LiteToken: Removing Intermediate Merge Residues From BPE Tokenizers

Tokenization is fundamental to how language models represent and process text, yet the behavior of widely used BPE tokenizers has received far less study than model architectures and training. In this paper, we investigate intermediate merge residues in BPE vocabularies: tokens that are frequent during merge learning so that retained in the final vocabulary, but are mostly further merged and rarely emitted when tokenizing the corpus during tokenizer usage. Such low-frequency tokens not only waste vocabulary capacity but also increase vulnerability to adversarial or atypical inputs. We present a systematic empirical characterization of this phenomenon across commonly used tokenizers and introduce LiteToken, a simple method for removing residue tokens. Because the affected tokens are rarely used, pretrained models can often accommodate the modified tokenizer without additional fine-tuning. Experiments show that LiteToken reduces token fragmentation, reduces parameters, and improves robustness to noisy or misspelled inputs, while preserving overall performance.

Proof-RM: A Scalable and Generalizable Reward Model for Math Proof

While Large Language Models (LLMs) have demonstrated strong math reasoning abilities through Reinforcement Learning with *Verifiable Rewards* (RLVR), many advanced mathematical problems are proof-based, with no guaranteed way to determine the authenticity of a proof by simple answer matching. To enable automatic verification, a Reward Model (RM) capable of reliably evaluating full proof processes is required. In this work, we design a *scalable* data-construction pipeline that, with minimal human effort, leverages LLMs to generate a large quantity of high-quality "**question-proof-check**" triplet data. By systematically varying problem sources, generation methods, and model configurations, we create diverse problem-proof pairs spanning multiple difficulty levels, linguistic styles, and error types, subsequently filtered through hierarchical human review for label alignment. Utilizing these data, we train a proof-checking RM, incorporating additional process reward and token weight balance to stabilize the RL process. Our experiments validate the model's scalability and strong performance from multiple perspectives, including reward accuracy, generalization ability and test-time guidance, providing important practical recipes and tools for strengthening LLM mathematical capabilities.