Abstract

In this paper, we extend the mathematical framework of **non-commutative scalar fields** and numerical techniques discussed previously to build a foundation for **AI-based reasoning systems**. The goal is to enable AI to operate over **symbolic hierarchies, semantic transformations**, and **large-scale infinite or non-commutative domains**. Inspired by quantum tensor field operations, we integrate reasoning over symbolic, numeric, and approximate representations into machine learning pipelines. This work leverages concepts from numerical techniques for non-commutative mixed derivatives, recur- sive tensor calculus, and symbolic transformation logic to build **deep learning architectures** capable of representing and reasoning with structured, hierarchical complexity.