Daily Papers

The inference of topological principles is a key problem in structured reconstruction. We observe that wrongly predicted topological relationships are often incurred by the lack of holistic geometry clues in low-level features. Inspired by the fact that massive signals can be compactly described with frequency analysis, we experimentally explore the efficiency and tendency of learning structure geometry in the frequency domain. Accordingly, we propose a frequency-domain feature learning strategy (F-Learn) to fuse scattered geometric fragments holistically for topology-intact structure reasoning. Benefiting from the parsimonious design, the F-Learn strategy can be easily deployed into a deep reconstructor with a lightweight model modification. Experiments demonstrate that the F-Learn strategy can effectively introduce structure awareness into geometric primitive detection and topology inference, bringing significant performance improvement to final structured reconstruction. Code and pre-trained models are available at https://github.com/Geo-Tell/F-Learn.

With the development of large language models (LLMs), particularly with the introduction of the long reasoning chain technique, the reasoning ability of LLMs in complex problem-solving has been significantly enhanced. While acknowledging the power of long reasoning chains, we cannot help but wonder: Why do different reasoning chains perform differently in reasoning? What components of the reasoning chains play a key role? Existing studies mainly focus on evaluating reasoning chains from a functional perspective, with little attention paid to their structural mechanisms. To address this gap, this work is the first to analyze and evaluate the quality of the reasoning chain from a structural perspective. We apply persistent homology from Topological Data Analysis (TDA) to map reasoning steps into semantic space, extract topological features, and analyze structural changes. These changes reveal semantic coherence, logical redundancy, and identify logical breaks and gaps. By calculating homology groups, we assess connectivity and redundancy at various scales, using barcode and persistence diagrams to quantify stability and consistency. Our results show that the topological structural complexity of reasoning chains correlates positively with accuracy. More complex chains identify correct answers sooner, while successful reasoning exhibits simpler topologies, reducing redundancy and cycles, enhancing efficiency and interpretability. This work provides a new perspective on reasoning chain quality assessment and offers guidance for future optimization.

Spatial reasoning depends not only on metric properties such as distance, angle, and shape, but also on topological relations that remain invariant under continuous deformation. Cognitive science identifies these relations as foundational to spatial understanding, yet foundation-model evaluations largely focus on metric or viewpoint-dependent relations. We introduce MindTopo, a benchmark of topological intuition across five properties grounded in cognitive science and formal topology: continuity, separation, order, enclosure, and knots. MindTopo evaluates each property at two cognitive levels. Reasoning asks a model to identify topological relations or infer how they change. Planning instantiates a foundation model as a closed-loop agent whose policy selects environment actions. MindTopo contains 11,030 instances across 13 procedurally generated task types with controllable difficulty. We benchmark 14 MLLMs and study agent configurations augmented with image and video generation, including 3 video generative models in planning settings. Every MLLM performs better on reasoning than on planning, and the best-performing model remains far below observed human performance. On Qwen3-VL-2B-Instruct, supervised fine-tuning and reinforcement learning improve reasoning more than planning. Generated observations retain local cues and reach plausible endpoints, but audited rollouts do not reliably follow environment dynamics or preserve topology across transitions. Our website is at https://mind-topo.github.io/

Structure reasoning is a fundamental capability of large language models (LLMs), enabling them to reason about structured commonsense and answer multi-hop questions. However, existing benchmarks for structure reasoning mainly focus on horizontal and coordinate structures (e.g. graphs), overlooking the hierarchical relationships within them. Hierarchical structure reasoning is crucial for human cognition, particularly in memory organization and problem-solving. It also plays a key role in various real-world tasks, such as information extraction and decision-making. To address this gap, we propose HiBench, the first framework spanning from initial structure generation to final proficiency assessment, designed to benchmark the hierarchical reasoning capabilities of LLMs systematically. HiBench encompasses six representative scenarios, covering both fundamental and practical aspects, and consists of 30 tasks with varying hierarchical complexity, totaling 39,519 queries. To evaluate LLMs comprehensively, we develop five capability dimensions that depict different facets of hierarchical structure understanding. Through extensive evaluation of 20 LLMs from 10 model families, we reveal key insights into their capabilities and limitations: 1) existing LLMs show proficiency in basic hierarchical reasoning tasks; 2) they still struggle with more complex structures and implicit hierarchical representations, especially in structural modification and textual reasoning. Based on these findings, we create a small yet well-designed instruction dataset, which enhances LLMs' performance on HiBench by an average of 88.84\% (Llama-3.1-8B) and 31.38\% (Qwen2.5-7B) across all tasks. The HiBench dataset and toolkit are available here, https://github.com/jzzzzh/HiBench, to encourage evaluation.

Large language models (LLMs) achieve promising performance, yet their ability to reason remains poorly understood. Existing evaluations largely emphasize task-level accuracy, often conflating pattern matching with reasoning capability. We present X-RAY, an explainable reasoning analysis system that maps the LLM reasoning capability using calibrated, formally verified probes. We model reasoning capability as a function of extractable structure, operationalized through formal properties such as constraint interaction, reasoning depth, and solution-space geometry. X-Ray generates probes via formal tools with controlled structural variations, enabling precise isolation of incremental structural information through formal calibration and verification. We evaluate state-of-the-art LLMs on problems ranging from junior-level to advanced in mathematics, physics, and chemistry. Our analysis reveals a systematic asymmetry in LLM reasoning: models are relatively robust to constraint refinement, where additional conditions shrink an existing solution space, but degrade sharply under solution-space restructuring, where modifications alter the underlying structural form of the solution manifold. Moreover, calibrated formal probes differentiate models that appear indistinguishable on standard benchmarks and reveal failure modes that are structurally interpretable rather than opaque. Beyond evaluation, our framework is contamination-free and supports the training and testing of reasoning models.

Scientific inquiry requires systems-level reasoning that integrates heterogeneous experimental data, cross-domain knowledge, and mechanistic evidence into coherent explanations. While Large Language Models (LLMs) offer inferential capabilities, they often depend on retrieval-augmented contexts that lack structural depth. Traditional Knowledge Graphs (KGs) attempt to bridge this gap, yet their pairwise constraints fail to capture the irreducible higher-order interactions that govern emergent physical behavior. To address this, we introduce a methodology for constructing hypergraph-based knowledge representations that faithfully encode multi-entity relationships. Applied to a corpus of ~1,100 manuscripts on biocomposite scaffolds, our framework constructs a global hypergraph of 161,172 nodes and 320,201 hyperedges, revealing a scale-free topology (power law exponent ~1.23) organized around highly connected conceptual hubs. This representation prevents the combinatorial explosion typical of pairwise expansions and explicitly preserves the co-occurrence context of scientific formulations. We further demonstrate that equipping agentic systems with hypergraph traversal tools, specifically using node-intersection constraints, enables them to bridge semantically distant concepts. By exploiting these higher-order pathways, the system successfully generates grounded mechanistic hypotheses for novel composite materials, such as linking cerium oxide to PCL scaffolds via chitosan intermediates. This work establishes a "teacherless" agentic reasoning system where hypergraph topology acts as a verifiable guardrail, accelerating scientific discovery by uncovering relationships obscured by traditional graph methods.

Multimodal Large Language Models (MLLMs) excel at recognizing individual visual elements and reasoning over simple linear diagrams. However, when faced with complex topological structures involving branching paths, converging flows, and cyclic dependencies, their reasoning capabilities degrade sharply, even on tasks as basic as counting endpoints. Existing benchmarks fail to probe this gap, focusing on semantic comprehension rather than structural reasoning. We introduce ReactBench, a benchmark that reveals fundamental limitations in structural reasoning through chemical reaction diagrams. These real-world scientific diagrams offer an ideal testbed because they naturally span diverse structures from linear chains to cyclic graphs, while requiring both precise local recognition and coherent global reasoning. Our benchmark comprises 1,618 expert-annotated QA pairs across four hierarchical task dimensions. Extensive evaluation across 17 MLLMs reveals a significant performance gap exceeding 30% between anchor-based tasks and holistic structural reasoning tasks. Controlled ablations confirm this bottleneck lies in reasoning, not perception. These findings expose a fundamental deficit in structural understanding and establish directions for advancing visual reasoning.

Vision-language models (VLMs) have made significant strides in reasoning, yet they often struggle with complex multimodal tasks and tend to generate overly verbose outputs. A key limitation is their reliance on chain-of-thought (CoT) reasoning, despite many tasks benefiting from alternative topologies like trees or graphs. To address this, we introduce STELAR-Vision, a training framework for topology-aware reasoning. At its core is TopoAug, a synthetic data pipeline that enriches training with diverse topological structures. Using supervised fine-tuning and reinforcement learning, we post-train Qwen2VL models with both accuracy and efficiency in mind. Additionally, we propose Frugal Learning, which reduces output length with minimal accuracy loss. On MATH-V and VLM-S2H, STELAR-Vision improves accuracy by 9.7% over its base model and surpasses the larger Qwen2VL-72B-Instruct by 7.3%. On five out-of-distribution benchmarks, it outperforms Phi-4-Multimodal-Instruct by up to 28.4% and LLaMA-3.2-11B-Vision-Instruct by up to 13.2%, demonstrating strong generalization. Compared to Chain-Only training, our approach achieves 4.3% higher overall accuracy on in-distribution datasets and consistently outperforms across all OOD benchmarks. We have released datasets, and code will be available.

Large reasoning models (LRMs) are often evaluated using metrics such as final-answer accuracy or token count. However, identical scores on these metrics can hide fundamentally different reasoning structures. To address this limitation, we introduce a scalable LRM benchmark of logic puzzles and a pipeline that converts unstructured traces into verifiable reasoning graphs of claims and dependencies. This turns reasoning into a structured, measurable object whose topology can be quantitatively analyzed. Building on this, we define a reasoning efficiency metric that quantifies how concentrated the model's logical flow is. Our analysis on open-source reasoning models shows that structural measurements separate behaviors that token count and accuracy conflate, providing a practical tool for diagnosing failure modes and comparing how reasoning scales with puzzle difficulty.

Reverse Chain-of-Thought Generation (RCG) synthesizes reasoning traces from query-answer pairs, but runs the risk of producing post-hoc rationalizations: when models can see the answer during generation, the answer serves as a cognitive anchor that shapes the entire explanation. We formalize this phenomenon through a three-level measurement hierarchy: lexical, entropic, and probabilistic anchoring, each captures surface artifacts, entropy dynamics, and latent answer dependence, respectively. We analyze semantic suppression, the intuitive mitigation strategy that instructs models to ignore the answer, to find out its counterproduction: while it reduces lexical overlap, it paradoxically increases entropic and probabilistic anchoring. Drawing on Ironic Process Theory from cognitive psychology, we attribute this failure to active monitoring of the forbidden answer, which inadvertently deepens dependence on it. To break this cycle, we propose Structural Skeleton-guided Reasoning (SSR), a two-phase approach that first generates an answer-invariant functional skeleton structure, then uses this skeleton to guide full trace generation. By redirecting the information flow to structural planning rather than answer monitoring, SSR consistently reduces anchoring across all three levels. We further introduce Distilled SSR (SSR-D), which fine-tunes models on teacher-generated SSR traces to ensure reliable structural adherence. Experiments across open-ended reasoning benchmarks demonstrate that SSR-D achieves up to 10% improvement over suppression baselines while preserving out-of-distribution (OOD) generalization.

We introduce CurveBench, a benchmark for hierarchical topological reasoning from visual input. CurveBench consists of 756 images of pairwise non-intersecting Jordan curves across easy, polygonal, topographic-inspired, maze-like, and dense counting configurations. Each image is annotated with a rooted tree encoding the containment relations between planar regions. We formulate the task as structured prediction: given an image, a model must recover the full rooted containment tree induced by the curves. Despite the visual simplicity of the task, the strongest evaluated model, Gemini 3.1 Pro, achieves only 71.1\% tree-generation accuracy on CurveBench-Easy and 19.1\% on CurveBench-Hard. We further demonstrate benchmark utility through RLVR-style fine-tuning of open-weight vision-language models. Our trained Qwen3-VL-8B model improves over Qwen-3-VL-8B-Thinking from 2.8\% to 33.3\% tree-generation accuracy on CurveBench-Easy, exceeding GPT-5.4 and Claude Opus 4.5 under our evaluation protocol. The remaining gap, especially on CurveBench-Hard, shows that exact topology-aware visual reasoning remains far from solved.

Topology reasoning aims to comprehensively understand road scenes and present drivable routes in autonomous driving. It requires detecting road centerlines (lane) and traffic elements, further reasoning their topology relationship, i.e., lane-lane topology, and lane-traffic topology. In this work, we first present that the topology score relies heavily on detection performance on lane and traffic elements. Therefore, we introduce a powerful 3D lane detector and an improved 2D traffic element detector to extend the upper limit of topology performance. Further, we propose TopoMLP, a simple yet high-performance pipeline for driving topology reasoning. Based on the impressive detection performance, we develop two simple MLP-based heads for topology generation. TopoMLP achieves state-of-the-art performance on OpenLane-V2 benchmark, i.e., 41.2% OLS with ResNet-50 backbone. It is also the 1st solution for 1st OpenLane Topology in Autonomous Driving Challenge. We hope such simple and strong pipeline can provide some new insights to the community. Code is at https://github.com/wudongming97/TopoMLP.

Solving topological grid puzzles requires reasoning over global spatial invariants such as connectivity, loop closure, and region symmetry and remains challenging for even the most powerful large language models (LLMs). To study these abilities under controlled settings, we introduce TopoBench, a benchmark of six puzzle families across three difficulty levels. We evaluate strong reasoning LLMs on TopoBench and find that even frontier models solve fewer than one quarter of hard instances, with two families nearly unsolved. To investigate whether these failures stem from reasoning limitations or from difficulty extracting and maintaining spatial constraints, we annotate 750 chain of thought traces with an error taxonomy that surfaces four candidate causal failure modes, then test them with targeted interventions simulating each error type. These interventions show that certain error patterns like premature commitment and constraint forgetting have a direct impact on the ability to solve the puzzle, while repeated reasoning is a benign effect of search. Finally we study mitigation strategies including prompt guidance, cell-aligned grid representations and tool-based constraint checking, finding that the bottleneck lies in extracting constraints from spatial representations and not in reasoning over them. Code and data are available at github.com/mayug/topobench-benchmark.

We introduce a novel dataset designed to benchmark the physical and spatial reasoning capabilities of Large Language Models (LLM) based on topology optimization, a method for computing optimal material distributions within a design space under prescribed loads and supports. In this dataset, LLMs are provided with conditions such as 2D boundary, applied forces and supports, and must reason about the resulting optimal material distribution. The dataset includes a variety of tasks, ranging from filling in masked regions within partial structures to predicting complete material distributions. Solving these tasks requires understanding the flow of forces and the required material distribution under given constraints, without access to simulation tools or explicit physical models, challenging models to reason about structural stability and spatial organization. Our dataset targets the evaluation of spatial and physical reasoning abilities in 2D settings, offering a complementary perspective to traditional language and logic benchmarks.

