# Recoverability Has a Law: The ERR Measure for Tool-Augmented Agents

arXiv: https://arxiv.org/abs/2601.22352 · HTML: https://mlsanity.com/paper/2601.22352/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Tool learning and function calling, one of 16 directions in LLM agents and tool use.
- Builds on 7 of its 15 references that are in the map.
- First posted on arXiv 29 Jan 2026.

## Placed in

- LLM agents and tool use / [Tool learning and function calling](https://mlsanity.com/agents/tool-learning-and-function-calling/)

## Builds on (most cited first, 7 of 7)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Reflexion: Language Agents with Verbal Reinforcement Learning](https://mlsanity.com/paper/2303.11366/) (2023)
- [Theoretically Principled Trade-off between Robustness and Accuracy](https://mlsanity.com/paper/1901.08573/) (2019)
- [Improving language models by retrieving from trillions of tokens](https://mlsanity.com/paper/2112.04426/) (2021)
- [CRITIC: Large Language Models Can Self-Correct with Tool-Interactive Critiquing](https://mlsanity.com/paper/2305.11738/) (2023)
- [API-Bank: A Comprehensive Benchmark for Tool-Augmented LLMs](https://mlsanity.com/paper/2304.08244/) (2023)

## Built on by (most cited first, 1 of 1)

- [Verified Tool Calls Improve LLM Agent Reliability Under Non-Atomic Failures](https://mlsanity.com/paper/2608.02645/) (2026)

## Abstract (arXiv)

Language model agents often appear capable of self-recovery after failing tool call executions, yet this behavior lacks a formal explanation. We present a predictive theory that resolves this gap by showing that recoverability follows a measurable law. To elaborate, we formalize recoverability through Expected Recovery Regret (ERR), which quantifies the deviation of a recovery policy from the optimal one under stochastic execution noise, and derive a first-order relationship between ERR and an empirical observable quantity, the Efficiency Score (ES). This yields a falsifiable first-order quantitative law of recovery dynamics in tool-using agents. We empirically validate the law across five tool-use benchmarks spanning controlled perturbations, diagnostic reasoning, and real-world APIs. Across model scales, perturbation regimes, and recovery horizons, predicted regret under the ERR-ES law closely matched observed post-failure regret measured from Monte Carlo rollouts, within delta less than or equal to 0.05. Our results reveal that recoverability is not an artifact of model scale or architecture, but a governed property of interaction dynamics, providing a theoretical foundation for execution-level robustness in language agents.