park-2025-natural-concepts-simplices
IN premise — summaries/2026/08/24/park-2024-categorical-hierarchical-concepts-sR-references-chunk-1.md
Created 2026-08-25T02:58:25+00:00
Proposition 9 states that 'natural' categorical concepts (where the model can freely manipulate any output marginal) are represented as (kâ1)-simplices rather than general polytopes.
Summary
When a model can freely adjust any individual category's probability without constraint, the geometry of that concept space is forced into the simplest possible shape â a simplex (think triangle or tetrahedron) â rather than some irregular, more complex polygon. This means "natural" categories carry a tight structural constraint on how they can be represented internally, which narrows the room for interpretation and tells us the model's geometry is minimal, not arbitrary.