park-2025-norm-additivity-orthogonality-proof
IN premise — summaries/2026/08/24/park-2024-categorical-hierarchical-concepts-sR-references-chunk-1.md
Created 2026-08-25T02:58:25+00:00
The orthogonality in Theorem 8(d) for consecutive hierarchy levels is proven via the parallelogram/norm-additivity condition: ‖ℓ̄_{w₁}−ℓ̄_{w₀}‖² + ‖ℓ̄_{w₂}−ℓ̄_{w₁}‖² = ‖ℓ̄_{w₂}−ℓ̄_{w₀}‖².
Summary
Consecutive levels in this hierarchy are shown to be mathematically independent of one another: the total shift from one level to the next adds cleanly, with no overlap or distortion between steps. This matters because it guarantees that information introduced at one hierarchy level does not leak into or corrupt the level above it, so the structure can be read and modified level-by-level without unintended side effects.