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Large Language Model (LLM) agents are now widely deployed in Ambient Intelligence (AmI) environments, where autonomous agents must sense, act, and coordinate at scale. As agent capabilities and interdependence increase, traditional reliability strategies such as isolated adaptive control, anomaly detection, or trust modeling have proven inadequate due to their fragmented and scenario-specific nature. Comprehensive architectures that enable integrated self-management, collective anomaly response, robust information dissemination, and privacy-preserving adaptation remain scarce. We propose a bio-autonomic framework for decentralized resilience in multi-agent LLM systems where a unified architecture systematically applies principles from biological autonomic systems to LLM-based multi-agent environments. Specifically, each agent implements an autonomic control loop, formally structured as Monitor-Analyze-Plan-Execute over a shared Knowledge base (MAPE-K), for self-regulation. At the system level, the framework integrates immune-inspired anomaly detection using the Dendritic Cell Algorithm, probabilistic computational trust, decentralized gossip for robust information sharing, and federated learning with homomorphic encryption for collaborative, privacy-preserving adaptation. This holistic approach enables LLM agent ecosystems to self-organize, detect and isolate faults, and collectively adapt as system complexity increases. Empirical evaluations show that our framework achieves substantially improved resilience and recovery compared to state-of-the-art multi-agent baselines.