Memory systems are essential for personalized Large Language Models (LLMs). However, existing retrieval methods in these systems primarily rely on semantic similarity, potentially missing logically critical memories with limited semantic overlap. Current benchmarks remain inadequate for evaluating this problem. To address this gap, we construct IMLogic, the first high-quality benchmark targeting implicit logical memory retrieval in long-dialogue scenarios. Motivated by this challenge, we introduce root memory, a structured, decision-preserving representation that distills reusable personalized logic from long-term user histories. We then propose RootMem, a plug-and-play framework that first distills raw histories into structured root memories and then uses an LLM-based router to activate logically relevant ones, complementing semantic retrieval with personalized decision logic. Extensive experiments demonstrate that RootMem significantly outperforms the strongest retrieval baselines and consistently boosts the accuracy of existing memory agents. Our benchmark and codes will be available at https://anonymous.4open.science/r/IMLogic-DBB3.

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