A Finite E-Group of Nilpotency Class Three

A group is an E-group if every element commutes with each of its endomorphic images. Caranti asked whether a finite E-group can have nilpotency class three. We prove that the $3$-group of order $3^{84}$ introduced by Abdollahi, Faghihi, and Mohammadi Hassanabadi, and later shown by Abdollahi, Faghihi, Linton, and O'Brien to have the corresponding automorphism property, is an E-group. Let $P$ denote this group and put $V=P/Φ(P)\cong \mathbb{F}_3^9$. The nine power relations of $P$ determine a linear map $q:V\longrightarrowΛ^2 V$. We prove that $q$ has no nonzero proper subspace $U$ satisfying $q(U)\subseteqΛ^2 U$. Since the image induced by any endomorphism of $P$ on $V$ has precisely this closure property, every endomorphism acts on $V$ either invertibly or trivially. The invertible case is the known A-group case. In the trivial case the image first lies in $Φ(P)=P'$, and the power relations then force it into $Ω_1(P')=Z(P)$. Thus every element commutes with every endomorphic image. The tensor rigidity is reduced to an exact finite calculation on the $9841$ points of $\mathrm{PG}(8,3)$.

An Exact Counterexample to Carlson's Associated-Prime Depth Conjecture from a Group of Order 128

In Question~3.1 of his 1995 paper on depth and transfer, Carlson asked whether the depth of a finite-group cohomology ring is always realized by the dimension of one of its associated primes. We give a negative answer. Let \[ G=\SG{128}{859},\qquad k=\kbar. \] An exact presentation certificate proves that $\depth H^*(G;k)=2$. Okuyama's associated-prime theorem would convert an associated prime of dimension two into a rank-two elementary abelian subgroup $E\leq G$ satisfying $\depth H^*(C_G(E);k)=2$. We enumerate all $75$ rank-two elementary abelian subgroups of $G$ and obtain six centralizer types. Duflot's theorem gives depth at least three for four types, while exact ideal-quotient certificates exhibit regular sequences of length three for the remaining two. Hence every rank-two centralizer has cohomological depth at least three, so $H^*(G;k)$ has no associated prime of dimension two. The finite group presentation, the three cohomology-ring presentations, the enumeration summary, and the exact algebraic certificates are included for independent verification.

An Explicit Counterexample to Stanley's Rankwise Lower-Bound Conjecture for Differential Posets

In Problem~6 of his 1988 paper on differential posets, Stanley asked for the least possible cardinality of a fixed rank of an $r$-differential poset and suggested that the minimum should be attained by $Y^r$, the $r$-fold Cartesian power of Young's lattice. We disprove the resulting universal coefficientwise lower bound. For every $r\geq 3$, we construct an infinite $r$-differential poset $P^{(r)}$ satisfying \[ \card{P^{(r)}_4} =\card{(Y^r)_4}-\left\lfloor\frac r3\right\rfloor. \] For $r=3$, the construction replaces thirteen rank-four lower-cover blocks of $Y^3$ by twelve blocks with the same point and pair incidence multiplicities, producing the initial rank sequence $1,3,9,22,50$ instead of $1,3,9,22,51$. A reflection extension then yields an infinite differential poset. The construction does not address the cases $r=1$ and $r=2$.

RecRec: Latent Interests Recursive Reasoning for Sequential Recommendation

Sequential recommender systems rely on a single forward pass to encode user interaction histories and predict the next item. Increasing inference-time computation through latent reasoning, with the model proceeding step by step before the final prediction, has been recently explored in sequential recommendation with promising results. However, how to structure the reasoning process for sequential recommendation remains an open question. Existing approaches couple reasoning and prediction in a single $d$-dimensional state, limiting reasoning depth and often relying on multi-stage pipelines with reinforcement learning (RL). We propose RecRec (Recursive Reasoning for Recommendation), an RL-free framework that decouples reasoning from prediction, overcoming the fixed $d$-dimensional state bottleneck of prior methods. RecRec consists of a Context Compressor and a Recursive Reasoner, trained in two simple supervised stages. The Context Compressor distills the backbone's hidden states into a small set of latent interests, with an Interest Diversity Regularizer encouraging each interest to capture a distinct aspect of user behavior. The Recursive Reasoner then refines these interests by reasoning in a separate intermediate latent space. Deep supervision lets the reasoning depth be freely adjusted at inference without retraining. On four real-world datasets, RecRec outperforms state-of-the-art reasoning-enhanced methods, and on three of four datasets, gains extend past the training-time depth. Our findings point to a decoupled, multi-vector recipe that unleashes latent reasoning from the single-state bottleneck of prior methods, suggesting reasoning-state structure as a design axis to explore further in sequential recommendation.