Time series reasoning treats time as a first-class axis and incorporates intermediate evidence directly into the answer. This survey defines the problem and organizes the literature by reasoning topology with three families: direct reasoning in one step, linear chain reasoning with explicit intermediates, and branch-structured reasoning that explores, revises, and aggregates. The topology is crossed with the main objectives of the field, including traditional time series analysis, explanation and understanding, causal inference and decision making, and time series generation, while a compact tag set spans these axes and captures decomposition and verification, ensembling, tool use, knowledge access, multimodality, agent loops, and LLM alignment regimes. Methods and systems are reviewed across domains, showing what each topology enables and where it breaks down in faithfulness or robustness, along with curated datasets, benchmarks, and resources that support study and deployment (https://github.com/blacksnail789521/Time-Series-Reasoning-Survey). Evaluation practices that keep evidence visible and temporally aligned are highlighted, and guidance is distilled on matching topology to uncertainty, grounding with observable artifacts, planning for shift and streaming, and treating cost and latency as design budgets. We emphasize that reasoning structures must balance capacity for grounding and self-correction against computational cost and reproducibility, while future progress will likely depend on benchmarks that tie reasoning quality to utility and on closed-loop testbeds that trade off cost and risk under shift-aware, streaming, and long-horizon settings. Taken together, these directions mark a shift from narrow accuracy toward reliability at scale, enabling systems that not only analyze but also understand, explain, and act on dynamic worlds with traceable evidence and credible outcomes.

The field of natural language processing (NLP) has witnessed significant progress in recent years, with a notable focus on improving large language models' (LLM) performance through innovative prompting techniques. Among these, prompt engineering coupled with structures has emerged as a promising paradigm, with designs such as Chain-of-Thought, Tree of Thoughts, or Graph of Thoughts, in which the overall LLM reasoning is guided by a structure such as a graph. As illustrated with numerous examples, this paradigm significantly enhances the LLM's capability to solve numerous tasks, ranging from logical or mathematical reasoning to planning or creative writing. To facilitate the understanding of this growing field and pave the way for future developments, we devise a general blueprint for effective and efficient LLM reasoning schemes. For this, we conduct an in-depth analysis of the prompt execution pipeline, clarifying and clearly defining different concepts. We then build the first taxonomy of structure-enhanced LLM reasoning schemes. We focus on identifying fundamental classes of harnessed structures, and we analyze the representations of these structures, algorithms executed with these structures, and many others. We refer to these structures as reasoning topologies, because their representation becomes to a degree spatial, as they are contained within the LLM context. Our study compares existing prompting schemes using the proposed taxonomy, discussing how certain design choices lead to different patterns in performance and cost. We also outline theoretical underpinnings, relationships between prompting and others parts of the LLM ecosystem such as knowledge bases, and the associated research challenges. Our work will help to advance future prompt engineering techniques.

Structure-property relationships are foundational to biology, chemistry and materials science, where function, reactivity and physical response emerge from spatial, chemical and periodic organization. Mechanistically explaining these relationships requires interpreting structural evidence through scientific principles and physical constraints, from stereochemistry and bonding to symmetry, energetics and periodic order. However, applying artificial intelligence to this process presents a joint challenge of representation and reasoning: models must preserve domain-native structural information while showing how specific evidence supports predictions under these constraints. Here we introduce SciReasoner, a multimodal scientific foundation model for native structural reasoning across proteins, small molecules and inorganic crystals. SciReasoner discretizes coordinates, topologies and periodic connectivities into a unified structure-aware vocabulary, treating structural tokens as addressable evidence units during reasoning. In homology-controlled Gene Ontology prediction, SciReasoner improves Cellular Component annotation for low-homology and orphan-like proteins, increasing F_{max} from 0.42 to 0.55. In chemistry, it raises single-step retrosynthesis accuracy from 0.63 to 0.72 while generating fragment-level disconnection and precursor-verification traces. In materials science, its representations separate elemental and compound phases and resolve high- and low-band-gap regimes. Across 86 benchmarks, SciReasoner achieves state-of-the-art performance on 67 tasks. Double-blind expert evaluation rates its reasoning traces as preferred or at least comparable to those of a frontier large language model in 98% of cases. By making structure an inspectable substrate for reasoning under scientific constraints, SciReasoner connects accurate prediction with interpretable scientific inference.

Structural topology optimization (TO) is central to engineering design but remains computationally intensive due to complex physics and hard constraints. Existing deep-learning methods are limited to fixed square grids, a few hand-coded boundary conditions, and post-hoc optimization, preventing general deployment. We introduce Optimize Any Topology (OAT), a foundation-model framework that directly predicts minimum-compliance layouts for arbitrary aspect ratios, resolutions, volume fractions, loads, and fixtures. OAT combines a resolution- and shape-agnostic autoencoder with an implicit neural-field decoder and a conditional latent-diffusion model trained on OpenTO, a new corpus of 2.2 million optimized structures covering 2 million unique boundary-condition configurations. On four public benchmarks and two challenging unseen tests, OAT lowers mean compliance up to 90% relative to the best prior models and delivers sub-1 second inference on a single GPU across resolutions from 64 x 64 to 256 x 256 and aspect ratios as high as 10:1. These results establish OAT as a general, fast, and resolution-free framework for physics-aware topology optimization and provide a large-scale dataset to spur further research in generative modeling for inverse design. Code & data can be found at https://github.com/ahnobari/OptimizeAnyTopology.

Understanding the topology of decision regions is central to explaining the inner workings of deep neural networks. Prior empirical work has provided evidence that these regions are path connected. We study a stronger topological question: whether closed loops inside a decision region can be contracted without leaving that region. To this end, we propose an iterative quad-mesh filling procedure that constructs a finite-resolution label-preserving surface bounded by a given loop and lying entirely within the same decision region. We further connect this construction to natural Coons patches in order to quantify its deviation from a canonical geometric interpolation of the loop. By evaluating our method across several modern image-classification models, we provide empirical evidence supporting the hypothesis that decision regions in deep neural networks are not only path connected, but also simply connected.

The natural world is full of complex systems characterized by intricate relations between their components: from social interactions between individuals in a social network to electrostatic interactions between atoms in a protein. Topological Deep Learning (TDL) provides a comprehensive framework to process and extract knowledge from data associated with these systems, such as predicting the social community to which an individual belongs or predicting whether a protein can be a reasonable target for drug development. TDL has demonstrated theoretical and practical advantages that hold the promise of breaking ground in the applied sciences and beyond. However, the rapid growth of the TDL literature has also led to a lack of unification in notation and language across Topological Neural Network (TNN) architectures. This presents a real obstacle for building upon existing works and for deploying TNNs to new real-world problems. To address this issue, we provide an accessible introduction to TDL, and compare the recently published TNNs using a unified mathematical and graphical notation. Through an intuitive and critical review of the emerging field of TDL, we extract valuable insights into current challenges and exciting opportunities for future development.

Representing and navigating hierarchy is a fundamental primitive of reasoning. Large language models have demonstrated proficiency in a wide variety of tasks requiring hierarchical reasoning, but there exists limited analysis on how the models geometrically represent the necessary latent constructions for such thinking. To this end, we develop H-probes, a collection of linear probes that extract hierarchical structure, specifically depth and pairwise distance, from latent representations. In synthetic tree traversal tasks, the H-probes robustly find the subspaces containing hierarchical structure necessary to complete the tasks; furthermore, in comprehensive ablation experiments, we show that these hierarchy-containing subspaces are low-dimensional, causally important for high task performance, and generalize within- and out-of-domain. Furthermore, we find analogous, though weaker, hierarchical structure in real-world hierarchical contexts such as mathematical reasoning traces. These results demonstrate that models represent hierarchy not only at the level of syntax and concepts, but at deeper levels of abstraction -- including the reasoning process itself.

Topological data analysis (TDA) is an area of data science that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for geometric data sets such as point clouds. One of the most important such descriptors is {\em persistent homology}, which encodes the change in shape as a filtration parameter changes; a typical parameter is the feature scale. For many data sets, it is useful to simultaneously vary multiple filtration parameters, for example feature scale and density. While the theoretical properties of single parameter persistent homology are well understood, less is known about the multiparameter case. In particular, a central question is the problem of representing multiparameter persistent homology by elements of a vector space for integration with standard machine learning algorithms. Existing approaches to this problem either ignore most of the multiparameter information to reduce to the one-parameter case or are heuristic and potentially unstable in the face of noise. In this article, we introduce a new general representation framework that leverages recent results on {\em decompositions} of multiparameter persistent homology. This framework is rich in information, fast to compute, and encompasses previous approaches. Moreover, we establish theoretical stability guarantees under this framework as well as efficient algorithms for practical computation, making this framework an applicable and versatile tool for analyzing geometric and point cloud data. We validate our stability results and algorithms with numerical experiments that demonstrate statistical convergence, prediction accuracy, and fast running times on several real data sets.

Recent text-to-CAD approaches have shown promising results by leveraging large language models, but they often struggle with maintaining structural consistency in complex designs and accurately grounding geometric parameters. To address these issues, we propose HierCAD, a hierarchical text-to-CAD framework that improves both structural reasoning and parameter prediction. HierCAD reformulates CAD generation as progressive reasoning by decomposing CAD construction trees into object-level procedural reasoning and part-level topology reasoning trajectories. To further improve generation fidelity, we introduce a unified Structure Alignment and Parameter Grounding (SAPG) learning strategy. Structure alignment aligns topology reasoning trajectories with their corresponding parametric CAD spans, while parameter grounding mitigates shortcut learning through structure-preserving parameter perturbations and ranking-based supervision. Experiments demonstrate that HierCAD outperforms prior state-of-the-art methods on both CAD sequence generation and reconstructed CAD model evaluation. Our code is available at https://github.com/Collab-Gen/HierCAD.

Adversarially perturbed images of text can cause sophisticated OCR systems to produce misleading or incorrect transcriptions from seemingly invisible changes to humans. Some of these perturbations even survive physical capture, posing security risks to high-stakes applications such as document processing, license plate recognition, and automated compliance systems. Existing defenses, such as adversarial training, input preprocessing, or post-recognition correction, are often model-specific, computationally expensive, and affect performance on unperturbed inputs while remaining vulnerable to unseen or adaptive attacks. To address these challenges, TopoReformer is introduced, a model-agnostic reformation pipeline that mitigates adversarial perturbations while preserving the structural integrity of text images. Topology studies properties of shapes and spaces that remain unchanged under continuous deformations, focusing on global structures such as connectivity, holes, and loops rather than exact distance. Leveraging these topological features, TopoReformer employs a topological autoencoder to enforce manifold-level consistency in latent space and improve robustness without explicit gradient regularization. The proposed method is benchmarked on EMNIST, MNIST, against standard adversarial attacks (FGSM, PGD, Carlini-Wagner), adaptive attacks (EOT, BDPA), and an OCR-specific watermark attack (FAWA).

Diagram-to-graph topology extraction aims to extract a graph of entities and their connections from a structural diagram. This task remains challenging for current vision-language models because it requires both fine-grained perceptual grounding and topology-aware reasoning with global consistency. We present TopoBench-180, a human-verified benchmark for diagram-to-graph topology extraction, and TopoAgent, a structure-aware perception-to-reasoning framework for reliable topology extraction using large vision-language models. TopoBench-180 contains 180 structural diagrams spanning Web-style and Network-style categories, paired with canonical graph annotations. TopoAgent progressively extracts the target graph by combining grounded perception, global structural priors, canonical node inventory construction, node-centric local-to-global relation reasoning, and topological consistency enforcement. Experiments on TopoBench-180 show that TopoAgent outperforms strong vision-language model baselines and recent visual reasoning frameworks, especially on edge extraction. More broadly, this work fills an important gap in multimodal structured understanding by establishing a benchmark and framework for diagram-to-graph topology extraction. The benchmark and associated resources will be publicly released at https://huggingface.co/datasets/WayneGuo0011/TopoBench-180.

We algorithmically determine the regions and facets of all dimensions of the canonical polyhedral complex, the universal object into which a ReLU network decomposes its input space. We show that the locations of the vertices of the canonical polyhedral complex along with their signs with respect to layer maps determine the full facet structure across all dimensions. We present an algorithm which calculates this full combinatorial structure, making use of our theorems that the dual complex to the canonical polyhedral complex is cubical and it possesses a multiplication compatible with its facet structure. The resulting algorithm is numerically stable, polynomial time in the number of intermediate neurons, and obtains accurate information across all dimensions. This permits us to obtain, for example, the true topology of the decision boundaries of networks with low-dimensional inputs. We run empirics on such networks at initialization, finding that width alone does not increase observed topology, but width in the presence of depth does. Source code for our algorithms is accessible online at https://github.com/mmasden/canonicalpoly.

Segmentation algorithms are prone to make topological errors on fine-scale structures, e.g., broken connections. We propose a novel method that learns to segment with correct topology. In particular, we design a continuous-valued loss function that enforces a segmentation to have the same topology as the ground truth, i.e., having the same Betti number. The proposed topology-preserving loss function is differentiable and we incorporate it into end-to-end training of a deep neural network. Our method achieves much better performance on the Betti number error, which directly accounts for the topological correctness. It also performs superiorly on other topology-relevant metrics, e.g., the Adjusted Rand Index and the Variation of Information. We illustrate the effectiveness of the proposed method on a broad spectrum of natural and biomedical datasets.

Persistent homology, a technique from computational topology, has recently shown strong empirical performance in the context of graph classification. Being able to capture long range graph properties via higher-order topological features, such as cycles of arbitrary length, in combination with multi-scale topological descriptors, has improved predictive performance for data sets with prominent topological structures, such as molecules. At the same time, the theoretical properties of persistent homology have not been formally assessed in this context. This paper intends to bridge the gap between computational topology and graph machine learning by providing a brief introduction to persistent homology in the context of graphs, as well as a theoretical discussion and empirical analysis of its expressivity for graph learning tasks.

In this paper we introduce a novel family of attributed graphs for the purpose of shape discrimination. Our graphs typically arise from variations on the Mapper graph construction, which is an approximation of the Reeb graph for point cloud data. Our attributions enrich these constructions with (persistent) homology in ways that are provably stable, thereby recording extra topological information that is typically lost in these graph constructions. We provide experiments which illustrate the use of these invariants for shape representation and classification. In particular, we obtain competitive shape classification results when using our topologically attributed graphs as inputs to a simple graph neural network classifier.

Combinatorial complexes have unified set-based (e.g., graphs, hypergraphs) and part-whole (e.g., simplicial, cellular complexes) structures into a common topological framework. Existing topological neural networks and Weisfeiler-Lehman variants remain fragmented, lacking a unified theoretical foundation for topological deep learning. In this work, we introduce the Combinatorial Complex Weisfeiler-Lehman (CCWL) test, an axiomatic-style extension of the WL test to combinatorial complexes. CCWL formalizes topological message passing through four types of neighborhood relation and provides a unified perspective on the expressive power of higher-order variants. We further prove that upper and lower neighborhoods are sufficient among the four adjacent WL tests to reach the expressivity of the full CCWL framework across topological structures of combinatorial complexes. Building on this framework, we also propose the Combinatorial Complex Isomorphism Network (CCIN) and evaluate it on synthetic and real-world benchmarks. Experimental results indicate CCIN outperforms baseline methods and offers a generalized expressive framework for topological deep learning.

