mass on reasoning paths the base model already produces. Recent work therefore seeks RLVR's gains without needing reward signals, by sampling from the power distribution $\pi_\alpha \propto \pi_{\text{base}}^\alpha$ via Markov chain Monte Carlo or sequential Monte Carlo. We show that these samplers fail to converge to $\pi_\alpha$ under realistic compute budgets, and that their gains are largely recreated via simple Best-of-$N$ with appropriate grading . This suggests they succeed by selecting high-probability sequences, not by approximating $\pi_\alpha$, raising a natural question: is the power distribution itself a worthwhile target? We test this hypothesis directly by fine-tuning the base model to match $\pi_\alpha$, sidestepping sampler convergence issues entirely. Despite approximating the target far more faithfully than existing samplers, this strategy yields gains that are noisy and specific to certain models and sampling tasks. Together, our results suggest that the value of power sampling lies not in faithfully targeting $\pi_\alpha$, but in the simpler goal of extracting sequences the base model already deems likely.