Do We Really Need Multimodal Emotion Language Models Larger Than 1B Parameters?

Recent advances in multimodal large language models (MLLMs) have significantly improved the performance of multimodal emotion recognition (MER) and enabled interpretable description generation by jointly modeling video, audio, and language, etc. However, these performance improvements are often accompanied by an increase in model parameter size (e.g, at least 7B), which simultaneously incurs high computational costs and reduces inference efficiency, thereby hindering real-time deployment on resource-constrained platforms such as robots and mobile devices. This raises a fundamental question: do we really need the multimodal MER model larger than 1B parameters for high-quality MER? In this paper, we challenge the assumption that larger models are inherently necessary and proposes a lightweight MER framework (called Light-MER), which achieves better and faster multimodal sentiment understanding and recognition through knowledge distillation. It can transfer knowledge from a strong, large-scale teacher model to a lightweight sub-billion-parameter student model, aiming to preserve rich multimodal emotion reasoning and recognition while substantially improving deployment efficiency. Specifically, we introduce two new optimization strategies to enhance knowledge transfer: (1) a new optimal transport loss that combines Sliced Wasserstein Distance with hidden-state alignment, and (2) a new multi-reward optimization strategy based on GRPO that balances MER performance and efficiency, aimed at further enhancing the learning capabilities of student models. Extensive experiments on nine benchmark datasets demonstrate that Light-MER achieves state-of-the-art performance while significantly improving inference efficiency. This highlights the strong potential of small multimodal emotion language models for future research. Code is available at https://github.com/GAIR-Lab/Light-MER.

Stream-aware Side Adaptation for Large Pre-trained Multimodal Embedding Models in Sequential Recommendation

Recently, large pretrained multimodal embedding models such as Qwen3-VL Embedding have shown strong promise for sequential recommendation, as they provide reusable semantic item representations across modalities and domains. However, directly using these embeddings often leads to suboptimal performance because of domain misalignment. Efficient side adaptation is therefore an attractive solution. Although adapting all backbone layers should help, existing side adapters often degrade with depth, prompting layer dropping despite the loss of useful hidden states. This is due to two major challenges: (1) the lack of modeling in selecting fused representations during residual addition, and (2) the insufficient preservation of earlier representations during progressive sigmoid fusion. This paper therefore asks a practical question: How can we design a side adaptation approach that effectively unlocks the potential of large pre-trained multimodal embedding models? To address this question, we propose Stresa, a stream-aware side-adaptation framework for frozen large pre-trained multimodal embedding models in sequential recommendation. Stresa introduces Stream-aware Hidden-Adapter Fusion (SHAF) to preserve historical side memory during fusion and Residual Stream Adapter (ReSA) to produce selective residual updates across layers. Empirically, Stresa consistently outperforms standard side adapters and state-of-the-art baselines on public datasets across multiple backbone embedding models. These results highlight the promise of adapting large embedding models for sequential recommendation. Our code is publicly available at https://github.com/GAIR-Lab/Stresa.

SkillAudit: From Fixed-Suite Benchmarking to Skill-Centered Assessment

Agent skills have become a practical way to extend large language model agents, but the growing skill ecosystem still lacks a reliable way to judge whether a skill is worth deploying. Existing evaluation methods remain largely anchored to fixed task suites, assessing skills through performance on predefined tasks and environments. As skill marketplaces expand, this paradigm becomes inadequate: fixed suites can conflate a skill's marginal contribution with backbone strength and miss its value when tasks fall outside the skill's intended scope. We introduce SkillAudit, an end-to-end framework for skill-centered assessment that takes an arbitrary agent skill as input and automatically generates a comprehensive, multi-dimensional evaluation report spanning utility, efficiency/cost, and safety. SkillAudit focuses on the skill artifact itself and constructs capability-aligned evaluation tasks directly from the skill package. The generated tasks are conducted in isolated sandbox environments to collect execution evidence, followed by automated checks with LLM-based judging to produce auditable results. To dissect the agent skills, we propose the baseline comparison principle to measure utility and efficiency/cost, and introduce a two-stage detection paradigm combining static semantic analysis with dynamic runtime verification to assess safety risks. After scanning top-ranked real-world skill packages spanning 23 occupational categories, we found that over 7% of skills are at risky status.

