C og A tom: From Cognitive Atoms to Olympiad-level Mathematical Reasoning in Large Language Models (2025)

Abstract: Mathematical reasoning poses significant challenges for Large Language Models (LLMs) due to its demand for multi-step reasoning and abstract conceptual integration. While recent test-time scaling techniques rely heavily on high-quality, challenging problems, the scarcity of Olympiad-level math problems remains a bottleneck. We introduce CogAtom, a novel cognitive atom-based framework for synthesizing mathematically rigorous and cognitively diverse problems. Unlike prior approaches, CogAtom models problem construction as a process of selecting and recombining fundamental reasoning units, cognitive atoms, extracted from human-authored solutions. A diversity-promoting random walk algorithm enables exploration of the cognitive atom space, while a constraint-based recombination mechanism ensures logical soundness and structural validity. The combinatorial nature of the graph structure provides a near-infinite space of reasoning paths, and the walk algorithm systematically explores this space to achieve large-scale synthesis of high-quality problems; meanwhile, by controlling the number of cognitive atoms, we can precisely adjust problem difficulty, ensuring diversity, scalability, and controllability of the generated problems. Experimental results demonstrate that CogAtom outperforms existing methods in accuracy, reasoning depth, and diversity, generating problems that closely match the difficulty of AIME while exceeding it in structural variation. Our work offers a cognitively grounded pathway toward scalable, high-quality math problem generation.Our code is publicly available at https://github.com/Icarus-1111/CogAtom.

## introduction reasoning abilities are core cognitive mechanisms underlying human problem-solving @xcite . among them, mathematical reasoning stands out for its unique cognitive complexity, requiring abstract concept comprehension, logical inference across domains, and multi-step solution strategies @xcite @xcite . as a result, mathematical reasoning has become a key benchmark for evaluating the progress of large language models (llms) toward artificial general intelligence (agi) @xcite . with the paradigm of llms shifting from training-time compute scaling toward test-time compute scaling @xcite @xcite , it heavily depends on the high-quality, multi-step mathematical problems as these high-quality problems are essential for effectively exposing model deficiencies in multi-step reasoning and validating the performance boundaries of test-time optimization strategies @xcite . this shift has generated a substantial demand for data that is simultaneously: (1) large-scale and scalable, to support compute-intensive optimization strategies; (2) highly challenging, to drive models beyond their current capabilities; and (3) cognitively diverse, to mitigate overfitting to familiar solution patterns. however, the existing collection of human-authored, olympiad-level problems is static, limited in size, and does not fulfill these requirements. this pronounced scarcity of suitable data has become a primary obstacle to the rigorous development and evaluation of advanced reasoning systems @xcite @xcite . the most straightforward solution to this data scarcity problem is to synthesize challenging problems automatically. existing synthesis methods can be categorized as prompt engineering-based methods @xcite @xcite , corpus mining methods @xcite , evolutionary and transfer-based methods @xcite and knowledge-driven synthesis methods @xcite @xcite . however, these methods fail to model fundamental units of thought-cognitive atoms-and their combinatorial principles from a cognitive science perspective. in contrast, when mathematics experts design olympiad problems, they carefully craft cognitive associations between concepts and construct rigorous logical frameworks that demand multi-level reasoning. current methods are unable to effectively emulate this process, leading to substantial gaps between generated problems and human-authored olympiad questions in terms of accuracy, logical coherence, and cognitive diversity. to overcome these structural limitations, we introduce cogatom, a framework that implements a paradigm shift from linear generation to structured synthesis, centered on the cognitive association graph. this framework operates through a systematic, three-stage generation process. first, reasoning atoms are extracted from a curated seed set and assembled into the global graph; its vast combinatorial nature provides a near-infinite space of potential reasoning paths, directly enabling scalable generation. next, a diversity-promoting random walk algorithm explores this structured space to sample long and intricate reasoning paths, the topological complexity of which forms the basis for problems of high difficulty. finally, these paths are transformed by a constraint-based recombination mechanism, driven by three cognitive transfer operators, which ensures the final combination is logically coherent while exhibiting high conceptual diversity.our contributions are summarized as follows: ## methodology as shown in figure 1 , our approach presents a comprehensive framework for mathematical problem synthesis, encompassing three crucial stages: (1) the extraction of reasoning atoms, (2) the construction of a cognitive association graph, and (3) the synthesis of challenging mathematical problems. ## conclusion in this paper, we introduced cogatom, a novel framework for mathematical problem synthesis that integrates reasoning atoms and cognitive association graphs to generate high-quality training data. our extensive experiments demonstrated that models fine-tuned on cogatom-generated problems achieve substantial improvements in mathematical reasoning capabilities, particularly on advanced multi-step reasoning tasks, outperforming existing methods by significant margins on challenging benchmarks including math500 and aime. the effectiveness of our approach highlights the importance of cognitive science principles in designing synthetic training data for enhancing reasoning abilities in foundation models.