Researchers introduce mathematical primitives and a four-dimension benchmark, identify Discovery as the dominant bottleneck in LLM mathematical reasoning, and improve reasoning with a primitive-privileged self-distillation framework

The paper introduces the notion of Mathematical Primitive to probe whether large language models possess the structural mathematical understanding underlying their solutions, and proposes a new benchmark that evaluates mathematical reasoning along four distinct dimensions: Discovery, Generation, Digestion, and Execution. Prior evaluation of LLM mathematical ability largely rests on final solution accuracy; this work separates mathematical understanding into four dimensions that can be examined individually, distinguishing 'getting the answer right' from 'having structural understanding'. Evidence comes from the benchmark the authors construct and the systematic diagnosis performed on it, described in the abstract as a 'systematic diagnosis'; the number of items, the model list, and the statistics are not given in the abstract.

The systematic diagnosis finds that solution accuracy masks distinct capability profiles, that primitives unlock substantial latent execution capacity, and that Discovery is the dominant bottleneck in mathematical reasoning. These three points decompose 'ability' under a single accuracy metric into distinguishable profiles and locate the bottleneck in discovery rather than execution, indicating a priority direction for subsequent improvement. Evidence comes from the authors' diagnostic evaluation of models on the four-dimension benchmark; the abstract states that 'primitives unlock substantial latent execution capacity' and that 'Discovery is the dominant bottleneck', while specific effect sizes are not given in the abstract.

Post-training analysis shows that discovery-limited failures are particularly amenable to repair; the resulting primitive-privileged self-distillation framework selectively transfers primitive-guided reasoning into the student model and consistently improves mathematical reasoning over baselines across model scales and challenging benchmarks. Rather than distilling entire reasoning traces indiscriminately, the framework selectively transfers primitive-guided reasoning in line with the diagnosis, aligning the post-training intervention with the identified bottleneck. Evidence comes from comparative experiments across model scales and multiple challenging benchmarks, which the abstract describes as 'consistently improves mathematical reasoning over baselines'; benchmark names, model scales, and improvement magnitudes are not given in the abstract.

The work addresses researchers and practitioners working on mathematical reasoning in large language models, and applies to evaluation settings that need to separate 'correct solution' from 'structural understanding', as well as to training pipelines that translate diagnostic findings into post-training improvements. The scope of its conclusions is bounded by the four-dimension benchmark described in the abstract and by the model scales and challenging benchmarks tested; the primitive-privileged self-distillation framework is described as selectively transferring primitive-guided reasoning into the student model, so it is positioned as a post-training method rather than a pretraining or inference-time method.

Readers would still want to know how each of the four dimensions is operationalized in item form and judged; whether the diagnostic conclusions hold across model families and scales; under what conditions transferring primitive-guided reasoning yields the largest gains; and whether the claim that Discovery is the dominant bottleneck has finer quantitative support beyond the abstract. Because the available text is abstract-level and lacks figures and experimental tables, these questions cannot be confirmed from the present material and remain open questions for the original paper.