Extending FEM Diversity to Higher Dimensions Without Augmentation

Verifying that finite element diversity holds in D=1,2,3 for both augmented and vanilla (non-augmented) sample sets.

Diversity Results Across Dimensions

DM (grid)AugmentedDiv. ProbabilityMean dim(C)Status
18Yes1.0001.0DIVERSE
18No1.0001.0DIVERSE
24Yes1.0001.0DIVERSE
24No1.0001.0DIVERSE
33Yes1.0001.0DIVERSE
33No1.0001.0DIVERSE
All configurations achieve diversity probability 1.0 with centralizer dimension exactly 1 (trivial). Vanilla (non-augmented) sample sets perform identically to augmented ones across all dimensions.

Diversity by Dimension

Augmented vs Vanilla

Key Insight: Mesh Connectivity

Higher dimensions have richer vertex connectivity in simplicial meshes: D=1 has 2 neighbors per node, D=2 has 5-7, D=3 has 12+. This increased algebraic structure makes the centralizer trivial even without augmentation.
FEM Diversity in Higher Dimensions — Computational Study