We present TopoMortar, a brick wall dataset that is the first dataset specifically designed to evaluate topology-focused image segmentation methods, such as topology loss functions. TopoMortar enables to investigate in two ways whether methods incorporate prior topological knowledge. First, by eliminating challenges seen in real-world data, such as small training set, noisy labels, and out-of-distribution test-set images, that, as we show, impact the effectiveness of topology losses. Second, by allowing to assess in the same dataset topology accuracy across dataset challenges, isolating dataset-related effects from the effect of incorporating prior topological knowledge. In these two experiments, it is deliberately difficult to improve topology accuracy without actually using topology information, thus, permitting to attribute an improvement in topology accuracy to the incorporation of prior topological knowledge. To this end, TopoMortar includes three types of labels (accurate, noisy, pseudo-labels), two fixed training sets (large and small), and in-distribution and out-of-distribution test-set images. We compared eight loss functions on TopoMortar, and we found that clDice achieved the most topologically accurate segmentations, Skeleton Recall loss performed best particularly with noisy labels, and the relative advantageousness of the other loss functions depended on the experimental setting. Additionally, we show that simple methods, such as data augmentation and self-distillation, can elevate Cross entropy Dice loss to surpass most topology loss functions, and that those simple methods can enhance topology loss functions as well. clDice and Skeleton Recall loss, both skeletonization-based loss functions, were also the fastest to train, making this type of loss function a promising research direction. TopoMortar and our code can be found at https://github.com/jmlipman/TopoMortar

As current Multimodal Large Language Models rapidly saturate canonical visual reasoning benchmarks, a key question emerges: do these strong scores genuinely reflect robust visual understanding? We identify a pervasive vulnerability, the Cartesian Shortcut: visual reasoning benchmarks prevalently build on orthogonal grid-based layouts that can be readily discretized into explicit textual coordinates. Models systematically exploit this property, heavily leveraging text-based deductive reasoning to assist visual problem-solving. To systematically dismantle this shortcut, we introduce Polaris-Bench, which re-formulates 53 visual reasoning tasks in Polar coordinate space with paired Cartesian counterparts as reference, while preserving consistent logical constraints and task semantics -- thus fundamentally breaking the orthogonal prior that models exploit. Comprehensive evaluation across 14 state-of-the-art MLLMs reveals that frontier models achieving 70--83% on Cartesian layouts collapse to 31--39% on Polar equivalents, with degradation persisting even under complete logical equivalence. Moreover, reasoning gains observed on Cartesian layouts are severely diminished on Polar equivalents. These findings expose a critical deficiency in current MLLMs: the lack of topology-invariant visual reasoning.

Accurate road topology reasoning is critical for autonomous driving, as it requires both perceiving road elements and understanding how lanes connect to each other (L2L) and to traffic elements (L2T). Existing methods often focus on either perception or L2L reasoning, leaving L2T underexplored and fall short of jointly optimizing perception and reasoning. Moreover, although topology prediction inherently involves relations, relational modeling itself is seldom incorporated into feature extraction or supervision. As humans naturally leverage contextual relationships to recognize road element and infer their connectivity, we posit that relational modeling can likewise benefit both perception and reasoning, and that these two tasks should be mutually enhancing. To this end, we propose RelTopo, a multi-level relational modeling approach that systematically integrates relational cues across three levels: 1) perception-level: a relation-aware lane detector with geometry-biased self-attention and curve-guided cross-attention enriches lane representations; 2) reasoning-level: relation-enhanced topology heads, including a geometry-enhanced L2L head and a cross-view L2T head, enhance topology inference via relational cues; and 3) supervision-level: a contrastive InfoNCE strategy regularizes relational embeddings. This design enables perception and reasoning to be optimized jointly. Extensive experiments on OpenLane-V2 demonstrate that RelTopo significantly improves both detection and topology reasoning, with gains of +3.1 in DET_l, +5.3 in TOP_{ll}, +4.9 in TOP_{lt}, and +4.4 overall in OLS, setting a new state-of-the-art. Code will be released.

Understanding how Large Language Models (LLMs) perform complex reasoning and their failure mechanisms is a challenge in interpretability research. To provide a measurable geometric analysis perspective, we define the concept of the Reasoning Manifold, a latent low-dimensional geometric structure formed by the internal representations corresponding to all correctly reasoned generations. This structure can be conceptualized as the embodiment of the effective thinking paths that the model has learned to successfully solve a given task. Based on this concept, we build REMA, a framework that explains the origins of failures by quantitatively comparing the spatial relationships of internal model representations corresponding to both erroneous and correct reasoning samples. Specifically, REMA first quantifies the geometric deviation of each erroneous representation by calculating its k-nearest neighbors distance to the approximated manifold formed by correct representations, thereby providing a unified failure signal. It then localizes the divergence points where these deviations first become significant by tracking this deviation metric across the model's layers and comparing it against a baseline of internal fluctuations from correct representations, thus identifying where the reasoning chain begins to go off-track. Our extensive experiments on diverse language and multimodal models and tasks demonstrate the low-dimensional nature of the reasoning manifold and the high separability between erroneous and correct reasoning representations. The results also validate the effectiveness of the REMA framework in analyzing the origins of reasoning failures. This research connects abstract reasoning failures to measurable geometric deviations in representations, providing new avenues for in-depth understanding and diagnosis of the internal computational processes of black-box models.

Text-to-image generation has reached photorealistic quality, yet state-of-the-art systems remain unreliable at producing scientific diagrams, whose value depends not on appearance but on physical faithfulness: correct force directions, valid coordinate systems, consistent thermodynamic states, and equations matching the depicted scenario. Trained on web imagery with physically shallow captions, generic models produce diagrams that look plausible but are physically wrong, harmful in education and scientific communication. We present Princigram, a physics-faithful scientific-diagram generator, and its data pipeline. Our central advance is Structured Physical Chain-of-Thought (SP-CoT): a per-subdiscipline schema that decomposes a physics diagram into an explicit multi-step reasoning chain across six subdisciplines, from scene identification through force or process analysis to governing laws and synthesis. Unlike free-form chain-of-thought, SP-CoT follows a fixed schema with strict fidelity rules that separate visually grounded facts from physically inferred reasoning and type all mathematics symbolically; it serves both as dense training supervision and, at inference, as a structured "thinking" prompt. With it we curate and structurally annotate 4.3 million physics images, of which 115,037 carry expert-level annotation, and adapt a unified multimodal backbone. We further introduce VeriphyT2IBench, whose questions are derived from each held-out diagram's own structured annotation: each diagram becomes an item-specific bank of binary questions about its objects, forces, and states, so a judge model's score decomposes into named physical facts rather than one holistic number. On the physics subset of GenExam and on VeriphyT2IBench, Princigram shows that explicit physics-structured supervision improves the physical faithfulness of generated scientific diagrams.

The manifold hypothesis, which assumes that data lies on or close to an unknown manifold of low intrinsic dimension, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibits distinct non-manifold structures, i.e. singularities, that can lead to erroneous findings. Detecting such singularities is therefore crucial as a precursor to interpolation and inference tasks. We address this issue by developing a topological framework that (i) quantifies the local intrinsic dimension, and (ii) yields a Euclidicity score for assessing the 'manifoldness' of a point along multiple scales. Our approach identifies singularities of complex spaces, while also capturing singular structures and local geometric complexity in image data.

CoT has significantly enhanced the reasoning ability of LLMs while it faces challenges when extended to multimodal domains, particularly in mathematical tasks. Existing MLLMs typically perform textual reasoning solely from a single static mathematical image, overlooking dynamic visual acquisition during reasoning. In contrast, humans repeatedly examine visual image and employ step-by-step reasoning to prove intermediate propositions. This strategy of decomposing the problem-solving process into key logical nodes adheres to Miller's Law in cognitive science. Inspired by this insight, we propose a ViRC framework for multimodal mathematical tasks, introducing a Reason Chunking mechanism that structures multimodal mathematical CoT into consecutive Critical Reasoning Units (CRUs) to simulate human expert problem-solving patterns. CRUs ensure intra-unit textual coherence for intermediate proposition verification while integrating visual information across units to generate subsequent propositions and support structured reasoning. To this end, we present CRUX dataset by using three visual tools and four reasoning patterns to provide explicitly annotated CRUs across multiple reasoning paths for each mathematical problem. Leveraging the CRUX dataset, we propose a progressive training strategy inspired by human cognitive learning, which includes Instructional SFT, Practice SFT, and Strategic RL, aimed at further strengthening the Reason Chunking ability of the model. The resulting ViRC-7B model achieves a 18.8% average improvement over baselines across multiple mathematical benchmarks. Code is available at https://github.com/Leon-LihongWang/ViRC.

We present an agentic, autonomous graph expansion framework that iteratively structures and refines knowledge in situ. Unlike conventional knowledge graph construction methods relying on static extraction or single-pass learning, our approach couples a reasoning-native large language model with a continually updated graph representation. At each step, the system actively generates new concepts and relationships, merges them into a global graph, and formulates subsequent prompts based on its evolving structure. Through this feedback-driven loop, the model organizes information into a scale-free network characterized by hub formation, stable modularity, and bridging nodes that link disparate knowledge clusters. Over hundreds of iterations, new nodes and edges continue to appear without saturating, while centrality measures and shortest path distributions evolve to yield increasingly distributed connectivity. Our analysis reveals emergent patterns, such as the rise of highly connected 'hub' concepts and the shifting influence of 'bridge' nodes, indicating that agentic, self-reinforcing graph construction can yield open-ended, coherent knowledge structures. Applied to materials design problems, we present compositional reasoning experiments by extracting node-specific and synergy-level principles to foster genuinely novel knowledge synthesis, yielding cross-domain ideas that transcend rote summarization and strengthen the framework's potential for open-ended scientific discovery. We discuss other applications in scientific discovery and outline future directions for enhancing scalability and interpretability.

The drive for predictable LLM reasoning in their integration with compound systems has popularized structured outputs, yet concerns remain about performance trade-offs compared to unconstrained natural language. At the same time, training on unconstrained Chain of Thought (CoT) traces has brought about a new class of strong reasoning models that nevertheless present novel compute budget and faithfulness challenges. This paper introduces iSelf-Discover, an instance-level adaptation of the Self-Discover framework, and using it compares dynamically generated structured JSON reasoning with its unstructured counterpart. Our empirical evaluation across diverse benchmarks using state-of-the-art open-source models supports a consistent advantage for unstructured reasoning. Notably, on the complex MATH benchmark, unstructured plans achieved relative performance improvements of up to 18.90\% over structured approaches. Zero-shot unstructured iSelf-Discover variants are also shown to outperform their five-shot structured counterparts, underscoring the significance of this gap, even when structured plans are dynamically generated to ensure reasoning precedes the final answer. We further demonstrate that the optimal granularity of plan generation (instance-level vs. task-level) is context-dependent. These findings invite re-evaluation of the reliance on structured formats for complex problem-solving and how compound systems should be organized.

Boundary representation (B-Rep) generation is a fundamental task in computer-aided design, yet the direct synthesis of high-fidelity and structurally valid B-Reps remains a major challenge. Existing deep generative methods suffer from two forms of brittleness: representation brittleness, caused by padding noise and feature contamination in the latent space, and generation brittleness, stemming from sequential error propagation and a train-inference mismatch due to non-differentiable validity enforcement. We propose HiFi-BRep, a novel framework that addresses these limitations through two synergistic contributions. First, a topology-aware encoder constructs a high-fidelity latent representation by eliminating padding via learnable queries and preventing feature contamination with topology-guided attention. Second, a single-stage decoder jointly predicts geometry and topology in parallel, embedding core manifold constraints as a differentiable learning objective. This design ensures mutual guidance between geometry and topology while avoiding cascaded errors. Extensive experiments show that HiFi-BRep significantly outperforms state-of-the-art methods in both structural validity and geometric fidelity, providing a robust solution for high-quality B-Rep synthesis. Code and models are publicly available at https://github.com/1nnoh/HiFi-BRep.

Test-Time Scaling (TTS) has significantly enhanced the capabilities of Large Reasoning Models (LRMs) but introduces a critical side-effect known as Overthinking. We conduct a preliminary study to rethink this phenomenon from a fine-grained perspective. We observe that LRMs frequently conduct repetitive self-verification without revision even after obtaining the final answer during the reasoning process. We formally define this specific position where the answer first stabilizes as the Reasoning Anchor. By analyzing pre- and post-anchor reasoning behaviors, we uncover the structural redundancy fixed in LRMs: the meaningless repetitive verification after deriving the first complete answer, which we term the Answer-Stable Tail (AST). Motivated by this observation, we propose Anchor-based Process Reward (APR), a structure-aware reward shaping method that localizes the reasoning anchor and penalizes exclusively the post-anchor AST. Leveraging the policy optimization algorithm suitable for length penalties, our APR models achieved the performance-efficiency Pareto frontier at 1.5B and 7B scales averaged across five mathematical reasoning datasets while requiring substantially fewer computational resources for RL training.

We introduce Diagram of Thought (DoT), a framework that models iterative reasoning in large language models (LLMs) as the construction of a directed acyclic graph (DAG) within a single model. Unlike traditional approaches that represent reasoning as linear chains or trees, DoT organizes propositions, critiques, refinements, and verifications into a cohesive DAG structure, allowing the model to explore complex reasoning pathways while maintaining logical consistency. Each node in the diagram corresponds to a proposition that has been proposed, critiqued, refined, or verified, enabling the LLM to iteratively improve its reasoning through natural language feedback. By leveraging auto-regressive next-token prediction with role-specific tokens, DoT facilitates seamless transitions between proposing ideas and critically evaluating them, providing richer feedback than binary signals. Furthermore, we formalize the DoT framework using Topos Theory, providing a mathematical foundation that ensures logical consistency and soundness in the reasoning process. This approach enhances both the training and inference processes within a single LLM, eliminating the need for multiple models or external control mechanisms. DoT offers a conceptual framework for designing next-generation reasoning-specialized models, emphasizing training efficiency, robust reasoning capabilities, and theoretical grounding. The code is available at https://github.com/diagram-of-thought/diagram-of-thought.

In this paper, we address the challenging task of multimodal mathematical reasoning by incorporating the ability of "slow thinking" into multimodal large language models (MLLMs). Our core idea is that different levels of reasoning abilities can be combined dynamically to tackle questions with different complexity. To this end, we propose a paradigm of Self-structured Chain of Thought (SCoT), which is composed of minimal semantic atomic steps. Different from existing methods that rely on structured templates or free-form paradigms, our method can not only generate cognitive CoT structures for various complex tasks but also mitigates the phenomenon of overthinking. To introduce structured reasoning capabilities into visual understanding models, we further design a novel AtomThink framework with four key modules, including (i) a data engine to generate high-quality multimodal reasoning paths; (ii) a supervised fine-tuning process with serialized inference data; (iii) a policy-guided multi-turn inference method; and (iv) an atomic capability metric to evaluate the single step utilization rate. We conduct extensive experiments to show that the proposed AtomThink significantly improves the performance of baseline MLLMs, achieving more than 10\% average accuracy gains on MathVista and MathVerse. Compared to state-of-the-art structured CoT approaches, our method not only achieves higher accuracy but also improves data utilization by 5 times and boosts inference efficiency by 85.3\%. Our code is now public available in https://github.com/Quinn777/AtomThink.

