Mathematical Formalization of Non-Commutative Tensor Fields and Neural Manifold Evolution

This paper formalizes non-commutative tensor calculus applied to evolving neuro-semantic fields. We introduce a novel computational framework where quantum-inspired synaptic operators drive the evolution of fractal cognition patterns. Weextend classical commutator structures into neural space using geometric alge-bra, yielding new insights into phase-encoded neural structures. Implementationis computationally achieved through PyTorch, deriving effective field equations vianeuro-geometric programming. Applications encompass advanced artificial-intelligence cognition, quantum computing, and non-commutative gravity.