park-2025-hierarchical-orthogonality-theorem-8
IN premise — summaries/2026/08/24/park-2024-categorical-hierarchical-concepts-s2-using-this-result-we-show-that-semantic-hierarchy-between-co-chunk-1.md
Created 2026-08-25T02:58:24+00:00
Theorem 8 states that for subordinate concepts z ≺ w, the parent vector is orthogonal to the child-minus-parent residual: ℓ̄_w ⊥ (ℓ̄_z − ℓ̄_w), and sibling contrast vectors are mutually orthogonal.
Summary
In a concept hierarchy, each level of specificity lives in its own independent direction, completely separate from the parent level and from sibling distinctions, so you can add, remove, or query detail without disturbing the broader category or confusing one sibling with another. This gives the system a clean, non-overlapping structure where hierarchical and lateral differences never bleed into each other, making reasoning and comparison at any level reliable.
Dependents
These beliefs depend on this one:
- OUT hierarchical-orthogonality-genuine-structure — The observed parent⊥(child−parent) orthogonality in LLM representation spaces reflects genuine hierarchical semantics rather than a trivial set-inclusion artifact.
- IN shuffling-contradiction-resolution — The apparent contradiction between the 2025 empirical control (shuffling unembeddings *destroys* the full orthogonality structure) and the set-inclusion finding (shuffled unembeddings *reproduce* the child-parent⊥parent sub-orthogonality) is resolved by scope: the *full* four-condition Theorem 8 structure is genuine and shuffle-destroyed, while a *specific* sub-condition (consecutive-level parent⊥child-parent) is a set-inclusion artifact—meaning the geometric framework captures the genuine full structure, not the spurious sub-structure.