Reasoning ability is one of the most crucial capabilities of a foundation model, signifying its capacity to address complex reasoning tasks. Chain-of-Thought (CoT) technique is widely regarded as one of the effective methods for enhancing the reasoning ability of foundation models and has garnered significant attention. However, the reasoning process of CoT is linear, step-by-step, similar to personal logical reasoning, suitable for solving general and slightly complicated problems. On the contrary, the thinking pattern of an expert owns two prominent characteristics that cannot be handled appropriately in CoT, i.e., high-order multi-hop reasoning and multimodal comparative judgement. Therefore, the core motivation of this paper is transcending CoT to construct a reasoning paradigm that can think like an expert. The hyperedge of a hypergraph could connect various vertices, making it naturally suitable for modelling high-order relationships. Inspired by this, this paper innovatively proposes a multimodal Hypergraph-of-Thought (HoT) reasoning paradigm, which enables the foundation models to possess the expert-level ability of high-order multi-hop reasoning and multimodal comparative judgement. Specifically, a textual hypergraph-of-thought is constructed utilizing triple as the primary thought to model higher-order relationships, and a hyperedge-of-thought is generated through multi-hop walking paths to achieve multi-hop inference. Furthermore, we devise a visual hypergraph-of-thought to interact with the textual hypergraph-of-thought via Cross-modal Co-Attention Graph Learning for multimodal comparative verification. Experimentations on the ScienceQA benchmark demonstrate the proposed HoT-based T5 outperforms CoT-based GPT3.5 and chatGPT, which is on par with CoT-based GPT4 with a lower model size.

Recent advances in search-augmented large reasoning models (LRMs) enable the retrieval of external knowledge to reduce hallucinations in multistep reasoning. However, their ability to operate on graph-structured data, prevalent in domains such as e-commerce, social networks, and scientific citations, remains underexplored. Unlike plain text corpora, graphs encode rich topological signals that connect related entities and can serve as valuable priors for retrieval, enabling more targeted search and improved reasoning efficiency. Yet, effectively leveraging such structure poses unique challenges, including the difficulty of generating graph-expressive queries and ensuring reliable retrieval that balances structural and semantic relevance. To address this gap, we introduce GraphSearch, the first framework that extends search-augmented reasoning to graph learning, enabling zero-shot graph learning without task-specific fine-tuning. GraphSearch combines a Graph-aware Query Planner, which disentangles search space (e.g., 1-hop, multi-hop, or global neighbors) from semantic queries, with a Graph-aware Retriever, which constructs candidate sets based on topology and ranks them using a hybrid scoring function. We further instantiate two traversal modes: GraphSearch-R, which recursively expands neighborhoods hop by hop, and GraphSearch-F, which flexibly retrieves across local and global neighborhoods without hop constraints. Extensive experiments across diverse benchmarks show that GraphSearch achieves competitive or even superior performance compared to supervised graph learning methods, setting state-of-the-art results in zero-shot node classification and link prediction. These findings position GraphSearch as a flexible and generalizable paradigm for agentic reasoning over graphs.

Real-world phenomena do not generate arbitrary variability: their signals concentrate on compact, low-variability subsets of functional space, enabling rapid generalization from few examples. A small child can recognize a dog after extremely limited exposure because the perceptual manifold of "dog" is compact, structured, and low-dimensional. We formalize this principle through a deterministic functional-topological framework in which the set of valid realizations produced by a physical process forms a compact subset of a Banach space, endowed with stable invariants, a finite Hausdorff radius, and an induced continuous perceptual functional. This geometry provides explicit limits on knowledge, conditions for identifiability, and guarantees for generalization from sparse evidence -- properties fundamental to both natural and artificial intelligence. Across electromechanical, electrochemical, and physiological domains, we show that real-world processes consistently generate compact perceptual manifolds with the same geometric characteristics. Their boundaries can be discovered in a fully self-supervised manner as the empirical radius saturates with increasing sampling, even when the governing equations are unknown. These results demonstrate that deterministic functional topology offers a unified mathematical foundation for perception, representation, and world-model construction. It provides a geometric explanation for why biological learners and self-supervised AI systems can generalize from few observations, and establishes compact perceptual manifolds as a fundamental building block for future AI architectures. Finally, this work unifies biological perception and modern self-supervised models under a single geometric principle: both derive their generalization ability from the compactness and invariants of real-world perceptual manifolds.

Structural fidelity is essential to scientific methodology diagrams. To communicate research logic, these diagrams must faithfully render components, directional relations, and textual annotations. Since a single error, such as a reversed arrow or an unreadable equation, can invalidate the entire figure, structural fidelity is inherently conjunctive: correctness on one axis cannot compensate for failure on another. Current open-source models fail to satisfy this criterion. Supervised fine-tuning (SFT) learns plausible layouts but cannot reliably ensure structural correctness, while scalar reward-based post-training obscures which structural dimension has failed. To address this, we introduce SciForma, a framework for the structure faithful generation of scientific methodology diagrams. Specifically, SciForma decomposes diagram quality into three structural axes: Component, Arrow, and Text, guided by a structural inventory. Built on this foundation, we curate SciFormaData-700K for structured training and SciFormaBench-2K for logic-verified evaluation. To close the gap left by SFT, we develop Multi-Dimensional Conjunctive Preference Optimization (M-DPO), which enforces simultaneous correctness across all axes and adaptively routes gradients to the most deficient dimension in post-training. The same structural inventory also enables iterative editing at inference time to correct residual errors. This combination allows SciForma-9B to exceed all open-source baselines and GPT-Image-1.5 on both SciFormaBench-2K and AIBench, bringing open scientific diagram generation close to proprietary-level structural fidelity. Our code and data will be available at: https://github.com/microsoft/SciForma.

Large language models place structured concepts on geometrically faithful manifolds: weekdays lie on a circle, months on another, usually taken to be a fixed world-model the network stores and looks up. We show that context is king: the structure a model actually uses is set by the in-context specification. A declarative rule fixes not only which relations the geometry encodes but its topology type: the same tokens form a cycle or a branching tree on command, built even on arbitrary, meaning-free tokens with no prior to inherit, which a relabeled stored shape cannot do. When the specification conflicts with a strong pretrained prior, the context-set geometry dominates it in capable models, read from the same activations (representational similarity 0.6--0.9 to the imposed structure versus near-zero to the prior), across the priors we test and both families we study (Gemma, Qwen). Activation patching shows the map is causally used, not a probe correlate: swapping one entity's activation for another's makes the model answer with the other entity's successor under the imposed order. A rough map forms readily, present even in small and base models; what scale gates is using it cleanly: clean dominance and the causal crossover emerge only in the larger models (up to Gemma-31B and Qwen-27B) and weaken or reverse below, so a mechanism present in a large model can be absent in a smaller one of the same family. Whether the model builds this geometry anew or reconfigures a stored one we leave open; operationally, the geometry it uses is the one the context specifies.

The structure of topology underpins much of the research on performance and robustness, yet available topology data are typically scarce, necessitating the generation of synthetic graphs with desired properties for testing or release. Prior diffusion-based approaches either embed conditions into the diffusion model, requiring retraining for each attribute and hindering real-time applicability, or use classifier-based guidance post-training, which does not account for topology scale and practical constraints. In this paper, we show from a discrete perspective that gradients from a pre-trained graph-level classifier can be incorporated into the discrete reverse diffusion posterior to steer generation toward specified structural properties. Based on this insight, we propose Classifier-guided Conditional Topology Generation with Persistent Homology (CoPHo), which builds a persistent homology filtration over intermediate graphs and interprets features as guidance signals that steer generation toward the desired properties at each denoising step. Experiments on four generic/network datasets demonstrate that CoPHo outperforms existing methods at matching target metrics, and we further validate its transferability on the QM9 molecular dataset.

Graph foundation models represent a transformative paradigm for learning transferable representations across diverse graph domains. Recent methods leverage large language models to unify graph and text modalities into a shared representation space using contrastive learning. However, systematic evaluations reveal significant performance degradation at structural boundaries where distinct topological patterns converge, with accuracy losses exceeding 20 percentage points. This issue arises from a key limitation: current methods assume all graph structures can be encoded within a single Euclidean space. In reality, tree structures require hyperbolic geometry to preserve hierarchical branching, while cyclic patterns depend on spherical geometry for closure properties. At structural boundaries, nodes experience conflicting geometric constraints that uniform encoding spaces cannot resolve. This raises a crucial challenge: Can alignment frameworks be designed to respect the intrinsic geometric diversity of graph structures? We introduce GraphShaper, a geometry-aware framework that enhances graph encoding through multi-geometric specialization. Our approach employs expert networks tailored to different geometric spaces, dynamically computing fusion weights to adaptively integrate geometric properties based on local structural characteristics. This adaptive fusion preserves structural integrity before alignment with text embeddings. Extensive experiments demonstrate that GraphShaper achieves 9.47\% accuracy improvements on citation networks and 7.63\% on social networks in zero-shot settings.

Machine learning for point clouds has been attracting much attention, with many applications in various fields, such as shape recognition and material science. For enhancing the accuracy of such machine learning methods, it is often effective to incorporate global topological features, which are typically extracted by persistent homology. In the calculation of persistent homology for a point cloud, we choose a filtration for the point cloud, an increasing sequence of spaces. Since the performance of machine learning methods combined with persistent homology is highly affected by the choice of a filtration, we need to tune it depending on data and tasks. In this paper, we propose a framework that learns a filtration adaptively with the use of neural networks. In order to make the resulting persistent homology isometry-invariant, we develop a neural network architecture with such invariance. Additionally, we show a theoretical result on a finite-dimensional approximation of filtration functions, which justifies the proposed network architecture. Experimental results demonstrated the efficacy of our framework in several classification tasks.

This paper introduces a novel method for merging open-domain terminological knowledge. It takes advantage of the Region Connection Calculus (RCC5), a formalism used to represent regions in a topological space and to reason about their set-theoretic relationships. To this end, we first propose a faithful translation of terminological knowledge provided by several and potentially conflicting sources into region spaces. The merging is then performed on these spaces, and the result is translated back into the underlying language of the input sources. Our approach allows us to benefit from the expressivity and the flexibility of RCC5 while dealing with conflicting knowledge in a principled way.

The pursuit of automated scientific discovery has fueled progress from symbolic logic to modern AI, forging new frontiers in reasoning and pattern recognition. Transformers function as potential systems, where every possible relationship remains latent potentiality until tasks impose constraints, akin to measurement. Yet, refining their sampling requires more than probabilistic selection: solutions must conform to specific structures or rules, ensuring consistency and the invocation of general principles. We present Graph-PReFLexOR (Graph-based Preference-based Recursive Language Modeling for Exploratory Optimization of Reasoning), a framework that combines graph reasoning with symbolic abstraction to dynamically expand domain knowledge. Inspired by reinforcement learning, Graph-PReFLexOR defines reasoning as a structured mapping, where tasks yield knowledge graphs, abstract patterns, and ultimately, final answers. Inspired by category theory, it encodes concepts as nodes and their relationships as edges, supporting hierarchical inference and adaptive learning through isomorphic representations. Demonstrations include hypothesis generation, materials design, and creative reasoning, such as discovering relationships between mythological concepts like 'thin places' with materials science. We propose a 'knowledge garden growth' strategy that integrates insights across domains, promoting interdisciplinary connections. Results with a 3-billion-parameter Graph-PReFLexOR model show superior reasoning depth and adaptability, underscoring the potential for transparent, multidisciplinary AI-driven discovery. It lays the groundwork for general autonomous reasoning solutions.

Geometric data augmentation is widely used in segmentation workflows, but polygon annotations are often assumed to remain valid after transformation. This assumption can fail in structured domains such as architectural floorplan analysis, where a region may contain an interior void encoded as part of a single ordered polygon chain. Cropping or clipping can remove bridge vertices in this chain, causing one semantic region to split into disconnected components. We propose a lightweight topology-preserving augmentation strategy that repairs missing adjacency relations in index space while preserving the original vertex order. The method adds minimal overhead and can be integrated into existing preprocessing workflows. Experiments show that the proposed approach achieves near-perfect Cyclic Adjacency Preservation (CAP) across common geometric transformations and improves annotation consistency in polygon-based segmentation.

The use of graph neural networks has produced significant advances in point cloud problems, such as those found in high energy physics. The question of how to produce a graph structure in these problems is usually treated as a matter of heuristics, employing fully connected graphs or K-nearest neighbors. In this work, we elevate this question to utmost importance as the Topology Problem. We propose an attention mechanism that allows a graph to be constructed in a learned space that handles geometrically the flow of relevance, providing one solution to the Topology Problem. We test this architecture, called GravNetNorm, on the task of top jet tagging, and show that it is competitive in tagging accuracy, and uses far fewer computational resources than all other comparable models.

This paper introduces an innovative approach to analyzing and improving multi-hop reasoning in AI systems by drawing inspiration from Hamiltonian mechanics. We propose a novel framework that maps reasoning chains in embedding spaces to Hamiltonian systems, allowing us to leverage powerful analytical tools from classical physics. Our method defines a Hamiltonian function that balances the progression of reasoning (kinetic energy) against the relevance to the question at hand (potential energy). Using this framework, we analyze a large dataset of reasoning chains from a multi-hop question-answering task, revealing intriguing patterns that distinguish valid from invalid reasoning. We show that valid reasoning chains have lower Hamiltonian energy and move in ways that make the best trade-off between getting more information and answering the right question. Furthermore, we demonstrate the application of this framework to steer the creation of more efficient reasoning algorithms within AI systems. Our results not only provide new insights into the nature of valid reasoning but also open up exciting possibilities for physics-inspired approaches to understanding and improving artificial intelligence.

While Large Reasoning Models (LRMs) have demonstrated impressive capabilities in solving complex tasks through the generation of long reasoning chains, this reliance on verbose generation results in significant latency and computational overhead. To address these challenges, we propose CoSMo (Consistency-Guided Split-Merge Optimization), a framework designed to eliminate structural redundancy rather than indiscriminately restricting token volume. Specifically, CoSMo utilizes a split-merge algorithm that dynamically refines reasoning chains by merging redundant segments and splitting logical gaps to ensure coherence. We then employ structure-aligned reinforcement learning with a novel segment-level budget to supervise the model in maintaining efficient reasoning structures throughout training. Extensive experiments across multiple benchmarks and backbones demonstrate that CoSMo achieves superior performance, improving accuracy by 3.3 points while reducing segment usage by 28.7\% on average compared to reasoning efficiency baselines.

We propose a novel approach for preserving topological structures of the input space in latent representations of autoencoders. Using persistent homology, a technique from topological data analysis, we calculate topological signatures of both the input and latent space to derive a topological loss term. Under weak theoretical assumptions, we construct this loss in a differentiable manner, such that the encoding learns to retain multi-scale connectivity information. We show that our approach is theoretically well-founded and that it exhibits favourable latent representations on a synthetic manifold as well as on real-world image data sets, while preserving low reconstruction errors.

Large language models (LLMs) generalize smoothly across continuous semantic spaces, yet strict logical reasoning demands the formation of discrete decision boundaries. Prevailing theories relying on linear isometric projections fail to resolve this fundamental tension. In this work, we argue that task context operates as a non-isometric dynamical operator that enforces a necessary "topological distortion." By applying Gram-Schmidt decomposition to residual-stream activations , we reveal a dual-modulation mechanism driving this process: a class-agnostic topological preservation that anchors global structure to prevent semantic collapse, and a specific algebraic divergence that directionally tears apart cross-class concepts to forge logical boundaries. We validate this geometric evolution across a gradient of tasks, from simple mapping to complex primality testing. Crucially, targeted specific vector ablation establishes a strict causal binding between this topology and model function: algebraically erasing the divergence component collapses parity classification accuracy from 100% to chance levels (38.57%). Furthermore, we uncover a three-phase layer-wise geometric dynamic and demonstrate that under social pressure prompts, models fail to generate sufficient divergence. This results in a "manifold entanglement" that geometrically explains sycophancy and hallucination. Ultimately, our findings revise the linear-isometric presumption, demonstrating that the emergence of discrete logic in LLMs is purchased at an irreducible cost of topological deformation.

