Large language model (LLM) agents rely on memory systems to leverage long-term interactions, yet existing systems typically retrieve memories based on their relevance to the current query and use them as contextual evidence. However, relevance alone does not characterize how a memory contributes to task solving: the same memory may serve different roles across tasks, such as directly supporting the answer or linking to complementary memories with weaker semantic relevance but necessary information. We therefore reframe agent memory systems as Task-Oriented Role-Adaptive Workspaces, where memories are selected and organized by their task-oriented roles and relations. Based on this view, we propose TrawMem, which organizes interaction histories into local threads and constructs task-oriented workspaces by assigning roles and jointly selecting complementary memories. Experiments on LoCoMo and MemGallery with four open- and closed-source backbones show consistent improvements over representative memory systems, with macro-F1 gains of 2.23-3.14 points. On LoCoMo, TrawMem improves the LLM-Judge score over SimpleMem by 4.07 and 4.31 percentage points on Qwen3.5-9B and DeepSeek-V4-Flash, respectively, while requiring approximately 12.8% and 5.1% fewer context tokens, respectively.

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