When Source Correction Is Not Enough: A Theory of Mistake Influence in Agents

Correcting an agent’s past mistake may leave its influence on later decisions intact. We distinguish a mistake’s presence in context, its repetition probability, and its causal influence, then separate this influence into the original output’s direct effect and a trail effect carried by subsequent history. A three-state construction shows that removing direct influence gives no uniform bound on recovery time, even when influence eventually vanishes. We derive sufficient conditions for fading and finite-horizon persistence, and identify an explicit fading-to-persistence threshold in a specified nonlinear feedback model. Exact enumeration of four-step histories under recorded binary-action policies spans 300 arithmetic problems and three models: Qwen3-Coder, Llama 4 Maverick, and DeepSeek V4 Flash. At decision four, trail influence exceeds direct influence in magnitude in 90.3–98.4% of settings, although its magnitude and direction vary substantially. On six development problems, supplementing source correction with recomputation and a visible repair record reduces mean mistake reuse by 32.7–53.4 percentage points across two menu orders. A separate coding-agent study finds that reminders can reduce subsequent violations while leaving files created by earlier violations intact, with responses differing across models.