Accelerating materials discovery requires AI systems that can generate scientifically valid hypotheses through multi-step, domain-grounded reasoning. Standard large language models often produce fluent but weakly traceable responses to open-ended materials design problems, making it difficult to determine whether final answers are supported by coherent intermediate reasoning. We develop Graph-PRefLexOR, a family of graph-native reasoning models fine-tuned with Group Relative Policy Optimization (GRPO) to organize reasoning into explicit phases for mechanism exploration, graph construction, pattern extraction, and hypothesis synthesis. This design links neural language generation with symbolic relational structure, enabling causal connections to be constructed, inspected, and reused. On 100 open-ended questions from materials science and mechanics literature, Graph-PRefLexOR achieves 40-65% improvements over corresponding base models, with the largest gains in reasoning traceability. Embedding analyses show broader semantic exploration and approximately 2-3 times greater semantic diversity than baselines. Semantic backtracking and layer-wise hidden-state analyses further show stronger alignment between structured reasoning and final answers. Finally, test-time graph expansion reveals that additional compute primarily increases long-range conceptual recombination within a bounded semantic space, rather than simply expanding semantic coverage. These results establish graph-native reinforcement learning as a pathway toward interpretable AI systems for scientific hypothesis generation in materials design and other scientific applications.

Topology reasoning, which unifies perception and structured reasoning, plays a vital role in understanding intersections for autonomous driving. However, its performance heavily relies on the accuracy of lane detection, particularly at connected lane endpoints. Existing methods often suffer from lane endpoints deviation, leading to incorrect topology construction. To address this issue, we propose TopoPoint, a novel framework that explicitly detects lane endpoints and jointly reasons over endpoints and lanes for robust topology reasoning. During training, we independently initialize point and lane query, and proposed Point-Lane Merge Self-Attention to enhance global context sharing through incorporating geometric distances between points and lanes as an attention mask . We further design Point-Lane Graph Convolutional Network to enable mutual feature aggregation between point and lane query. During inference, we introduce Point-Lane Geometry Matching algorithm that computes distances between detected points and lanes to refine lane endpoints, effectively mitigating endpoint deviation. Extensive experiments on the OpenLane-V2 benchmark demonstrate that TopoPoint achieves state-of-the-art performance in topology reasoning (48.8 on OLS). Additionally, we propose DET_p to evaluate endpoint detection, under which our method significantly outperforms existing approaches (52.6 v.s. 45.2 on DET_p). The code is released at https://github.com/Franpin/TopoPoint.

The scarcity of high-quality, logically sound data is a critical bottleneck for advancing the mathematical reasoning of Large Language Models (LLMs). Our work confronts this challenge by turning decades of automated theorem proving research into a scalable data engine. Rather than relying on error-prone LLMs or complex proof-assistant syntax like Lean and Isabelle, our framework leverages E-prover's saturation capabilities on the vast TPTP axiom library to derive a massive, guaranteed-valid corpus of theorems. Our pipeline is principled and simple: saturate axioms, filter for "interesting" theorems, and generate tasks. With no LLMs in the loop, we eliminate factual errors by construction. This purely symbolic data is then transformed into three difficulty-controlled challenges: entailment verification, premise selection, and proof reconstruction. Our zero-shot experiments on frontier models reveal a clear weakness: performance collapses on tasks requiring deep, structural reasoning. Our framework provides both the diagnostic tool to measure this gap and a scalable source of symbolic training data to address it. We make the code and data publicly available. https://github.com/sileod/reasoning_core https://hf.co/datasets/reasoning-core/rc1

Recent advances in large multimodal models suggest that explicit reasoning mechanisms play a critical role in improving model reliability, interpretability, and cross-modal alignment. While such reasoning-centric approaches have been proven effective in language and vision tasks, their extension to 3D remains underdeveloped. CoRe3D introduces a unified 3D understanding and generation reasoning framework that jointly operates over semantic and spatial abstractions, enabling high-level intent inferred from language to directly guide low-level 3D content formation. Central to this design is a spatially grounded reasoning representation that decomposes 3D latent space into localized regions, allowing the model to reason over geometry in a compositional and procedural manner. By tightly coupling semantic chain-of-thought inference with structured spatial reasoning, CoRe3D produces 3D outputs that exhibit strong local consistency and faithful alignment with linguistic descriptions.

Mathematical reasoning is a hallmark of human intelligence, requiring logical deduction, symbolic manipulation, and abstract thinking. Recent multimodal large language models (MLLMs) have demonstrated strong performance on geometry problems through multi-step reasoning. To better emulate human problem-solving, intermediate steps can incorporate auxiliary visual constructions, such as additional lines or points, which improve geometric interpretation and educational clarity. In this work, we introduce the GeoMathCode, where programmatic representations serve as intermediate visual outputs. We further conduct an in-depth analysis of the underlying reasoning geometry. Experimental results show that reasoning and code generation steps can be disentangled in the latent space, while supervised fine-tuning (SFT) makes the reasoning manifold more structured and informative. Moreover, hierarchical syntactic code structures emerge as disentangled latent subspaces, and contain more mathematical symbolic information than visual representations.

We study how large language models (LLMs) ``think'' through their representation space. We propose a novel geometric framework that models an LLM's reasoning as flows -- embedding trajectories evolving where logic goes. We disentangle logical structure from semantics by employing the same natural deduction propositions with varied semantic carriers, allowing us to test whether LLMs internalize logic beyond surface form. This perspective connects reasoning with geometric quantities such as position, velocity, and curvature, enabling formal analysis in representation and concept spaces. Our theory establishes: (1) LLM reasoning corresponds to smooth flows in representation space, and (2) logical statements act as local controllers of these flows' velocities. Using learned representation proxies, we design controlled experiments to visualize and quantify reasoning flows, providing empirical validation of our theoretical framework. Our work serves as both a conceptual foundation and practical tools for studying reasoning phenomenon, offering a new lens for interpretability and formal analysis of LLMs' behavior.

Image-based 3D reconstruction has increasingly stunning results over the past few years with the latest improvements in computer vision and graphics. Geometry and topology are two fundamental concepts when dealing with 3D mesh structures. But the latest often remains a side issue in the 3D mesh-based reconstruction literature. Indeed, performing per-vertex elementary displacements over a 3D sphere mesh only impacts its geometry and leaves the topological structure unchanged and fixed. Whereas few attempts propose to update the geometry and the topology, all need to lean on costly 3D ground-truth to determine the faces/edges to prune. We present in this work a method that aims to refine the topology of any 3D mesh through a face-pruning strategy that extensively relies upon 2D alpha masks and camera pose information. Our solution leverages a differentiable renderer that renders each face as a 2D soft map. Its pixel intensity reflects the probability of being covered during the rendering process by such a face. Based on the 2D soft-masks available, our method is thus able to quickly highlight all the incorrectly rendered faces for a given viewpoint. Because our module is agnostic to the network that produces the 3D mesh, it can be easily plugged into any self-supervised image-based (either synthetic or natural) 3D reconstruction pipeline to get complex meshes with a non-spherical topology.

Mapping is crucial for spatial reasoning, planning and robot navigation. Existing approaches range from metric, which require precise geometry-based optimization, to purely topological, where image-as-node based graphs lack explicit object-level reasoning and interconnectivity. In this paper, we propose a novel topological representation of an environment based on "image segments", which are semantically meaningful and open-vocabulary queryable, conferring several advantages over previous works based on pixel-level features. Unlike 3D scene graphs, we create a purely topological graph with segments as nodes, where edges are formed by a) associating segment-level descriptors between pairs of consecutive images and b) connecting neighboring segments within an image using their pixel centroids. This unveils a "continuous sense of a place", defined by inter-image persistence of segments along with their intra-image neighbours. It further enables us to represent and update segment-level descriptors through neighborhood aggregation using graph convolution layers, which improves robot localization based on segment-level retrieval. Using real-world data, we show how our proposed map representation can be used to i) generate navigation plans in the form of "hops over segments" and ii) search for target objects using natural language queries describing spatial relations of objects. Furthermore, we quantitatively analyze data association at the segment level, which underpins inter-image connectivity during mapping and segment-level localization when revisiting the same place. Finally, we show preliminary trials on segment-level `hopping' based zero-shot real-world navigation. Project page with supplementary details: oravus.github.io/RoboHop/

Modern representation learning increasingly relies on unsupervised and self-supervised methods trained on large-scale unlabeled data. While these approaches achieve impressive generalization across tasks and domains, evaluating embedding quality without labels remains an open challenge. In this work, we propose Persistence, a topology-aware metric based on persistent homology that quantifies the geometric structure and topological richness of embedding spaces in a fully unsupervised manner. Unlike metrics that assume linear separability or rely on covariance structure, Persistence captures global and multi-scale organization. Empirical results across diverse domains show that Persistence consistently achieves top-tier correlations with downstream performance, outperforming existing unsupervised metrics and enabling reliable model and hyperparameter selection.

Leaf-lesion segmentation is topology-sensitive: small merges, splits, or false holes can be biologically meaningful descriptors of biochemical pathways, yet they are weakly penalized by standard pixel-wise losses in Euclidean latents. I explore HyperTopo-Adapters, a lightweight, parameter-efficient head trained on top of a frozen vision encoder, which embeds features on a product manifold -- hyperbolic + Euclidean + spherical (H + E + S) -- to encourage hierarchical separation (H), local linear detail (E), and global closure (S). A topology prior complements Dice/BCE in two forms: (i) persistent-homology (PH) distance for evaluation and selection, and (ii) a differentiable surrogate that combines a soft Euler-characteristic match with total variation regularization for stable training. I introduce warm-ups for both the hyperbolic contrastive term and the topology prior, per-sample evaluation of structure-aware metrics (Boundary-F1, Betti errors, PD distance), and a min-PD within top-K Dice rule for checkpoint selection. On a Kaggle leaf-lesion dataset (N=2,940), early results show consistent gains in boundary and topology metrics (reducing Delta beta_1 hole error by 9%) while Dice/IoU remain competitive. The study is diagnostic by design: I report controlled ablations (curvature learning, latent dimensions, contrastive temperature, surrogate settings), and ongoing tests varying encoder strength (ResNet-50, DeepLabV3, DINOv2/v3), input resolution, PH weight, and partial unfreezing of late blocks. The contribution is an open, reproducible train/eval suite (available at https://github.com/ChimdiWalter/HyperTopo-Adapters) that isolates geometric/topological priors and surfaces failure modes to guide stronger, topology-preserving architectures.

There is a growing body of work that leverages features extracted via topological data analysis to train machine learning models. While this field, sometimes known as topological machine learning (TML), has seen some notable successes, an understanding of how the process of learning from topological features differs from the process of learning from raw data is still limited. In this work, we begin to address one component of this larger issue by asking whether a model trained with topological features learns internal representations of data that are fundamentally different than those learned by a model trained with the original raw data. To quantify ``different'', we exploit two popular metrics that can be used to measure the similarity of the hidden representations of data within neural networks, neural stitching and centered kernel alignment. From these we draw a range of conclusions about how training with topological features does and does not change the representations that a model learns. Perhaps unsurprisingly, we find that structurally, the hidden representations of models trained and evaluated on topological features differ substantially compared to those trained and evaluated on the corresponding raw data. On the other hand, our experiments show that in some cases, these representations can be reconciled (at least to the degree required to solve the corresponding task) using a simple affine transformation. We conjecture that this means that neural networks trained on raw data may extract some limited topological features in the process of making predictions.

The ongoing development of Lean 4's Mathlib has produced a macroscopic structural complexity that interweaves logical, mathematical, and infrastructural dependencies. We present a network analysis of this library, extracting its dependency structure into a multilayer graph of 308,129 declarations, 8.4 million edges, and 7,563 modules. By introducing graph decompositions that isolate explicit edges from those synthesized by the compiler or driven by proofs, we quantify the structural properties of formalized mathematics. Our analysis reveals three findings. First, taxonomies designed by humans diverge from logical structures, exhibiting a 50.9% coupling across namespaces. Second, developers utilize a median of 1.6% of the imported scope. Third, formalization compresses semantic hierarchies, with network centrality capturing language infrastructure rather than mathematical relevance.

Several machine learning models are defined for inputs of any size, such as graphs with different numbers of nodes and point clouds containing varying numbers of points. The universality properties of such any-dimensional models remain poorly understood, as universality is traditionally studied for models accepting inputs of a fixed size, defined on a compact subset of their domain. In sharp contrast, any-dimensional models can be viewed as sequences of functions defined on growing-sized inputs, and it is not clear in which sense they can be universal. We develop a systematic approach to establish any-dimensional universality, by identifying any-dimensional functions with a unique function taking inputs in a suitable infinite-dimensional limit space containing inputs of all finite sizes as well as their limits. Using the symmetries of these inputs and relations between inputs of different sizes, we show that this limit space admits a natural topology with rich families of compact sets on which any-dimensional universality can be established. We illustrate our approach by showing that several existing architectures fail to be universal, and we propose simple modifications that restore universality.

Current Chain-of-Thought (CoT) verification methods predict reasoning correctness based on outputs (black-box) or activations (gray-box), but offer limited insight into why a computation fails. We introduce a white-box method: Circuit-based Reasoning Verification (CRV). We hypothesize that attribution graphs of correct CoT steps, viewed as execution traces of the model's latent reasoning circuits, possess distinct structural fingerprints from those of incorrect steps. By training a classifier on structural features of these graphs, we show that these traces contain a powerful signal of reasoning errors. Our white-box approach yields novel scientific insights unattainable by other methods. (1) We demonstrate that structural signatures of error are highly predictive, establishing the viability of verifying reasoning directly via its computational graph. (2) We find these signatures to be highly domain-specific, revealing that failures in different reasoning tasks manifest as distinct computational patterns. (3) We provide evidence that these signatures are not merely correlational; by using our analysis to guide targeted interventions on individual transcoder features, we successfully correct the model's faulty reasoning. Our work shows that, by scrutinizing a model's computational process, we can move from simple error detection to a deeper, causal understanding of LLM reasoning.

Mathematical reasoning remains challenging for LLMs due to complex logic and the need for precise computation. Existing methods enhance LLM reasoning by synthesizing datasets through problem rephrasing, but face issues with generation quality and problem complexity. To address this, we propose to extract structural information with generated problem-solving code from mathematical reasoning and guide data generation with structured solutions. Applied to MATH and GSM8K, our approach produces 39K problems with labeled intermediate steps and a 6.1K-problem benchmark of higher difficulty. Results on our benchmark show that model performance declines as reasoning length increases. Additionally, we conducted fine-tuning experiments using the proposed training data on a range of LLMs, and the results validate the effectiveness of our dataset. We hope the proposed method and dataset will contribute to future research in enhancing LLM reasoning capabilities. Our code and data are available at https://github.com/OpenCausaLab/StructuralGeneration.

