Association Rules Machine Learning complete intersection Calabi-Yau 5-Folds and 6-Folds

TL;DR

This work tackles the challenge of partially known topological invariants for complete intersection Calabi–Yau manifolds in dimensions five and six. It applies Apriori-based association-rule mining to encode $h^{i,j}$ values as transactional items and identify high-confidence rules relating different Hodge numbers. The study reports 60 rules for CICY5 and 160 rules for CICY6 across large datasets, including a striking universal pattern that $h^{1,2} = h^{1,3} = h^{1,4} = h^{2,3} = 0$ for all CICY6 entries. These findings provide predictive, interpretable insights to guide future cohomology computations and theoretical developments in higher-dimensional Calabi–Yau geometry.

Abstract

Association rule machine learning is applied to the dataset of complete intersection Calabi--Yau 5-folds and 6-folds in order to uncover hidden patterns among their Hodge numbers. These Hodge numbers -- six for the 5-folds and nine for the 6-folds -- serve as the items in our analysis. For the 5-folds, we discover 60 significant association rules. For example, within the dataset, if $h_{1,3} = 0$ and $h_{2,2} = 5$, then $h_{1,1} = 3$ with 99.43\% confidence. Similarly, if $h_{2,1} = 0$, $h_{1,3} = 0$, and $h_{2,2} = 5$, then $h_{1,1} = 3$ with 99.42\% confidence. For the 6-folds, we identify 160 association rules across a dataset of 1,482,022 examples. A particularly striking observation is that $h_{1,2} = h_{1,3} = h_{1,4} = h_{2,3} = 0$ for all entries in this dataset. These types of association rules are especially valuable because the Hodge numbers of complete intersection Calabi--Yau 5-folds have only been computed for approximately 53 percent of the dataset, while those of 6-folds remain largely undetermined. The discovered patterns provide predictive insights that can guide future computations and theoretical developments.