OpenRaiser/Intern-Atlas huggingface dataset:
Formalization of Amicable Numbers Theory
Venue: arXiv cs.LO 2026
Year: 2026

This paper presents a formalization of the theory of amicable numbers in the Lean~4 proof assistant. Two positive integers $m$ and $n$ are called an amicable pair if the sum of proper divisors of $m$ equals $n$ and the sum of proper divisors of $n$ equals $m$. Our formalization introduces the proper divisor sum function $\propersum(n) = σ(n) - n$, defines the concepts of amicable pairs and amicable numbers, and computationally verifies historically famous amicable pairs. Furthermore, we formalize basic structural theorems, including symmetry, non-triviality, and connections to abundant/deficient numbers. A key contribution is the complete formal proof of the classical Thābit formula (9th century), using index-shifting and the \texttt{zify} tactic. Additionally, we provide complete formal proofs of both Thābit's rule and Euler's generalized rule (1747), two fundamental theorems for generating amicable pairs. A major achievement is the first complete formalization of the Borho-Hoffmann breeding method (1986), comprising 540 lines with 33 theorems and leveraging automated algebra tactics (\texttt{zify} and \texttt{ring}) to verify complex polynomial identities. We also formalize extensions including sociable numbers (aliquot cycles), betrothed numbers (quasi-amicable pairs), parity constraint theorems, and computational search bounds for coprime pairs ($>10^{65}$). We verify the smallest sociable cycle of length 5 (Poulet's cycle) and computationally verify specific instances. The formalization comprises 2076 lines of Lean code organized into Mathlib-candidate and paper-specific modules, with 139 theorems and all necessary infrastructure for divisor sum multiplicativity and coprimality reasoning.
WaveMan: mmWave-Based Room-Scale Human Interaction Perception for Humanoid Robots
Venue: arXiv cs.RO 2026
Year: 2026

Reliable humanoid-robot interaction (HRI) in household environments is constrained by two fundamental requirements, namely robustness to unconstrained user positions and preservation of user privacy. Millimeter-wave (mmWave) sensing inherently supports privacy-preserving interaction, making it a promising modality for room-scale HRI. However, existing mmWave-based interaction-sensing systems exhibit poor spatial generalization at unseen distances or viewpoints. To address this challenge, we introduce WaveMan, a spatially adaptive room-scale perception system that restores reliable human interaction sensing across arbitrary user positions. WaveMan integrates viewpoint alignment and spectrogram enhancement for spatial consistency, with dual-channel attention for robust feature extraction. Experiments across five participants show that, under fixed-position evaluation, WaveMan achieves the same cross-position accuracy as the baseline with five times fewer training positions. In random free-position testing, accuracy increases from 33.00% to 94.33%, enabled by the proposed method. These results demonstrate the feasibility of reliable, privacy-preserving interaction for household humanoid robots across unconstrained user positions.
AntiPaSTO: Self-Supervised Honesty Steering via Anti-Parallel Representations
Venue: arXiv cs.LG 2026
Year: 2026

As models grow more capable, humans cannot reliably verify what they say. Scalable steering requires methods that are internal, self-supervised, and transfer out-of-distribution; existing methods satisfy some but not all three. We introduce AntiPaSTO, which separates representations along an antiparallel axis (+1/-1 produce opposite shifts), with coherence constraints preventing collapse. Human input is minimal: two contrasting words inserted into template sentences, no preference labels. Using 800 such pairs on Gemma-3-1B, AntiPaSTO beats prompting baselines by 6.9x on DailyDilemmas and maintains bidirectional control where prompting triggers refusal.
Anatomy Aware Cascade Network: Bridging Epistemic Uncertainty and Geometric Manifold for 3D Tooth Segmentation
Venue: arXiv cs.CV 2026
Year: 2026

