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Large language model (LLM) agents are usually improved by changing prompts, models, or hand-written workflows, while the execution harness around the model is treated as fixed infrastructure. We argue that this harness is itself a learnable control layer. We formalize harness operation as a finite-horizon Harness MDP, where a lightweight controller selects structural execution actions while the LLM executor remains frozen. The controller is trained from offline rollouts using advantage-weighted regression with only terminal task-rubric rewards. We also separate final task quality from a post-hoc Harness Maturity Score, which measures whether the harness follows reliable execution patterns rather than only whether the final answer is correct. This separation gives a finite-buffer view of harness learning: final-quality gains require high-return support in the offline buffer, while process behavior can shift whenever it aligns with advantage-weighted actions. Across six controlled domains and two public-benchmark adapters, the learned controller consistently improves verification behavior and selectively improves final task quality, with the largest gains on adapted tau-bench retail, adapted AgentBench DB-Bench, and coding with a calibrated structural verifier. Ablations against behavior cloning and Forced CHECK show that the gains are not explained by imitation or by simply adding checks. These results identify harness control as a learnable layer for frozen LLM agents, while showing that offline support limits when better process control becomes better final answers.