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This work presents a novel cybersecurity framework designed to address emerging threats posed by quantum computing in decentralized Edge Artificial Intelligence (Edge-AI) environments. As the development of cryptographically relevant quantum computers (CRQCs) accelerates, widely deployed public-key cryptographic systems such as RSA and Elliptic Curve Cryptography (ECC) face potential compromise due to quantum algorithms capable of solving integer factorization and discrete logarithm problems. These vulnerabilities pose significant risks to distributed Edge-AI infrastructures including Internet of Things (IoT) networks, autonomous drone swarms, and industrial cyber-physical systems. To address these challenges, this paper proposes a Neuromorphic Cyber-Twin (NCT) framework that integrates bio-inspired swarm intelligence, neuromorphic computing, and post-quantum cryptography to provide autonomous and energy-efficient security for distributed Edge-AI ecosystems. The proposed architecture combines Spiking Neural Networks (SNNs) for event-driven anomaly detection with Particle Swarm Optimization (PSO) for decentralized coordination among edge nodes. When malicious activity is detected, the framework autonomously activates post-quantum cryptographic protection using the CRYSTALS-Kyber key encapsulation mechanism, enabling secure communication resilient to quantum-enabled adversaries. The system was evaluated using two widely used intrusion detection datasets, NSL-KDD and CIC-IDS, to assess the effectiveness of the neuromorphic anomaly detection mechanism. Experimental results demonstrate a detection accuracy of 98.8% with an AUC-ROC score of 0.9991, indicating strong performance in identifying cyber threats such as Distributed Denial-of-Service (DDoS) and brute-force attacks while maintaining minimal false positives. In addition, the event-driven neuromorphic architecture significantly reduces computational overhead and energy consumption compared to conventional deep learning-based intrusion detection systems. The proposed Neuromorphic Cyber-Twin framework offers a scalable and autonomous defense architecture capable of protecting distributed Edge-AI systems against both classical cyber attacks and future quantum-enabled threats. This work contributes toward the development of quantum-resilient cybersecurity mechanisms for next-generation intelligent networks.