We give a conceptual treatment of the notion of joints, marginals, and independence in the setting of categorical probability. This is achieved by endowing the usual probability monads (like the Giry monad) with a monoidal and an opmonoidal structure, mutually compatible (i.e. a bimonoidal structure). If the underlying monoidal category is cartesian monoidal, a bimonoidal structure is given uniquely by a commutative strength. However, if the underlying monoidal category is not cartesian monoidal, a strength is not enough to guarantee all the desired properties of joints and marginals. A bimonoidal structure is then the correct requirement for the more general case. We explain the theory and the operational interpretation, with the help of the graphical calculus for monoidal categories. We give a definition of stochastic independence based on the bimonoidal structure, compatible with the intuition and with other approaches in the literature for cartesian monoidal categories. We then show as an example that the Kantorovich monad on the category of complete metric spaces is a bimonoidal monad for a non-cartesian monoidal structure.

We argue that progress toward AGI is theory limited rather than data or scale limited. Building on the critical rationalism of Popper and Deutsch, we challenge the Platonic Representation Hypothesis. Observationally equivalent worlds can diverge under interventions, so observational adequacy alone cannot guarantee interventional competence. We begin by laying foundations, definitions of knowledge, learning, intelligence, counterfactual competence and AGI, and then analyze the limits of observational learning that motivate an error centric shift. We recast the problem as three questions about how explicit and implicit errors evolve under an agent's actions, which errors are unreachable within a fixed hypothesis space, and how conjecture and criticism expand that space. From these questions we propose Causal Mechanics, a mechanisms first program in which hypothesis space change is a first class operation and probabilistic structure is used when useful rather than presumed. We advance structural principles that make error discovery and correction tractable, including a differential Locality and Autonomy Principle for modular interventions, a gauge invariant form of Independent Causal Mechanisms for separability, and the Compositional Autonomy Principle for analogy preservation, together with actionable diagnostics. The aim is a scaffold for systems that can convert unreachable errors into reachable ones and correct them.

Accurately segmenting thin tubular structures, such as vessels, nerves, roads or concrete cracks, is a crucial task in computer vision. Standard deep learning-based segmentation loss functions, such as Dice or Cross-Entropy, focus on volumetric overlap, often at the expense of preserving structural connectivity or topology. This can lead to segmentation errors that adversely affect downstream tasks, including flow calculation, navigation, and structural inspection. Although current topology-focused losses mark an improvement, they introduce significant computational and memory overheads. This is particularly relevant for 3D data, rendering these losses infeasible for larger volumes as well as increasingly important multi-class segmentation problems. To mitigate this, we propose a novel Skeleton Recall Loss, which effectively addresses these challenges by circumventing intensive GPU-based calculations with inexpensive CPU operations. It demonstrates overall superior performance to current state-of-the-art approaches on five public datasets for topology-preserving segmentation, while substantially reducing computational overheads by more than 90%. In doing so, we introduce the first multi-class capable loss function for thin structure segmentation, excelling in both efficiency and efficacy for topology-preservation.

Recently, studies exemplified by Hyper-Connections (HC) have extended the ubiquitous residual connection paradigm established over the past decade by expanding the residual stream width and diversifying connectivity patterns. While yielding substantial performance gains, this diversification fundamentally compromises the identity mapping property intrinsic to the residual connection, which causes severe training instability and restricted scalability, and additionally incurs notable memory access overhead. To address these challenges, we propose Manifold-Constrained Hyper-Connections (mHC), a general framework that projects the residual connection space of HC onto a specific manifold to restore the identity mapping property, while incorporating rigorous infrastructure optimization to ensure efficiency. Empirical experiments demonstrate that mHC is effective for training at scale, offering tangible performance improvements and superior scalability. We anticipate that mHC, as a flexible and practical extension of HC, will contribute to a deeper understanding of topological architecture design and suggest promising directions for the evolution of foundational models.

Online scene perception and topology reasoning are critical for autonomous vehicles to understand their driving environments, particularly for mapless driving systems that endeavor to reduce reliance on costly High-Definition (HD) maps. However, recent advances in online scene understanding still face limitations, especially in long-range or occluded scenarios, due to the inherent constraints of onboard sensors. To address this challenge, we propose a Standard-Definition (SD) Map Enhanced scene Perception and Topology reasoning (SEPT) framework, which explores how to effectively incorporate the SD map as prior knowledge into existing perception and reasoning pipelines. Specifically, we introduce a novel hybrid feature fusion strategy that combines SD maps with Bird's-Eye-View (BEV) features, considering both rasterized and vectorized representations, while mitigating potential misalignment between SD maps and BEV feature spaces. Additionally, we leverage the SD map characteristics to design an auxiliary intersection-aware keypoint detection task, which further enhances the overall scene understanding performance. Experimental results on the large-scale OpenLane-V2 dataset demonstrate that by effectively integrating SD map priors, our framework significantly improves both scene perception and topology reasoning, outperforming existing methods by a substantial margin.

Geometric Problem Solving (GPS) remains at the heart of enhancing mathematical reasoning in large language models because it requires the combination of diagrammatic understanding, symbolic manipulation and logical inference. In existing literature, researchers have chiefly focused on synchronising the diagram descriptions with text literals and solving the problem. In this vein, they have either taken a neural, symbolic or neuro-symbolic approach. But this solves only the first two of the requirements, namely diagrammatic understanding and symbolic manipulation, while leaving logical inference underdeveloped. The logical inference is often limited to one chain-of-thought (CoT). To address this weakness in hitherto existing models, this paper proposes MARS-GPS, that generates multiple parallel reasoning rollouts augmented with Python code execution for numerical verification, ranks them using token-level entropy as a confidence signal, and aggregates answers through a multi-stage voting and self-verification pipeline. Empirical results show that MARS-GPS with 8 parallel rollouts achieves 88.8% on Geometry3K, a nearly +11% improvement over the prior state-of-the-art, with accuracy scaling consistently as the number of rollouts increases from 1 to 16 (+6.0% on ablation subset). We provide our code and data in an anonymous repository: https://anonymous.4open.science/r/MARS-GPS-DE55.

Human knowledge is inherently structured and interdependent: mastery of a concept requires prior mastery of its prerequisites, a principle formalized by Knowledge Space Theory (KST). While LLMs achieve strong performance on complex reasoning tasks, it remains unclear whether they exhibit coherent, human-like knowledge structure. We introduce a KST-grounded framework for evaluating LLM knowledge structure in mathematical reasoning, using it as a normative framework to analyze whether LLM behavior adheres to principled knowledge dependencies. Evaluating eight open- and closed-source LLMs against real human learners, we find that (1) LLMs do not adhere to human knowledge structure -- they frequently violate knowledge dependencies and fail to leverage related knowledge provided in context to improve performance on dependent questions; (2) LLMs do not share a consistent knowledge structure among themselves, as reflected by low overlap in their knowledge distributions. Furthermore, these structural deficiencies remain largely invisible to accuracy-based and LLM-as-judge evaluations. Together, our results provide behavioral evidence that current LLMs knowledge does not follow a human-like structure.

Dramatic progress has been witnessed in basic vision tasks involving low-level perception, such as object recognition, detection, and tracking. Unfortunately, there is still an enormous performance gap between artificial vision systems and human intelligence in terms of higher-level vision problems, especially ones involving reasoning. Earlier attempts in equipping machines with high-level reasoning have hovered around Visual Question Answering (VQA), one typical task associating vision and language understanding. In this work, we propose a new dataset, built in the context of Raven's Progressive Matrices (RPM) and aimed at lifting machine intelligence by associating vision with structural, relational, and analogical reasoning in a hierarchical representation. Unlike previous works in measuring abstract reasoning using RPM, we establish a semantic link between vision and reasoning by providing structure representation. This addition enables a new type of abstract reasoning by jointly operating on the structure representation. Machine reasoning ability using modern computer vision is evaluated in this newly proposed dataset. Additionally, we also provide human performance as a reference. Finally, we show consistent improvement across all models by incorporating a simple neural module that combines visual understanding and structure reasoning.

Leveraging generative Artificial Intelligence (AI), we have transformed a dataset comprising 1,000 scientific papers into an ontological knowledge graph. Through an in-depth structural analysis, we have calculated node degrees, identified communities and connectivities, and evaluated clustering coefficients and betweenness centrality of pivotal nodes, uncovering fascinating knowledge architectures. The graph has an inherently scale-free nature, is highly connected, and can be used for graph reasoning by taking advantage of transitive and isomorphic properties that reveal unprecedented interdisciplinary relationships that can be used to answer queries, identify gaps in knowledge, propose never-before-seen material designs, and predict material behaviors. We compute deep node embeddings for combinatorial node similarity ranking for use in a path sampling strategy links dissimilar concepts that have previously not been related. One comparison revealed structural parallels between biological materials and Beethoven's 9th Symphony, highlighting shared patterns of complexity through isomorphic mapping. In another example, the algorithm proposed a hierarchical mycelium-based composite based on integrating path sampling with principles extracted from Kandinsky's 'Composition VII' painting. The resulting material integrates an innovative set of concepts that include a balance of chaos/order, adjustable porosity, mechanical strength, and complex patterned chemical functionalization. We uncover other isomorphisms across science, technology and art, revealing a nuanced ontology of immanence that reveal a context-dependent heterarchical interplay of constituents. Graph-based generative AI achieves a far higher degree of novelty, explorative capacity, and technical detail, than conventional approaches and establishes a widely useful framework for innovation by revealing hidden connections.

Persistent homology (PH) provides topological descriptors for geometric data, such as weighted graphs, which are interpretable, stable to perturbations, and invariant under, e.g., relabeling. Most applications of PH focus on the one-parameter case -- where the descriptors summarize the changes in topology of data as it is filtered by a single quantity of interest -- and there is now a wide array of methods enabling the use of one-parameter PH descriptors in data science, which rely on the stable vectorization of these descriptors as elements of a Hilbert space. Although the multiparameter PH (MPH) of data that is filtered by several quantities of interest encodes much richer information than its one-parameter counterpart, the scarceness of stability results for MPH descriptors has so far limited the available options for the stable vectorization of MPH. In this paper, we aim to bring together the best of both worlds by showing how the interpretation of signed barcodes -- a recent family of MPH descriptors -- as signed measures leads to natural extensions of vectorization strategies from one parameter to multiple parameters. The resulting feature vectors are easy to define and to compute, and provably stable. While, as a proof of concept, we focus on simple choices of signed barcodes and vectorizations, we already see notable performance improvements when comparing our feature vectors to state-of-the-art topology-based methods on various types of data.

The adaptation of large language models (LLMs) to chemistry has shown promising performance in molecular understanding tasks, such as generating a text description from a molecule. However, proper reasoning based on molecular structural information remains a significant challenge, e.g., even advanced LLMs such as GPT-4o struggle to identify functional groups which are crucial for inferring the molecular property of interest. To address this limitation, we propose StructCoT, a structure-aware chain-of-thought (CoT) that enhances LLMs' understanding of molecular structures by explicitly injecting the key structural features of molecules. Moreover, we introduce two fine-tuning frameworks for adapting the existing LLMs to use our StructCoT. Our experiments demonstrate that incorporating StructCoT with our fine-tuning frameworks leads to consistent improvements in both molecular understanding tasks.

Mathematical geometric reasoning is essential for scientific discovery and educational development, requiring precise logic and rigorous formal verification. While recent advances in Multimodal Large Language Models (MLLMs) have improved reasoning tasks, existing models typically struggle with formal geometric reasoning, particularly when dynamically constructing and verifying auxiliary geometric elements. To address these challenges, we introduce Geoint-R1, a multimodal reasoning framework designed to generate formally verifiable geometric solutions from textual descriptions and visual diagrams. Geoint-R1 uniquely integrates auxiliary elements construction, formal reasoning represented via Lean4, and interactive visualization. To systematically evaluate and advance formal geometric reasoning, we propose the Geoint benchmark, comprising 1,885 rigorously annotated geometry problems across diverse topics such as plane, spatial, and solid geometry. Each problem includes structured textual annotations, precise Lean4 code for auxiliary constructions, and detailed solution steps verified by experts. Extensive experiments demonstrate that Geoint-R1 significantly surpasses existing multimodal and math-specific reasoning models, particularly on challenging problems requiring explicit auxiliary element constructions.

LLMs have demonstrated strong mathematical reasoning abilities by leveraging reinforcement learning with long chain-of-thought, yet they continue to struggle with theorem proving due to the lack of clear supervision signals when solely using natural language. Dedicated domain-specific languages like Lean provide clear supervision via formal verification of proofs, enabling effective training through reinforcement learning. In this work, we propose Seed-Prover, a lemma-style whole-proof reasoning model. Seed-Prover can iteratively refine its proof based on Lean feedback, proved lemmas, and self-summarization. To solve IMO-level contest problems, we design three test-time inference strategies that enable both deep and broad reasoning. Seed-Prover proves 78.1% of formalized past IMO problems, saturates MiniF2F, and achieves over 50\% on PutnamBench, outperforming the previous state-of-the-art by a large margin. To address the lack of geometry support in Lean, we introduce a geometry reasoning engine Seed-Geometry, which outperforms previous formal geometry engines. We use these two systems to participate in IMO 2025 and fully prove 5 out of 6 problems. This work represents a significant advancement in automated mathematical reasoning, demonstrating the effectiveness of formal verification with long chain-of-thought reasoning.

Knowledge Graph-based Question Answering (KGQA) plays a pivotal role in complex reasoning tasks but remains constrained by two persistent challenges: the structural heterogeneity of Knowledge Graphs(KGs) often leads to semantic mismatch during retrieval, while existing reasoning path retrieval methods lack a global structural perspective. To address these issues, we propose Structure-Tracing Evidence Mining (STEM), a novel framework that reframes multi-hop reasoning as a schema-guided graph search task. First, we design a Semantic-to-Structural Projection pipeline that leverages KG structural priors to decompose queries into atomic relational assertions and construct an adaptive query schema graph. Subsequently, we execute globally-aware node anchoring and subgraph retrieval to obtain the final evidence reasoning graph from KG. To more effectively integrate global structural information during the graph construction process, we design a Triple-Dependent GNN (Triple-GNN) to generate a Global Guidance Subgraph (Guidance Graph) that guides the construction. STEM significantly improves both the accuracy and evidence completeness of multi-hop reasoning graph retrieval, and achieves State-of-the-Art performance on multiple multi-hop benchmarks.

We apply a generalized piecewise-linear (PL) version of Morse theory due to Grunert-Kuhnel-Rote to define and study new local and global notions of topological complexity for fully-connected feedforward ReLU neural network functions, F: R^n -> R. Along the way, we show how to construct, for each such F, a canonical polytopal complex K(F) and a deformation retract of the domain onto K(F), yielding a convenient compact model for performing calculations. We also give a construction showing that local complexity can be arbitrarily high.

Human reasoning relies on constructing and manipulating mental models-simplified internal representations of situations that we use to understand and solve problems. Conceptual diagrams (for example, sketches drawn by humans to aid reasoning) externalize these mental models, abstracting irrelevant details to efficiently capture relational and spatial information. In contrast, Large Language Models (LLMs) and Large Multimodal Models (LMMs) predominantly reason through textual representations, limiting their effectiveness in complex multi-step combinatorial and planning tasks. In this paper, we propose a zero-shot fully automatic framework that enables LMMs to reason through multiple chains of self-generated intermediate conceptual diagrams, significantly enhancing their combinatorial planning capabilities. Our approach does not require any human initialization beyond a natural language description of the task. It integrates both textual and diagrammatic reasoning within an optimized graph-of-thought inference framework, enhanced by beam search and depth-wise backtracking. Evaluated on multiple challenging PDDL planning domains, our method substantially improves GPT-4o's performance (for example, from 35.5% to 90.2% in Blocksworld). On more difficult planning domains with solution depths up to 40, our approach outperforms even the o1-preview reasoning model (for example, over 13% improvement in Parking). These results highlight the value of conceptual diagrams as a complementary reasoning medium in LMMs.

