Large Language Models (LLMs) are increasingly deployed as long-term interactive agents, yet their limited context windows make it difficult to sustain coherent behavior over extended interactions. Existing memory systems for LLMs often store isolated records and retrieve fragments, limiting their ability to consolidate evolving experience and resolve conflicts. We introduce EverMemOS, a self-organizing memory operating system that implements an engram-inspired lifecycle for computational memory. First, Episodic Trace Formation converts dialogue streams into MemCells that capture episodic traces, atomic facts, and time-bounded foresight. Second, Semantic Consolidation organizes MemCells into thematic MemScenes, distilling stable semantic structures and updating user profiles. Finally, Reconstructive Recollection performs MemScene-guided agentic retrieval to compose the necessary and sufficient context for downstream reasoning. Experiments on LoCoMo, LongMemEval, and PersonaMem-v2 show that EverMemOS significantly outperforms state-of-the-art methods on memory-augmented reasoning tasks.