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

TL;DR

This work formalizes recoverability for tool-augmented agents through Expected Recovery Regret (ERR) and links it to an observable Efficiency Score (ES), yielding a first-order law ERR ≈ (1/(1−γ))(1−ES) with small higher-order corrections. The ERR–ES coupling is validated across five benchmarks, multiple model scales, and diverse perturbations, revealing a low-dimensional efficiency–regret manifold that governs recovery dynamics independent of architecture. The framework introduces observable surrogates (RR, CSR, ES), variance-reduction techniques via retrieval conditioning, and a minimal recovery mechanism (FORTIFY) to enable falsifiability and diagnostic use. The findings imply that robustness in tool-using agents is a governed property of interaction dynamics, enabling predictive planning, safe execution, and curriculum design based on recoverability signals rather than relying solely on model scale or design. Together, these contributions provide a principled, testable theory of execution-level robustness with practical implications for planning, safety, and evaluation of multi-step AI systems.

Abstract

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.

Figures (7)

• Figure 1: Recoverability follows a measurable law under execution noise. Observed recovery regret remains predictable in a low-variance regime and breaks sharply as execution instability increases. This phase transition motivates the ERR formulation and its efficiency-based predictor.
• Figure 2: Efficiency--regret frontier. Recovery trajectories collapse onto a one-dimensional structure predicted by the recovery law.
• Figure 3: RR–CSR tradeoff showing that recovery mechanisms move systems along a shared efficiency–regret frontier rather than inducing new regimes.
• Figure 4: Scaling behavior of recovery efficiency.
• Figure 5: CSR degradation over steps illustrating early-step dominance predicted by the recovery law

Theorems & Definitions (13)

• Theorem 3.1: Uniqueness of the Efficiency Surrogate
• Corollary 3.2
• Corollary 3.3: Linear Regime
• Corollary 3.4: Early-Step Stability
• Lemma 1.1: Linearized Loss Gap
• proof : Sketch
• Lemma 1.2: Variance-Induced Bias Term
• Lemma 1.3: Boundedness and Vanishing of $\beta$
• Corollary 1.4: Model-Scale Alignment
• proof : Sketch