Spectral Sheaf Heuristics for Consistency Detection in Multi-Agent Systems

detecting global inconsistencies in distributed networks remains computationally expensive. We introduce the Phronesis Index (Φ), a computationally efficient spectral heuristic that quantifies consistency by approximating topological obstructions in cellular sheaves. Our method achieves O(N log N) complexity (versus O(N^3) for exact cohomology) with provable error bounds. We validate Φ across four scenarios: Logic Maze anomaly detection, safe reinforcement learning via Bellman consistency monitoring, multi-robot coordination, and scalability tests up to 50,000 agents. In the safe RL scenario, Φ-based reward shaping reduces cumulative safety violations compared to standard Q-learning (Welch t-test computed automatically by the experiment script). We provide comprehensive guidance for sheaf construction, robustness under noise, and practical deployment considerations. All code and data are publicly available at https://github.com/sepehrbayat/phronesis-index-nmi .