Accurate three-dimensional (3D) tooth segmentation from Cone-Beam Computed Tomography (CBCT) is a prerequisite for digital dental workflows. However, achieving high-fidelity segmentation remains challenging due to adhesion artifacts in naturally occluded scans, which are caused by low contrast and indistinct inter-arch boundaries. To address these limitations, we propose the Anatomy Aware Cascade Network (AACNet), a coarse-to-fine framework designed to resolve boundary ambiguity while maintaining global structural consistency. Specifically, we introduce two mechanisms: the Ambiguity Gated Boundary Refiner (AGBR) and the Signed Distance Map guided Anatomical Attention (SDMAA). The AGBR employs an entropy based gating mechanism to perform targeted feature rectification in high uncertainty transition zones. Meanwhile, the SDMAA integrates implicit geometric constraints via signed distance map to enforce topological consistency, preventing the loss of spatial details associated with standard pooling. Experimental results on a dataset of 125 CBCT volumes demonstrate that AACNet achieves a Dice Similarity Coefficient of 90.17 \% and a 95\% Hausdorff Distance of 3.63 mm, significantly outperforming state-of-the-art methods. Furthermore, the model exhibits strong generalization on an external dataset with an HD95 of 2.19 mm, validating its reliability for downstream clinical applications such as surgical planning. Code for AACNet is available at https://github.com/shiliu0114/AACNet.
Judging Against the Reference: Uncovering Knowledge-Driven Failures in LLM-Judges on QA Evaluation
Venue: arXiv cs.CL 2026
Year: 2026

While large language models (LLMs) are increasingly used as automatic judges for question answering (QA) and other reference-conditioned evaluation tasks, little is known about their ability to adhere to a provided reference. We identify a critical failure mode of such reference-based LLM QA evaluation: when the provided reference conflicts with the judge model's parametric knowledge, the resulting scores become unreliable, substantially degrading evaluation fidelity. To study this phenomenon systematically, we introduce a controlled swapped-reference QA framework that induces reference-belief conflicts. Specifically, we replace the reference answer with an incorrect entity and construct diverse pairings of original and swapped references with correspondingly aligned candidate answers. Surprisingly, grading reliability drops sharply under swapped references across a broad set of judge models. We empirically show that this vulnerability is driven by judges' over-reliance on parametric knowledge, leading judges to disregard the given reference under conflict. Finally, we find that this failure persists under common prompt-based mitigation strategies, highlighting a fundamental limitation of LLM-as-a-judge evaluation and motivating reference-based protocols that enforce stronger adherence to the provided reference.
Principal ideal problem and ideal shortest vector over rational primes in power-of-two cyclotomic fields
Venue: arXiv cs.CR 2026
Year: 2026

The shortest vector problem (SVP) over ideal lattices is closely related to the Ring-LWE problem, which is widely used to build post-quantum cryptosystems. Power-of-two cyclotomic fields are frequently adopted to instantiate Ring-LWE. Pan et al. (EUROCRYPT~2021) explored the SVP over ideal lattices via the decomposition fields and, in particular determined the length of the shortest vector in prime ideals lying over rational primes $p\equiv3,5\pmod{8}$ in power-of-two cyclotomic fields via explicit construction of reduced lattice bases. In this work, we first provide a new method (different from analyzing lattice bases) to analyze the length of the shortest vector in prime ideals in $\mathbb{Z}[ζ_{2^{n+1}}]$ when $p\equiv3,5\pmod{8}$. Then we precisely characterize the length of the shortest vector in the cases of $p\equiv7,9\pmod{16}$. Furthermore, we derive a new upper bound $\sqrt[4]{2^{2n+1}p}$ for this length, which is tighter than the bound $2^n\sqrt[4]{p}$ obtained from Minkowski's theorem. Our key technique is to investigate whether a generator of a principal ideal can achieve the shortest length after embedding as a vector. If this holds for the ideal, finding the shortest vector in this ideal can be reduced to finding its shortest generator.
A Parity-Consistent Decomposition Method for the Weight Distribution of Pre-Transformed Polar Codes
Venue: arXiv cs.IT 2026
Year: 2026

