Automated Proving of Shannon-Type Entropy Inequalities via Fine-Tuned Language Models and Guided Tree Search

Proving Shannon-type entropy inequalities is a fundamental task in information theory that often requires constructing non-trivial linear combinations of known constraints, which is a combinatorial search problem that scales poorly with the number of random variables. We investigate whether small-scale large language models (0.6B--1.7B parameters), fine-tuned on atomic proof steps and combined with guided beam search, can automate this process. On a held-out test set of 60 inequalities spanning n=10 to 15 variables, our 0.6B fine-tuned model achieves an 85\% proof success rate with tree search. GPT-5.5 solves 1.7\% samples under zero-shot prompting while Psitip solves 33.3\% samples. A systematic ablation study across training context length (4096 vs.\ 8192 tokens) and data distribution (n=9-skewed vs not skewed) reveals that a 4096-token not skewed training distribution yields the best performance, with extended context and skewed data providing no marginal benefit. We further identify two dominant failure modes -- format failures and step quality degradation -- and verify that the beam-scoring heuristic is essential via a controlled ablation (random scoring reduces success from 83\% to 23\%).

Routing-Aligned Fine-Tuning for Multilingual Downstream Tasks in Mixture-of-Experts Models

Mixture-of-Experts (MoE) models have emerged as a dominant paradigm for efficient LLM scaling, yet adapting them to non-English downstream tasks remains challenging. Existing fine-tuning approaches treat MoE models as monolithic learners, ignoring the heterogeneous routing structure that develops during pretraining. We validate across multiple MoE models and downstream tasks that middle layers form a language-universal alignment zone where routing divergence strongly predicts per-language task performance gaps. Building on this observation, we propose RA-MoE (Routing-Aligned MoE Fine-Tuning), a three-stage framework that categorizes parallel task examples into a four-way taxonomy (cc/ci/ic/ii) based on correctness in English and the target language, identifies task-relevant experts in the middle layers, and augments standard SFT with a routing alignment loss that encourages target-language routing on ci-type examples to follow the English task-expert activation pattern. Experiments across three MoE models, three tasks, and six target languages demonstrate that RA-MoE consistently outperforms standard SFT and strong baselines including Routing Steering and RISE, with the ci proportion of a task-language pair serving as a reliable predictor of alignment benefit.