Researchers introduce the concept of cliff tokens to identify specific single-token failure triggers in large language models during mathematical reasoning tasks. Unlike prior work that analyzes failures at step or sentence levels, this method pinpoints the exact token where potential drops significantly using an adaptive threshold based on a z-test. The study evaluates seven models across three benchmarks: GSM1K, MATH500, and AIME 2025. Deleting the first cliff token and resampling allows recovery of pass@64 to 1.0, whereas keeping it limits recovery between 0.71 and 1.00. The authors propose a taxonomy classifying cliffs as deterministic, uncertain, or sampled-off based on greedy choice and token entropy. This classification generalizes across different model scales and exhibits distinct probabilistic characteristics for each type. Furthermore, the team validates this taxonomy through single-token preference optimization known as Cliff-DPO. Trained on GSM8K, Cliff-DPO improves accuracy by up to +6.6 across benchmarks. Optimization proves effective for uncertain and sampled-off cliffs but yields no improvement for deterministic ones.