Summary: We introduce the Zero-Error Horizon (ZEH) concept for dependable language models, defining the longest sequence a model can process flawlessly. Although ZEH is straightforward, assessing it in top-tier LLMs reveals valuable findings. For instance, testing GPT-5.2's ZEH shows it struggles with basic tasks like determining the parity of the sequence 11000 or checking if the parentheses in ((((()))))) are properly matched. These shortcomings are unexpected given GPT-5.2's advanced performance. Such errors on elementary problems highlight critical considerations for deploying LLMs in high-stakes environments. Applying ZEH to Qwen2.5 and performing in-depth examination, we observe that ZEH relates to precision but exhibits distinct patterns, offering insights into the development of algorithmic skills. Additionally, while ZEH calculation demands substantial resources, we explore methods to reduce this burden, achieving nearly tenfold acceleration through tree-based structures and online softmax techniques.
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