park-2024-causal-inner-product-whitening
IN premise — summaries/2026/08/24/park-2024-categorical-hierarchical-concepts-sR-references.md
Created 2026-08-25T02:58:26+00:00
The hierarchical orthogonality in Park et al. 2024 is only visible under a 'causal inner product' that includes both centering and whitening via an invertible matrix A; under the raw Euclidean inner product (centering only), the orthogonality breaks down.
Summary
The clean hierarchical separation that Park et al. report only appears when you rescale the data in a very specific way (centering plus whitening through a particular matrix), not from the raw geometry of the data itself. In practice, this means the structure is a lens-dependent finding: any system or argument that treats those layers as genuinely independent needs to carry the specific whitening assumption along with it, because under a plain centered Euclidean view the independence falls apart.