This paper introduces an efficient algorithm based on the Parity-Consistent Decomposition (PCD) method to determine the WD of pre-transformed polar codes. First, to address the bit dependencies introduced by the pre-transformation matrix, we propose an iterative algorithm to construct an \emph{Expanded Information Set}. By expanding the information bits within this set into 0s and 1s, we eliminate the correlations among information bits, thereby enabling the recursive calculation of the Hamming weight distribution using the \emph{PCD method}. Second, to further reduce computational complexity, we establish the theory of equivalence classes for pre-transformed polar codes. Codes within the same equivalence class share an identical weight distribution but correspond to different \emph{Expanded Information Set} sizes. By selecting the pre-transformation matrix that minimizes the \emph{Expanded Information Set} size within an equivalence class, we optimize the computation process. Numerical results demonstrate that the proposed method significantly reduces computational complexity compared to existing deterministic algorithms.
Sparse Point-wise Privacy Leakage: Mechanism Design and Fundamental Limits
Venue: arXiv cs.IT 2026
Year: 2026

We study an information-theoretic privacy mechanism design problem, where an agent observes useful data $Y$ that is arbitrarily correlated with sensitive data $X$, and design disclosed data $U$ generated from $Y$ (the agent has no direct access to $X$). We introduce \emph{sparse point-wise privacy leakage}, a worst-case privacy criterion that enforces two simultaneous constraints for every disclosed symbol $u\in\mathcal{U}$: (i) $u$ may be correlated with at most $N$ realizations of $X$, and (ii) the total leakage toward those realizations is bounded. In the high-privacy regime, we use concepts from information geometry to obtain a local quadratic approximation of mutual information which measures utility between $U$ and $Y$. When the leakage matrix $P_{X|Y}$ is invertible, this approximation reduces the design problem to a sparse quadratic maximization, known as the Rayleigh-quotient problem, with an $\ell_0$ constraint. We further show that, for the approximated problem, one can without loss of optimality restrict attention to a binary released variable $U$ with a uniform distribution. For small alphabet sizes, the exact sparsity-constrained optimum can be computed via combinatorial support enumeration, which quickly becomes intractable as the dimension grows. For general dimensions, the resulting sparse Rayleigh-quotient maximization is NP-hard and closely related to sparse principal component analysis (PCA). We propose a convex semidefinite programming (SDP) relaxation that is solvable in polynomial time and provides a tractable surrogate for the NP-hard design, together with a simple rounding procedure to recover a feasible leakage direction. We also identify a sparsity threshold beyond which the sparse optimum saturates at the unconstrained spectral value and the SDP relaxation becomes tight.
Estimators for Substitution Rates in Genomes from Read Data
Venue: arXiv cs.IT 2026
Year: 2026

We study the problem of estimating the mutation rate between two sequences from noisy sequencing reads. Existing alignment-free methods typically assume direct access to the full sequences. We extend these methods to the sequencing framework, where only noisy reads from the sequences are observed. We use a simple model in which both mutations and sequencing errors are substitutions. We propose multiple estimators, provide theoretical guarantees for one of them, and evaluate the others through simulations.
On the Sequence Reconstruction Problem for the Single-Deletion Two-Substitution Channel
Venue: arXiv cs.IT 2026
Year: 2026

The Levenshtein sequence reconstruction problem studies the reconstruction of a transmitted sequence from multiple erroneous copies of it. A fundamental question in this field is to determine the minimum number of erroneous copies required to guarantee correct reconstruction of the original sequence. This problem is equivalent to determining the maximum possible intersection size of two error balls associated with the underlying channel. Existing research on the sequence reconstruction problem has largely focused on channels with a single type of error, such as insertions, deletions, or substitutions alone. However, relatively little is known for channels that involve a mixture of error types, for instance, channels allowing both deletions and substitutions. In this work, we study the sequence reconstruction problem for the single-deletion two-substitution channel, which allows one deletion and at most two substitutions applied to the transmitted sequence. Specifically, we prove that if two $q$-ary length-$n$ sequences have the Hamming distance $d\geq 2$, where $q\geq 2$ is any fixed integer, then the intersection size of their error balls under the single-deletion two-substitution channel is upper bounded by $(q^2-1)n^2-(3q^2+5q-5)n+O_q(1)$, where $O_q(1)$ is a constant independent from $n$ but dependent on $q$. Moreover, we show that this upper bound is tight up to an additive constant.
VirtualEnv: A Platform for Embodied AI Research
Venue: arXiv cs.AI 2026
Year: 2026

