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Even GPT-5.2 Can't Count to Five: The Case for Zero-Error Horizons in Trustworthy LLMs

Ryoma Sato

TL;DR

This work introduces Zero-Error Horizon (ZEH) as a size-based boundary metric for evaluating trustworthy LLMs in safety-critical contexts. By fixing a model, task, and prompt, ZEH identifies the largest problem size that can be solved without error and highlights ZEH limiters at size $ZEH+1$ to expose failure modes, even in high-performing models like GPT-5.2. The paper demonstrates how ZEH correlates with but is distinct from accuracy, reveals the emergence of algorithmic reasoning, and provides practical speedups (e.g., teacher forcing, batching, and FlashTree) to mitigate computational costs. Through empirical analyses on GPT-5.2 and Qwen2.5 variants, the authors argue that ZEH offers a rigorous, less arbitrary, and safety-relevant perspective for evaluating LLM reliability and guiding trustworthy deployment. They also discuss limitations, including sensitivity to prompts and the potential need for formal methods to scale ZEH measurements in real-world settings.

Abstract

We propose Zero-Error Horizon (ZEH) for trustworthy LLMs, which represents the maximum range that a model can solve without any errors. While ZEH itself is simple, we demonstrate that evaluating the ZEH of state-of-the-art LLMs yields abundant insights. For example, by evaluating the ZEH of GPT-5.2, we found that GPT-5.2 cannot even compute the parity of a short string like 11000, and GPT-5.2 cannot determine whether the parentheses in ((((()))))) are balanced. This is surprising given the excellent capabilities of GPT-5.2. The fact that LLMs make mistakes on such simple problems serves as an important lesson when applying LLMs to safety-critical domains. By applying ZEH to Qwen2.5 and conducting detailed analysis, we found that while ZEH correlates with accuracy, the detailed behaviors differ, and ZEH provides clues about the emergence of algorithmic capabilities. Finally, while computing ZEH incurs significant computational cost, we discuss how to mitigate this cost by achieving up to one order of magnitude speedup using tree structures and online softmax.

Even GPT-5.2 Can't Count to Five: The Case for Zero-Error Horizons in Trustworthy LLMs

TL;DR

This work introduces Zero-Error Horizon (ZEH) as a size-based boundary metric for evaluating trustworthy LLMs in safety-critical contexts. By fixing a model, task, and prompt, ZEH identifies the largest problem size that can be solved without error and highlights ZEH limiters at size to expose failure modes, even in high-performing models like GPT-5.2. The paper demonstrates how ZEH correlates with but is distinct from accuracy, reveals the emergence of algorithmic reasoning, and provides practical speedups (e.g., teacher forcing, batching, and FlashTree) to mitigate computational costs. Through empirical analyses on GPT-5.2 and Qwen2.5 variants, the authors argue that ZEH offers a rigorous, less arbitrary, and safety-relevant perspective for evaluating LLM reliability and guiding trustworthy deployment. They also discuss limitations, including sensitivity to prompts and the potential need for formal methods to scale ZEH measurements in real-world settings.

Abstract

We propose Zero-Error Horizon (ZEH) for trustworthy LLMs, which represents the maximum range that a model can solve without any errors. While ZEH itself is simple, we demonstrate that evaluating the ZEH of state-of-the-art LLMs yields abundant insights. For example, by evaluating the ZEH of GPT-5.2, we found that GPT-5.2 cannot even compute the parity of a short string like 11000, and GPT-5.2 cannot determine whether the parentheses in ((((()))))) are balanced. This is surprising given the excellent capabilities of GPT-5.2. The fact that LLMs make mistakes on such simple problems serves as an important lesson when applying LLMs to safety-critical domains. By applying ZEH to Qwen2.5 and conducting detailed analysis, we found that while ZEH correlates with accuracy, the detailed behaviors differ, and ZEH provides clues about the emergence of algorithmic capabilities. Finally, while computing ZEH incurs significant computational cost, we discuss how to mitigate this cost by achieving up to one order of magnitude speedup using tree structures and online softmax.
Paper Structure (24 sections, 3 equations, 3 figures, 8 tables)

This paper contains 24 sections, 3 equations, 3 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Zero-Error Horizon
  3. Definition
  4. Qualitative Characteristics of ZEH
  5. Safety Guarantees and Warning Signals
  6. Evidence via ZEH Limiters
  7. Analyzing ZEH Limiters is Beneficial
  8. Automatically Obtaining Surprising Results
  9. Accuracy Has Scale Arbitrariness, but ZEH Does Not
  10. ZEH Does Not Become Obsolete as a Metric
  11. Instability
  12. Notes on Using External Tools
  13. Analysis
  14. ZEH of GPT-5.2
  15. Detailed Analysis of Qwen2.5
  16. ...and 9 more sections

Figures (3)

  • Figure 1: GPT-5.2 cannot compute the parity of the short string 11000, cannot determine whether the parentheses in ((((()))))) are balanced, and makes a calculation error on $127 \times 82$. These commands can be copied and pasted, and if you set your API key $OPENAI_API_KEY, you can try them right now. Please give it a try.
  • Figure 2: GPT-5.2-Thinking also fails to determine whether the parentheses in ((((()))))) are balanced.
  • Figure 3: Comparison of ZEH and accuracy on the multiplication task for Qwen2.5. Red dots indicate errors. From left to right, accuracy is shown for ranges $n \leq 20$, $n \leq 50$, and $n \leq 99$. The rightmost panel shows ZEH. Accuracy varies greatly depending on the choice of range, but ZEH does not require range selection, making it an objective metric.