The capability of autonomous exploration in complex, unknown environments is important in many robotic applications. While recent research on autonomous exploration have achieved much progress, there are still limitations, e.g., existing methods relying on greedy heuristics or optimal path planning are often hindered by repetitive paths and high computational demands. To address such limitations, we propose a novel exploration framework that utilizes the global topology information of observed environment to improve exploration efficiency while reducing computational overhead. Specifically, global information is utilized based on a skeletal topological graph representation of the environment geometry. We first propose an incremental skeleton extraction method based on wavefront propagation, based on which we then design an approach to generate a lightweight topological graph that can effectively capture the environment's structural characteristics. Building upon this, we introduce a finite state machine that leverages the topological structure to efficiently plan coverage paths, which can substantially mitigate the back-and-forth maneuvers (BFMs) problem. Experimental results demonstrate the superiority of our method in comparison with state-of-the-art methods. The source code will be made publicly available at: https://github.com/Haochen-Niu/STGPlanner.

LLM reasoning traces suffer from complex flaws -- *Step Internal Flaws* (logical errors, hallucinations, etc.) and *Step-wise Flaws* (overthinking, underthinking), which vary by sample. A natural approach would be to provide ground-truth labels to guide LLMs' reasoning. Contrary to intuition, we show that this yields no improvement in reasoning ability. We then propose CRAFT, a unified framework that mitigates both types of Step flaws, which builds a Reasoning Knowledge Graph (RKG) based on the consensus parts of multiple candidate traces, and synthesizes a high-quality trace through topological generation. Our approach improves label-prediction accuracy by 10+% on average, and consistently outperforms all baselines across both logical and mathematical reasoning benchmarks. Further, detailed benchmark evaluation proves that our method also improves the quality of LLMs' reasoning traces in multiple dimensions.

Though recent advances in vision-language models (VLMs) have achieved remarkable progress across a wide range of multimodal tasks, understanding 3D spatial relationships from limited views remains a significant challenge. Previous reasoning methods typically rely on pure text (e.g., topological cognitive maps) or on 2D visual cues. However, their limited representational capacity hinders performance in specific tasks that require 3D spatial imagination. To address this limitation, we propose 3DThinker, a framework that can effectively exploits the rich geometric information embedded within images while reasoning, like humans do. Our framework is the first to enable 3D mentaling during reasoning without any 3D prior input, and it does not rely on explicitly labeled 3D data for training. Specifically, our training consists of two stages. First, we perform supervised training to align the 3D latent generated by VLM while reasoning with that of a 3D foundation model (e.g., VGGT). Then, we optimize the entire reasoning trajectory solely based on outcome signals, thereby refining the underlying 3D mentaling. Extensive experiments across multiple benchmarks show that 3DThinker consistently outperforms strong baselines and offers a new perspective toward unifying 3D representations into multimodal reasoning. Our code will be available at https://github.com/zhangquanchen/3DThinker.

Current 3D spatial reasoning methods face a fundamental trade-off: neuro-symbolic 3D (NS3D) concept learners achieve interpretable reasoning through compositional programs but are constrained to closed-set concept vocabularies and simple programs; end-to-end 3D multi-modal LLMs (3D MLLMs) could handle complex natural language and open-vocabulary concepts but suffer from black-box reasoning without explicit spatial verification. We introduce APEIRIA, a neuro-symbolic 3D MLLM to bridge two paradigms by distilling symbolic reasoning patterns into MLLMs with natural language chain-of-thought. Our three-stage curriculum progressively builds reasoning capabilities: a) 3D perception alignment grounds object visual-geometric features to the LLM, b) CoT-SFT teaches query decomposition and stepwise verification from symbolic program traces, and c) CoT-RL extends reasoning patterns to open-set concepts and deeply nested instructions. By transferring reasoning patterns rather than concept-specific knowledge, APEIRIA preserves key NS3D virtues: transparent reasoning and modular interchangeability of planning and perception components. Evaluations on grounding, question answering, and captioning show that APEIRIA surpasses prior NS3D methods and matches state-of-the-art 3D MLLMs on 3D spatial reasoning datasets, unifying symbolic methods' systematic reasoning with MLLMs' flexibility. Code is available at https://github.com/oceanflowlab/APEIRIA.

Multimodal geometry reasoning requires VLMs to extract precise visual relations and preserve them through multi-step deduction. Existing free-form traces obscure the decisions that determine the answer, and trajectory-level reinforcement learning distributes a single terminal signal across the entire response. We introduce credit-addressable reasoning, in which the semantic units exposed during inference also define where learning compares alternatives and assigns credit. We instantiate this principle with Code-CoT, which retains the diagram, represents visual relations as line-addressable executable code, and organizes reasoning into typed events, and CE-GRPO, which selects event boundaries using structural priors and type-normalized entropy, samples complete continuations from shared prefixes, and converts outcome differences into localized advantages. Across nine geometry benchmarks, CE-GRPO achieves an average accuracy of 76.04, outperforming Qwen3-VL-8B and trajectory-level GRPO by 8.09 and 3.43 points, respectively. Its relative advantage increases with the number of intermediate events, demonstrating the value of representation--optimization co-design for long, dependency-heavy multimodal reasoning.

Vision-Language Models (VLMs) have achieved remarkable progress in integrating visual perception with language understanding. However, effective multimodal reasoning requires both accurate perception and robust reasoning, and weakness in either limits the performance of VLMs. Prior efforts to enhance reasoning often depend on high-quality chain-of-thought (CoT) data, obtained via labor-intensive human annotations, costly proprietary models, or self-training methods that overlook perception. To address these limitations, we propose a simple yet effective self-training framework called See-Think-Learn (STL). At its core, STL introduces a structured reasoning template that encourages the model to see before thinking, first extracting visual attributes in textual form, then using them to guide reasoning. The framework jointly improves perception and reasoning by having the model generate and learn from its own structured rationales in a self-training loop. Furthermore, we augment the training data with negative rationales, i.e. explanations that justify why certain answer choices are incorrect, to enhance the model's ability to distinguish between correct and misleading responses. This fosters more discriminative and robust learning. Experiments across diverse domains show that STL consistently outperforms baselines trained directly only on answers or self-generated reasoning, while qualitative analysis confirms the high quality of its rationales. STL thus provides a cost-effective solution to enhance multimodal reasoning ability of VLMs.

A common assumption holds that enough observational and interventional data, given to a strong enough predictor, suffices. We report a failure mode that contradicts it. Across hundreds of structural causal models, on identified quantities a strong predictor and a Bayesian baseline both succeed, but on unidentified quantities (the couplings between counterfactual worlds) the predictor collapses to a point, on 28% of models to one no valid model can produce, while the truth is an admissible interval more data never narrows. The gap is structural: prediction cannot represent uncertainty over counterfactual couplings. We cast a world model as a single positive semidefinite coupling kernel K(T,T') over admissible worlds, whose diagonal is the ordinary posterior (what a predictor recovers) and whose off-diagonal is the cross-world coupling it cannot, which every counterfactual reads. The paper is the theory of that off-diagonal. It is real: two states with identical posteriors differ on a cross-world query, and the off-diagonal is the coupling that fixes counterfactuals. It can be bounded: positive semidefiniteness is partial-identifying information the marginals lack, and enforcing it bounds counterfactuals in polynomial time where the exact response-type program is intractable. Logical structure sharpens it: ontology axioms tighten the bound by up to a third, propagating to couplings they never touch. It can be acquired: targeted scars, constraints learned from encountered infeasibilities, close the gap several times faster than untargeted ones. Its full reconstruction is approximate counting of the admissible worlds, tractable below the Sly-Sun threshold and inapproximable above; we do not claim to beat the worst case.

We present EuLearn, the first surface datasets equitably representing a diversity of topological types. We designed our embedded surfaces of uniformly varying genera relying on random knots, thus allowing our surfaces to knot with themselves. EuLearn contributes new topological datasets of meshes, point clouds, and scalar fields in 3D. We aim to facilitate the training of machine learning systems that can discern topological features. We experimented with specific emblematic 3D neural network architectures, finding that their vanilla implementations perform poorly on genus classification. To enhance performance, we developed a novel, non-Euclidean, statistical sampling method adapted to graph and manifold data. We also introduce adjacency-informed adaptations of PointNet and Transformer architectures that rely on our non-Euclidean sampling strategy. Our results demonstrate that incorporating topological information into deep learning workflows significantly improves performance on these otherwise challenging EuLearn datasets.

Complex reasoning problems often involve implicit spatial, geometric, and structural relationships that are not explicitly encoded in text. While recent reasoning models have achieved strong performance across many domains, purely text-based reasoning struggles to represent global structural constraints in complex settings. In this paper, we introduce FIGR, which integrates active visual thinking into multi-turn reasoning via end-to-end reinforcement learning. FIGR externalizes intermediate structural hypotheses by constructing visual representations during problem solving. By adaptively regulating when and how visual reasoning should be invoked, FIGR enables more stable and coherent reasoning over global structural properties that are difficult to capture from text alone. Experiments on challenging mathematical reasoning benchmarks demonstrate that FIGR outperforms strong text-only chain-of-thought baselines. In particular, FIGR improves the base model by 13.12% on AIME 2025 and 11.00% on BeyondAIME, highlighting the effectiveness of figure-guided multimodal reasoning in enhancing the stability and reliability of complex reasoning.

Geometry problem solving (GPS) is a high-level mathematical reasoning requiring the capacities of multi-modal fusion and geometric knowledge application. Recently, neural solvers have shown great potential in GPS but still be short in diagram presentation and modal fusion. In this work, we convert diagrams into basic textual clauses to describe diagram features effectively, and propose a new neural solver called PGPSNet to fuse multi-modal information efficiently. Combining structural and semantic pre-training, data augmentation and self-limited decoding, PGPSNet is endowed with rich knowledge of geometry theorems and geometric representation, and therefore promotes geometric understanding and reasoning. In addition, to facilitate the research of GPS, we build a new large-scale and fine-annotated GPS dataset named PGPS9K, labeled with both fine-grained diagram annotation and interpretable solution program. Experiments on PGPS9K and an existing dataset Geometry3K validate the superiority of our method over the state-of-the-art neural solvers. Our code, dataset and appendix material are available at https://github.com/mingliangzhang2018/PGPS.

Spatial intelligence is crucial for vision--language models (VLMs) in the physical world, yet many benchmarks evaluate largely unconstrained scenes where models can exploit 2D shortcuts. We introduce SSI-Bench, a VQA benchmark for spatial reasoning on constrained manifolds, built from complex real-world 3D structures whose feasible configurations are tightly governed by geometric, topological, and physical constraints. SSI-Bench contains 1,000 ranking questions spanning geometric and topological reasoning and requiring a diverse repertoire of compositional spatial operations, such as mental rotation, cross-sectional inference, occlusion reasoning, and force-path reasoning. It is created via a fully human-centered pipeline: ten researchers spent over 400 hours curating images, annotating structural components, and designing questions to minimize pixel-level cues. Evaluating 31 widely used VLMs reveals a large gap to humans: the best open-source model achieves 22.2% accuracy and the strongest closed-source model reaches 33.6%, while humans score 91.6%. Encouraging models to think yields only marginal gains, and error analysis points to failures in structural grounding and constraint-consistent 3D reasoning. Project page: https://ssi-bench.github.io.

Large language models have achieved near-expert performance in structured reasoning domains like mathematics and programming, yet their ability to perform compositional multi-hop reasoning in specialized scientific fields remains limited. We propose a bottom-up learning paradigm in which models are grounded in axiomatic domain facts and compose them to solve complex, unseen tasks. To this end, we present a post-training pipeline, based on a combination of supervised fine-tuning and reinforcement learning (RL), in which knowledge graphs act as implicit reward models. By deriving novel reward signals from knowledge graph paths, we provide verifiable, scalable, and grounded supervision that encourages models to compose intermediate axioms rather than optimize only final answers during RL. We validate this approach in the medical domain, training a 14B model on short-hop reasoning paths (1-3 hops) and evaluating its zero-shot generalization to complex multi-hop queries (4-5 hops). Our experiments show that path-derived rewards act as a "compositional bridge", enabling our model to significantly outperform much larger models and frontier systems like GPT-5.2 and Gemini 3 Pro, on the most difficult reasoning tasks. Furthermore, we demonstrate the robustness of our approach to adversarial perturbations against option-shuffling stress tests. This work suggests that grounding the reasoning process in structured knowledge is a scalable and efficient path toward intelligent reasoning.

Large language models (LLMs) excel at generating long chains of thought, but long reasoning traces are often verbose and memory-inefficient. In this work, we introduce Structured Thoughts, a framework that organizes reasoning into alternating <try> and <outcome> blocks: <try> captures exploratory scratch work, while <outcome> contains the distilled conclusion of that step. We construct a dataset of structured thoughts by segmenting reasoning traces into <try> blocks and prompting an LLM to summarize each step into its corresponding <outcome>. Fine-tuning pretrained foundation models on this reformatted data produces models that adopt the structured reasoning style, leading to performance gains of up to 8.08\% on reasoning benchmarks compared to standard SFT. The explicit structure also enables context pruning: after each <try>/<outcome> pair, the <try> can be pruned, allowing the model to retain conclusions without keeping the full scratch work in the context. A proof-of-concept pruning implementation achieves an average of 85\% memory / context savings with an 8.67\% performance drop across mathematical tasks.

We present ThinkProbe, a framework for structural analysis of LLM reasoning traces. ThinkProbe converts each trace into a Thought Graph a directed graph with cycles, 8 node types, and 6 edge types and derives a 19-metric five-dimensional cognitive profile (5D-CP: Breadth, Depth, Structure, Metacognitive, Efficiency) through a fully non-generative pipeline combining rule-based segmentation and discriminative semantic linking. Applied to 4{,}200 traces from 7 native reasoning models across 200 open-ended questions and 10 cognitive domains, ThinkProbe reveals that reasoning structure is a stable, model-level property: between-model variance exceeds between-domain variance by up to fourfold across four of five cognitive dimensions, with Structure showing genuine sensitivity to question domain, exposing qualitatively distinct cognitive profiles invisible to accuracy-based evaluation.

Can reinforcement learning with hard, verifiable rewards teach a compact language model to reason about physics, or does it primarily learn to pattern-match toward correct answers? We study this question by training a 1.5B-parameter reasoning model on beam statics, a classic engineering problem, using parameter-efficient RLVR with binary correctness rewards from symbolic solvers, without teacher-generated reasoning traces. The best BeamPERL checkpoint achieves a 66.7% improvement in Pass@1 over the base model. However, the learned competence is anisotropic: the model generalizes compositionally (more loads) but fails under topological shifts (moved supports) that require the same equilibrium equations. Intermediate checkpoints yield the strongest reasoning, while continued optimization degrades robustness while maintaining reward. These findings reveal a key limitation of outcome-level alignment: reinforcement learning with exact physics rewards induces procedural solution templates rather than internalization of governing equations. The precision of the reward signal - even when analytically exact - does not by itself guarantee transferable physical reasoning. Our results suggest that verifiable rewards may need to be paired with structured reasoning scaffolding to move beyond template matching toward robust scientific reasoning.