As large language models (LLMs) continue to improve in reasoning and decision-making, there is a growing need for realistic and interactive environments where their abilities can be rigorously evaluated. We present VirtualEnv, a next-generation simulation platform built on Unreal Engine 5 that enables fine-grained benchmarking of LLMs in embodied and interactive scenarios. VirtualEnv supports rich agent-environment interactions, including object manipulation, navigation, and adaptive multi-agent collaboration, as well as game-inspired mechanics like escape rooms and procedurally generated environments. We provide a user-friendly API built on top of Unreal Engine, allowing researchers to deploy and control LLM-driven agents using natural language instructions. We integrate large-scale LLMs and vision-language models (VLMs), such as GPT-based models, to generate novel environments and structured tasks from multimodal inputs. Our experiments benchmark the performance of several popular LLMs across tasks of increasing complexity, analyzing differences in adaptability, planning, and multi-agent coordination. We also describe our methodology for procedural task generation, task validation, and real-time environment control. VirtualEnv is released as an open-source platform, we aim to advance research at the intersection of AI and gaming, enable standardized evaluation of LLMs in embodied AI settings, and pave the way for future developments in immersive simulations and interactive entertainment.
FlyCo: Foundation Model-Empowered Drones for Autonomous 3D Structure Scanning in Open-World Environments
Venue: arXiv cs.RO 2026
Year: 2026

Autonomous 3D scanning of open-world target structures via drones remains challenging despite broad applications. Existing paradigms rely on restrictive assumptions or effortful human priors, limiting practicality, efficiency, and adaptability. Recent foundation models (FMs) offer great potential to bridge this gap. This paper investigates a critical research problem: What system architecture can effectively integrate FM knowledge for this task? We answer it with FlyCo, a principled FM-empowered perception-prediction-planning loop enabling fully autonomous, prompt-driven 3D target scanning in diverse unknown open-world environments. FlyCo directly translates low-effort human prompts (text, visual annotations) into precise adaptive scanning flights via three coordinated stages: (1) perception fuses streaming sensor data with vision-language FMs for robust target grounding and tracking; (2) prediction distills FM knowledge and combines multi-modal cues to infer the partially observed target's complete geometry; (3) planning leverages predictive foresight to generate efficient and safe paths with comprehensive target coverage. Building on this, we further design key components to boost open-world target grounding efficiency and robustness, enhance prediction quality in terms of shape accuracy, zero-shot generalization, and temporal stability, and balance long-horizon flight efficiency with real-time computability and online collision avoidance. Extensive challenging real-world and simulation experiments show FlyCo delivers precise scene understanding, high efficiency, and real-time safety, outperforming existing paradigms with lower human effort and verifying the proposed architecture's practicality. Comprehensive ablations validate each component's contribution. FlyCo also serves as a flexible, extensible blueprint, readily leveraging future FM and robotics advances. Code will be released.
GPU accelerated surface-based gaze mapping for XR experiences
Venue: arXiv cs.HC 2026
Year: 2026

Extended reality is a fast-growing domain for which there is an increasing need to analyze and understand user behavior. In particular, understanding human visual attention during immersive experiences is crucial for many applications. The visualization and analysis of visual attention are commonly done by building fixation density maps from eye-tracking data. Such visual attention mapping is well mastered for 3 degrees of freedom (3DoF) experiences (\textit{i.e.}, involving 360 images or videos) but much less so for 6DoFs data, when the user can move freely in the 3D space. In that case, the visual attention information has to be mapped onto the 3D objects themselves. Some solutions exist for constructing such surface-based 6DoFs attention maps, however, they own several drawbacks: processing time, strong dependence on mesh resolution and/or texture mapping, and/or unpractical data representation for further processing. In this context, we propose a novel GPU-based algorithm that resolves the issues above while being generated in interactive time and rendered in real-time. Experiment on a challenging scene demonstrates the accuracy and robustness of our approach. To stimulate research in this area, the source code is publicly released and integrated into PLUME for ease of use in XR experiments.
A Multimodal Dataset of Student Oral Presentations with Sensors and Evaluation Data
Venue: arXiv cs.HC 2026
Year: 2026

