When Does Memory Help Multi-Trajectory Inference for Tool-Use LLM Agents?

Multi-trajectory inference for tool-use LLM agents - generating multiple reasoning attempts and selecting among them - benefits from transferring knowledge across attempts so that later ones avoid the pitfalls of earlier ones. Existing cross-trajectory memory methods (trajectory-level reflection, atomic fact extraction, raw observation injection) are each evaluated under a single inference strategy on a single task, making it unclear whether reported gains reflect properties of the memory abstraction or of the inference method. We propose a unified framework that decomposes memory along two axes -- the scope of transfer (within an expansion vs. across trajectories) and the abstraction of the transferred content -- and evaluate four methods under three inference strategies (best-of-N, beam search, MCTS) on four tool-use benchmarks spanning SQL, knowledge-graph, and CLI environments, in a verifier-free setting that matches the deployment regime of practical agents. The experiment matrix identifies the inference method as a confound: the same memory method produces statistically distinct results under different inference strategies on the same examples. Reflection reaches significance only under MCTS (not under best-of-N); within-expansion injection (conditioning each candidate on prior siblings' outcomes) helps only diversity-starved beam search; and atomic fact extraction is accuracy-neutral but shortens trajectories by 19-26% on tasks with reusable environmental structure.