Cliff Tokens: Identifying Single-Token Failure Triggers in LLM Mathematical Reasoning

Large language models reach high accuracy in mathematical reasoning, but individual traces on the same problem diverge; some arrive at the correct answer while others fail. Prior work localizes such failures at the step, chunk, or sentence level, or identifies tokens where failure has already occurred. These approaches leave open which token triggers failure. We introduce the cliff token, a token at which the estimated probability of reaching the correct answer (success probability) drops beyond an adaptive threshold. Across seven models and three mathematical reasoning benchmarks (GSM1K, MATH500, AIME 2025), cliff tokens act as failure triggers. For incorrect traces containing cliff tokens, we compare resampling immediately before and after the first cliff token. Resampling before it shows higher pass@k at the same sample count. We further introduce a cliff taxonomy of deterministic, uncertain, and sampled-off cliffs, defined by greedy choice and token entropy. Additionally, we show that the three types differ as training signals. Using single-token preference optimization at cliff positions (Cliff-DPO), we find that uncertain and sampled-off cliffs show larger accuracy gains than deterministic cliffs on three evaluation benchmarks. We release token-level rollout data and source code to enable further analysis without regenerating costly rollouts: https://github.com/beaver-22/Cliff-token