One common function class in machine learning is the class of ReLU neural networks. ReLU neural networks induce a piecewise linear decomposition of their input space called the canonical polyhedral complex. It has previously been established that it is decidable whether a ReLU neural network is piecewise linear Morse. In order to expand computational tools for analyzing the topological properties of ReLU neural networks, and to harness the strengths of discrete Morse theory, we introduce a schematic for translating between a given piecewise linear Morse function (e.g. parameters of a ReLU neural network) on a canonical polyhedral complex and a compatible (``relatively perfect") discrete Morse function on the same complex. Our approach is constructive, producing an algorithm that can be used to determine if a given vertex in a canonical polyhedral complex corresponds to a piecewise linear Morse critical point. Furthermore we provide an algorithm for constructing a consistent discrete Morse pairing on cells in the canonical polyhedral complex which contain this vertex. We additionally provide some new realizability results with respect to sublevel set topology in the case of shallow ReLU neural networks.

Combinatorics is central to Olympiad-level mathematical problem solving, requiring deep discrete reasoning, creative constructions, and rigorous structural insight. Recent evidence suggests that even today's strongest frontier models remain uneven on Olympiad combinatorics, revealing a gap in creative mathematical reasoning. We introduce ComBench, an Olympiad-level combinatorics benchmark for evaluating and diagnosing the combinatorial reasoning capabilities of large language models. ComBench contains 100 human-annotated competition-level problems organized around two complementary settings: analysis-centric problems, which primarily require rigorous mathematical arguments, and construction-centric problems, which require explicit constructions in addition to correctness justifications. The evaluation protocol combines rubric-guided proof grading with deterministic construction verification, exposing cases where proof quality and construction validity diverge. Experiments on frontier open- and closed-source models show that ComBench is far from saturated: the strongest model reaches 65.4% overall Avg. and 75.3% overall Best@4. We further find that Rigorous Proof Reasoning and Constructive Realization are distinct capabilities: Kimi-K2.6 trails GPT-5.5 on analysis-centric proof grading but surpasses it on construction-centric Best@4, while Existence and Construction problems remain consistently hardest across representative frontier models.

We introduce Topological Neural Operators (TNOs), a principled framework for operator learning on cell complexes that lifts neural operators (NOs) from functions on points and/or edges to topological domains. TNOs represent data as features defined on cells of varying dimension and model their interactions through Discrete Exterior Calculus, enabling explicit cross-dimensional coupling via gradient-, curl-, and divergence-type operators. The key design principle is to decouple where information flows, as governed by fixed topological operators, from how it is transformed (which is learned), yielding models that respect the geometric support of physical quantities and expose conservation and compatibility structure. We further propose Hierarchical TNOs (HTNOs), which incorporate learned coarse complexes to propagate long-range and topology-dependent information. Our framework subsumes existing NOs as a special case, providing a unified perspective on operator learning across discretizations. Across a range of PDE benchmarks, including irregular-geometry flow problems, TNOs and HTNOs improve accuracy; controlled studies further isolate the benefits of native higher-rank and topological structure. Project page: https://circle-group.github.io/research/TNO

We propose a many-sorted modal logic for reasoning about knowledge in multi-agent systems. Our logic introduces a clear distinction between participating agents and the environment. This allows to express local properties of agents and global properties of worlds in a uniform way, as well as to talk about the presence or absence of agents in a world. The logic subsumes the standard epistemic logic and is a conservative extension of it. The semantics is given in chromatic hypergraphs, a generalization of chromatic simplicial complexes, which were recently used to model knowledge in distributed systems. We show that the logic is sound and complete with respect to the intended semantics. We also show a further connection of chromatic hypergraphs with neighborhood frames.

Large Language Models (LLMs) have made remarkable strides in reasoning tasks, yet their performance often falters on novel and complex problems. Domain-specific continued pretraining (CPT) methods, such as those tailored for mathematical reasoning, have shown promise but lack transferability to broader reasoning tasks. In this work, we pioneer the use of Graph Problem Reasoning (GPR) to enhance the general reasoning capabilities of LLMs. GPR tasks, spanning pathfinding, network analysis, numerical computation, and topological reasoning, require sophisticated logical and relational reasoning, making them ideal for teaching diverse reasoning patterns. To achieve this, we introduce GraphPile, the first large-scale corpus specifically designed for CPT using GPR data. Spanning 10.9 billion tokens across 23 graph tasks, the dataset includes chain-of-thought, program-of-thought, trace of execution, and real-world graph data. Using GraphPile, we train GraphMind on popular base models Llama 3 and 3.1, as well as Gemma 2, achieving up to 4.9 percent higher accuracy in mathematical reasoning and up to 21.2 percent improvement in non-mathematical reasoning tasks such as logical and commonsense reasoning. By being the first to harness GPR for enhancing reasoning patterns and introducing the first dataset of its kind, our work bridges the gap between domain-specific pretraining and universal reasoning capabilities, advancing the adaptability and robustness of LLMs.

Artificial intelligence in construction increasingly depends on structured representations such as Building Information Models and knowledge graphs, yet early-stage building designs are predominantly created as flexible boundary-representation (B-rep) models that lack explicit spatial, semantic, and performance structure. This paper presents a robust, fully automated framework that transforms unstructured B-rep geometry into knowledge-graph-based Building Information Models and further into executable Building Energy Models. The framework enables artificial intelligence to explicitly interpret building elements, spatial topology, and their associated thermal and performance attributes. It integrates automated geometry cleansing, multiple auto space-generation strategies, graph-based extraction of space and element topology, ontology-aligned knowledge modeling, and reversible transformation between ontology-based BIM and EnergyPlus energy models. Validation on parametric, sketch-based, and real-world building datasets demonstrates high robustness, consistent topological reconstruction, and reliable performance-model generation. By bridging design models, BIM, and BEM, the framework provides an AI-oriented infrastructure that extends BIM- and graph-based intelligence pipelines to flexible early-stage design geometry, enabling performance-driven design exploration and optimization by learning-based methods.

We introduce ProofGrid, a benchmark suite for evaluating LLM reasoning through machine-checkable proofs rather than final answers alone. ProofGrid contains 15 tasks spanning proof writing, proof checking, proof masking, and proof gap-filling. Tasks are expressed in minimal formal notation, especially NDL, a compact natural-deduction language that fits in short prompts and supports precise, auditable verification. This yields mechanical, reproducible, and fine-grained evaluation rather than judgments by humans or LLMs. ProofGrid covers a calibrated difficulty spectrum, from foundational reasoning tests to structurally rich challenge tasks that no current model solves, while minimizing reliance on domain knowledge, solver delegation, and long-context artifacts. We also develop a comparative framework for reasoning benchmarks and use it to situate ProofGrid relative to existing work in terms of representation, verification guarantees, and reasoning depth. Methodologically, we introduce an instrumented proof-checking pipeline that tolerates minor surface deviations while locating the first substantive reasoning failure, improving measurement resolution and separating proof planning from low-level execution noise. Using this pipeline, we evaluate a broad range of open and proprietary models. Results show rapid progress but substantial remaining limits: frontier models perform well on several foundational tasks, yet difficult tasks, especially those requiring global combinatorial reasoning or low-level proof synthesis, remain far from solved. We also identify epistemic instability, where models generate flawed proofs yet correctly reject those local inferences in isolation, and formalize this with an Epistemic Stability Index. Finally, we complement accuracy with 2PL IRT analyses, Wright maps, and a normalized task-discrimination measure based on Fisher information.

Topological Neural Networks (TNNs) incorporate higher-order relational information beyond pairwise interactions, enabling richer representations than Graph Neural Networks (GNNs). Concurrently, topological descriptors based on persistent homology (PH) are being increasingly employed to augment the GNNs. We investigate the benefits of integrating these two paradigms. Specifically, we introduce TopNets as a broad framework that subsumes and unifies various methods in the intersection of GNNs/TNNs and PH such as (generalizations of) RePHINE and TOGL. TopNets can also be readily adapted to handle (symmetries in) geometric complexes, extending the scope of TNNs and PH to spatial settings. Theoretically, we show that PH descriptors can provably enhance the expressivity of simplicial message-passing networks. Empirically, (continuous and E(n)-equivariant extensions of) TopNets achieve strong performance across diverse tasks, including antibody design, molecular dynamics simulation, and drug property prediction.

Recent advances in pretraining 3D point cloud encoders (e.g., Point-BERT, Point-MAE) have produced powerful models, whose abilities are typically evaluated on geometric or semantic tasks. At the same time, topological descriptors have been shown to provide informative summaries of a shape's multiscale structure. In this paper we pose the question whether topological information can be derived from features produced by 3D encoders. To address this question, we first introduce DONUT, a synthetic benchmark with controlled topological complexity, and propose FILTR (Filtration Transformer), a learnable framework to predict persistence diagrams directly from frozen encoders. FILTR adapts a transformer decoder to treat diagram generation as a set prediction task. Our analysis on DONUT reveals that existing encoders retain only limited global topological signals, yet FILTR successfully leverages information produced by these encoders to approximate persistence diagrams. Our approach enables, for the first time, data-driven extraction of persistence diagrams from raw point clouds through an efficient learnable feed-forward mechanism.

Mathematical reasoning has long served as a stringent test of machine intelligence; over the past decade, it has moved from a niche problem within NLP to one of the most consequential AI frontiers. This survey provides a unified account of the field's evolution, from early rule-based math word problem (MWP) solvers and template-driven geometry systems, through neural expression generation and LLM prompting, to contemporary reasoning models, multi-agent systems, neuro-symbolic theorem provers, and verified discovery workflows. We organize the landscape along four axes: (i) informal reasoning over text and diagrams, spanning MWP solving, multimodal geometry, and VLMs; (ii) formal reasoning in proof assistants, including autoformalization, tactic prediction, compiler-guided repair, and proof search; (iii) mathematical discovery, where systems propose constructions, improve bounds, or assist attacks on open problems; and (iv) the inference and training-time techniques, including CoT prompting, tool use, process reward models, and RLVR, that increasingly connect generation with verification. We catalog major benchmarks across grade-school arithmetic, competition mathematics, geometry, formal proving, multimodal and multilingual reasoning, and expert evaluation, and we examine benchmark saturation, contamination, reporting mismatches, and the distinction between pass@1, majority voting, and verifier-assisted pass@k. We critically assess failure modes: brittleness under perturbation, reward hacking, multimodal grounding failures, fragile formalization, and the energy cost of reasoning-scale inference. Drawing on recent perspectives from working mathematicians, we identify future directions centered on verified-discovery workflows, reasoning efficiency, and infrastructure to make AI-assisted formalization broadly usable. Companion materials: https://github.com/Starscream-11813/awesome-AI4Math.

Large Language Models (LLMs) have achieved impressive performance on complex reasoning tasks with Chain-of-Thought (CoT) prompting. However, conventional CoT relies on reasoning steps explicitly verbalized in natural language, introducing inefficiencies and limiting its applicability to abstract reasoning. To address this, there has been growing research interest in latent CoT reasoning, where inference occurs within latent spaces. By decoupling reasoning from language, latent reasoning promises richer cognitive representations and more flexible, faster inference. Researchers have explored various directions in this promising field, including training methodologies, structural innovations, and internal reasoning mechanisms. This paper presents a comprehensive overview and analysis of this reasoning paradigm. We begin by proposing a unified taxonomy from four perspectives: token-wise strategies, internal mechanisms, analysis, and applications. We then provide in-depth discussions and comparative analyses of representative methods, highlighting their design patterns, strengths, and open challenges. We aim to provide a structured foundation for advancing this emerging direction in LLM reasoning. The relevant papers will be regularly updated at https://github.com/EIT-NLP/Awesome-Latent-CoT.

Understanding the physical structure is essential for real-world applications such as embodied agents, interactive design, and long-horizon manipulation. Yet, prevailing Vision-Language Model (VLM) evaluations still center on structure-agnostic, single-turn setups (e.g., VQA), which fail to assess agents' ability to reason about how geometry, contact, and support relations jointly constrain what actions are possible in a dynamic environment. To address this gap, we introduce the Causal Hierarchy of Actions and Interactions (CHAIN) benchmark, an interactive 3D, physics-driven testbed designed to evaluate whether models can understand, plan, and execute structured action sequences grounded in physical constraints. CHAIN shifts evaluation from passive perception to active problem solving, spanning tasks such as interlocking mechanical puzzles and 3D stacking and packing. We conduct a comprehensive study of state-of-the-art VLMs and diffusion-based models under unified interactive settings. Our results show that top-performing models still struggle to internalize physical structure and causal constraints, often failing to produce reliable long-horizon plans and cannot robustly translate perceived structure into effective actions. The project is available at https://social-ai-studio.github.io/CHAIN/.

Multimodal Large Language Models (MLLMs) have achieved remarkable progress but continue to struggle with geometric reasoning, primarily due to the perception bottleneck regarding fine-grained visual elements. While formal languages have aided plane geometry understanding, solid geometry which requires spatial understanding remains largely unexplored. In this paper, we address this challenge by designing a unified formal language that integrates plane and solid geometry, comprehensively covering geometric structures and semantic relations. We construct GDP-29K, a large-scale dataset comprising 20k plane and 9k solid geometry samples collected from diverse real-world sources, each paired with its ground-truth formal description. To ensure syntactic correctness and geometric consistency, we propose a training paradigm that combines Supervised Fine-Tuning with Reinforcement Learning via Verifiable Rewards. Experiments show that our approach achieves state-of-the-art parsing performance. Furthermore, we demonstrate that our parsed formal descriptions serve as a critical cognitive scaffold, significantly boosting MLLMs' capabilities for downstream geometry reasoning tasks. Our data and code are available at Geoparsing.

Despite the success of test-time scaling, Large Reasoning Models (LRMs) frequently encounter repetitive loops that lead to computational waste and inference failure. In this paper, we identify a distinct failure mode termed Circular Reasoning. Unlike traditional model degeneration, this phenomenon manifests as a self-reinforcing trap where generated content acts as a logical premise for its own recurrence, compelling the reiteration of preceding text. To systematically analyze this phenomenon, we introduce LoopBench, a dataset designed to capture two distinct loop typologies: numerical loops and statement loops. Mechanistically, we characterize circular reasoning as a state collapse exhibiting distinct boundaries, where semantic repetition precedes textual repetition. We reveal that reasoning impasses trigger the loop onset, which subsequently persists as an inescapable cycle driven by a self-reinforcing V-shaped attention mechanism. Guided by these findings, we employ the Cumulative Sum (CUSUM) algorithm to capture these precursors for early loop prediction. Experiments across diverse LRMs validate its accuracy and elucidate the stability of long-chain reasoning.

Deep iterative chain-of-thought (CoT) reasoning enables LLMs to tackle complex tasks by progressively activating relevant pre-trained knowledge. However, it faces challenges in ensuring continual improvement and determining a stopping criterion. In this paper, we investigate whether the relevant knowledge that contributes directly to solving the given question can be activated from the initial reasoning path, thus circumventing the need for iterative refinement. Our experiments reveal that increasing the diversity of initial reasoning paths can achieve comparable or superior performance, a concept we term breadth reasoning. However, existing breadth reasoning approaches, such as self-consistency, offer limited diversity. To address this limitation, we propose a simple yet effective method that enhances reasoning breadth by integrating contextual exploration with reduced sampling randomness. Extensive experiments demonstrate that our approach significantly outperforms deep iterative reasoning. Our code is provided in https://github.com/zongqianwu/breadth.