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Recent advances in large language models have driven the emergence of intelligent agents operating in open-world, multimodal environments. To support long-term reasoning, such agents are typically equipped with external memory systems. However, most existing multimodal agent memories rely primarily on neural representations and vector-based retrieval, which are well-suited for inductive, intuitive reasoning but fundamentally limited in supporting analytical, deductive reasoning critical for real-world decision making. To address this limitation, we propose NS-Mem, a long-term neuro-symbolic memory framework designed to advance multimodal agent reasoning by integrating neural memory with explicit symbolic structures and rules. Specifically, NS-Mem is operated around three core components of a memory system: (1) a three-layer memory architecture that consists episodic layer, semantic layer and logic rule layer, (2) a memory construction and maintenance mechanism implemented by SK-Gen that automatically consolidates structured knowledge from accumulated multimodal experiences and incrementally updates both neural representations and symbolic rules, and (3) a hybrid memory retrieval mechanism that combines similarity-based search with deterministic symbolic query functions to support structured reasoning. Experiments on real-world multimodal reasoning benchmarks demonstrate that Neural-Symbolic Memory achieves an average 4.35% improvement in overall reasoning accuracy over pure neural memory systems, with gains of up to 12.5% on constrained reasoning queries, validating the effectiveness of NS-Mem.