We introduce LM-Lexicon, an innovative definition modeling approach that incorporates data clustering, semantic expert learning, and model merging using a sparse mixture-of-experts architecture. By decomposing the definition modeling task into specialized semantic domains, where small language models are trained as domain experts, LM-Lexicon achieves substantial improvements (+7% BLEU score compared with the prior state-of-the-art model) over existing methods on five widely used benchmarks. Empirically, we demonstrate that 1) the clustering strategy enables fine-grained expert specialization with nearly 10% improvement in definition quality; 2) the semantic-aware domain-level routing mechanism achieves higher expert efficacy (+1%) than conventional token-level routing; and 3) further performance gains can be obtained through test-time compute and semantic expert scaling. Our work advances definition modeling while providing insights into the development of efficient language models for semantic-intensive applications.

Large Language Models (LLMs) often suffer from "Reasoning Collapse" on challenging mathematical reasoning tasks, where stochastic sampling produces lexical variations of the same erroneous logic rather than genuine semantic exploration. We observe that failed reasoning traces are often associated with a low-rank bias manifold in the model’s hidden-state geometry, which reduces exploration toward corrective solution directions. To address this, we propose Spectral Orthogonal Exploration (SOE), a geometric inference framework under a "Student Guides Teacher" paradigm. Instead of using a weak auxiliary agent for imitation, SOE uses it as an orthogonal probe to introduce semantically heterogeneous reasoning signals into the teacher’s orthogonal complement of its dominant subspace. This intervention steers the teacher toward more diverse reasoning trajectories and improves exploration beyond standard sampling. Experiments on mathematical benchmarks show that SOE improves average accuracy by 62.4% and average sampling efficiency by 113.7% over baseline methods, suggesting that geometric interventions can be effective for mitigating reasoning collapse in mathematical reasoning. We further provide preliminary evidence that SOE is also effective on logic and code generation benchmarks. Code is available at https://github.com/dayuwang401/spectral-orthogonal-exploration.