The multi-domain, context-sensitive architecture of human cognition, when faithfully formally represented in hybrid intelligence, fosters systems that are interpretable, ethically aligned, and capable of meaningful collaboration. In this paper, we propose \(M_{\mu \nu }\) , a tensor-based ontological framework that formally captures this structure by representing cognitive states as a second-order tensor \(M_{\mu \nu } \in [0,1]^{n \times m}\) , where dimensions explicitly index cognitive domains \(\mathcal {C}\) and contextual factors \(\mathcal {E}\) . The dynamics are derived from first principles via a cognitive Lagrangian \(\mathcal {L}(M_{\mu \nu }, \dot{M}_{\mu \nu })\) , yielding stable update equations that formally model biologically plausible adaptation patterns. The framework formally captures scalability through CANDECOMP/PARAFAC decomposition and provides explicit interpretability via tensor slicing, geometric curvature analysis, and sensitivity metrics. Comprehensive simulations demonstrate that \(M_{\mu \nu }\) achieves stable dynamics with spectral radius \(\rho (\textbf{A}) = 0.9952\) , computational speedups of up to 6.6 \(\times \) via CP decomposition, and emergent cognitive functionality including 66.7% decision-making accuracy and 60.2% pattern completion performance. Comparative evaluation shows the framework achieves 40.0% overall accuracy while maintaining near-instantaneous inference (0.06 ms). The proposed framework establishes a mathematically rigorous foundation for explainable hybrid intelligence in applications ranging from medical diagnostics and human-robot collaboration to cognitive simulation and training systems.