Oral presentation skills are a critical component of higher education, yet comprehensive datasets capturing real-world student performance across multiple modalities remain scarce. To address this gap, we present SOPHIAS (Student Oral Presentation monitoring for Holistic Insights & Analytics using Sensors), a 12-hour multimodal dataset containing recordings of 50 oral presentations (10-15-minute presentation followed by 5-15-minute Q&A) delivered by 65 undergraduate and master's students at the Universidad Autonoma de Madrid. SOPHIAS integrates eight synchronized sensor streams from high-definition webcams, ambient and webcam audio, eye-tracking glasses, smartwatch physiological sensors, and clicker, keyboard, and mouse interactions. In addition, the dataset includes slides and rubric-based evaluations from teachers, peers, and self-assessments, along with timestamped contextual annotations. The dataset captures presentations conducted in real classroom settings, preserving authentic student behaviors, interactions, and physiological responses. SOPHIAS enables the exploration of relationships between multimodal behavioral and physiological signals and presentation performance, supports the study of peer assessment, and provides a benchmark for developing automated feedback and Multimodal Learning Analytics tools. The dataset is publicly available for research through GitHub and Science Data Bank.
Beyond Entangled Planning: Task-Decoupled Planning for Long-Horizon Agents
Venue: arXiv cs.AI 2026
Year: 2026

Recent advances in large language models (LLMs) have enabled agents to autonomously execute complex, long-horizon tasks, yet planning remains a primary bottleneck for reliable task execution. Existing methods typically fall into two paradigms: step-wise planning, which is reactive but often short-sighted; and one-shot planning, which generates a complete plan upfront yet is brittle to execution errors. Crucially, both paradigms suffer from entangled contexts, where the agent must reason over a monolithic history spanning multiple sub-tasks. This entanglement increases cognitive load and lets local errors propagate across otherwise independent decisions, making recovery computationally expensive. To address this, we propose Task-Decoupled Planning (TDP), a training-free framework that replaces entangled reasoning with task decoupling. TDP decomposes tasks into a directed acyclic graph (DAG) of sub-goals via a Supervisor. Using a Planner and Executor with scoped contexts, TDP confines reasoning and replanning to the active sub-task. This isolation prevents error propagation and corrects deviations locally without disrupting the workflow. Results on TravelPlanner, ScienceWorld, and HotpotQA show that TDP outperforms strong baselines while reducing token consumption by up to 82%, demonstrating that sub-task decoupling improves both robustness and efficiency for long-horizon agents.
OODEval: Evaluating Large Language Models on Object-Oriented Design
Venue: arXiv cs.SE 2026
Year: 2026

Recent advances in large language models (LLMs) have driven extensive evaluations in software engineering. however, most prior work concentrates on code-level tasks, leaving software design capabilities underexplored. To fill this gap, we conduct a comprehensive empirical study evaluating 29 LLMs on object-oriented design (OOD) tasks. Owing to the lack of standardized benchmarks and metrics, we introduce OODEval, a manually constructed benchmark comprising 50 OOD tasks of varying difficulty, and OODEval-Human, the first human-rated OOD benchmark, which includes 940 undergraduate-submitted class diagrams evaluated by instructors. We further propose CLUE (Class Likeness Unified Evaluation), a unified metric set that assesses both global correctness and fine-grained design quality in class diagram generation. Using these benchmarks and metrics, we investigate five research questions: overall correctness, comparison with humans, model dimension analysis, task feature analysis, and bad case analysis. The results indicate that while LLMs achieve high syntactic accuracy, they exhibit substantial semantic deficiencies, particularly in method and relationship generation. Among the evaluated models, Qwen3-Coder-30B achieves the best overall performance, rivaling DeepSeek-R1 and GPT-4o, while Gemma3-4B-IT outperforms GPT-4o-Mini despite its smaller parameter scale. Although top-performing LLMs nearly match the average performance of undergraduates, they remain significantly below the level of the best human designers. Further analysis shows that parameter scale, code specialization, and instruction tuning strongly influence performance, whereas increased design complexity and lower requirement readability degrade it. Bad case analysis reveals common failure modes, including keyword misuse, missing classes or relationships, and omitted methods.