Benchmarking Multimodal Large Language Models for Missing Modality Completion in Product Catalogues

Missing-modality information on e-commerce platforms, such as absent product images or textual descriptions, often arises from annotation errors or incomplete metadata, impairing both product presentation and downstream applications such as recommendation systems. Motivated by the multimodal generative capabilities of recent Multimodal Large Language Models (MLLMs), this work investigates a fundamental yet underexplored question: can MLLMs generate missing modalities for products in e-commerce scenarios? We propose the Missing Modality Product Completion Benchmark (MMPCBench), which consists of two sub-benchmarks: a Content Quality Completion Benchmark and a Recommendation Benchmark. We further evaluate six state-of-the-art MLLMs from the Qwen2.5-VL and Gemma-3 model families across nine real-world e-commerce categories, focusing on image-to-text and text-to-image completion tasks. Experimental results show that while MLLMs can capture high-level semantics, they struggle with fine-grained word-level and pixel- or patch-level alignment. In addition, performance varies substantially across product categories and model scales, and we observe no trivial correlation between model size and performance, in contrast to trends commonly reported in mainstream benchmarks. We also explore Group Relative Policy Optimization (GRPO) to better align MLLMs with this task. GRPO improves image-to-text completion but does not yield gains for text-to-image completion. Overall, these findings expose the limitations of current MLLMs in real-world cross-modal generation and represent an early step toward more effective missing-modality product completion.

GenCP: Towards Generative Modeling Paradigm of Coupled Physics

Real-world physical systems are inherently complex, often involving the coupling of multiple physics, making their simulation both highly valuable and challenging. Many mainstream approaches face challenges when dealing with decoupled data. Besides, they also suffer from low efficiency and fidelity in strongly coupled spatio-temporal physical systems. Here we propose GenCP, a novel and elegant generative paradigm for coupled multiphysics simulation. By formulating coupled-physics modeling as a probability modeling problem, our key innovation is to integrate probability density evolution in generative modeling with iterative multiphysics coupling, thereby enabling training on data from decoupled simulation and inferring coupled physics during sampling. We also utilize operator-splitting theory in the space of probability evolution to establish error controllability guarantees for this "conditional-to-joint" sampling scheme. We evaluate our paradigm on a synthetic setting and three challenging multi-physics scenarios to demonstrate both principled insight and superior application performance of GenCP. Code is available at this repo: github.com/AI4Science-WestlakeU/GenCP.

RealPDEBench: A Benchmark for Complex Physical Systems with Real-World Data

Predicting the evolution of complex physical systems remains a central problem in science and engineering. Despite rapid progress in scientific Machine Learning (ML) models, a critical bottleneck is the lack of expensive real-world data, resulting in most current models being trained and validated on simulated data. Beyond limiting the development and evaluation of scientific ML, this gap also hinders research into essential tasks such as sim-to-real transfer. We introduce RealPDEBench, the first benchmark for scientific ML that integrates real-world measurements with paired numerical simulations. RealPDEBench consists of five datasets, three tasks, eight metrics, and ten baselines. We first present five real-world measured datasets with paired simulated datasets across different complex physical systems. We further define three tasks, which allow comparisons between real-world and simulated data, and facilitate the development of methods to bridge the two. Moreover, we design eight evaluation metrics, spanning data-oriented and physics-oriented metrics, and finally benchmark ten representative baselines, including state-of-the-art models, pretrained PDE foundation models, and a traditional method. Experiments reveal significant discrepancies between simulated and real-world data, while showing that pretraining with simulated data consistently improves both accuracy and convergence. In this work, we hope to provide insights from real-world data, advancing scientific ML toward bridging the sim-to-real gap and real-world deployment. Our benchmark, datasets, and instructions are available at https://realpdebench.github.io/.