park-2024-proposition-9-simplices
IN premise — summaries/2026/08/24/park-2024-categorical-hierarchical-concepts-sR-references.md
Created 2026-08-25T02:58:26+00:00
Proposition 9 states that if k mutually exclusive attributes can independently have arbitrary joint distributions, their vector representations form a (k−1)-simplex.
Summary
When k mutually exclusive categories are allowed to combine freely, their vector encodings are locked into a fixed geometric shape with exactly k vertices and k-1 dimensions, like a line for two options, a triangle for three, a tetrahedron for four. This matters because it means the system cannot freely reshape or extend that representational space; the exclusion constraint dictates exactly what geometry is available for any reasoning over those categories.