This work presents experimental and machine learning applications of the relational torsion invariant framework introduced in the Z-Series research program. The paper investigates the statistical properties, geometric interpretations, and computational applications of relational torsion invariants derived from squared coordinate differences among triples of dataset elements. Large-scale Monte Carlo simulations are conducted to analyze invariant distributions and validate theoretical predictions. The study further explores geometric interpretations of the invariant as a quadratic dispersion energy over triangular structures. The framework is then applied to geometric deep learning models. In particular, relational invariant regularization is integrated into graph neural network training procedures to encourage preservation of local triangular structure in node embeddings. Experimental evaluation across multiple benchmark datasets including Cora, Citeseer, PubMed, and OGB-ArXiv demonstrates improved embedding stability and classification accuracy when invariant regularization is applied. These results suggest that relational torsion invariants provide a promising mathematical mechanism for incorporating higher-order geometric structure into machine learning models. This paper forms the experimental and application component of the Z-Series research program on algebraic frameworks for relational dataset geometry.
abhishek Chaudhary (Tue,) studied this question.