The Dual Bottleneck:A Composition Law for Judge_Reliability and Generator Capability in Self-Evolving_Agents

Self-evolving agents—systems that iteratively generate tasks, solve them, and distill experience—have shown promise in code generation and mathematical reasoning. Existing work predominantly focuses on the judge (the evaluator providing feedback), assuming that reliable judges suffice. Through 2,400 controlled experiments on Z3 constraint-solving tasks, we uncover a second, equally critical bottleneck: the generator (the LLM authoring new tasks). We formalize the Dual Bottleneck as a composition law on error budgets: 1−∆ global t = (1−∆ gen t )(1−∆valid t ), verified to machine precision (1.11e-16) on 2,400 real-API rounds. The law implies that the generator and judge bottlenecks share one budget: a condition with large ∆ gen t needs a more reliable judge for the same overall tolerance. If ∆ gen t alone exceeds the budget, no judge reliability can compensate. We empirically measure τc ≈ 0.55 on Level 1 tasks using the real DeepSeek V4 Flash API, and derive the parameter-free formula τc = √︁ 1− 2∆max t . We identify a pseudo difficulty trap: when the difficulty controller upgrades to Level 2/3, the LLM’stask-generation validity collapses from ∼89% to <10%, causing the system to spin without progress. We propose a four-layer constraint generation framework that restores validity to 100% at L1, 73.8% at L2, and 83.8% at L3 on real API.