park-2025-binary-contrast-vector-difference

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

Corollary 6 establishes that binary contrasts are vector differences of binary features: ℓ̄_{w0⇒w1} = ℓ̄_{w1} − ℓ̄_{w0}, enabled by representations having well-defined magnitudes (not just directions).

Summary

Changing a binary feature from off to on can be measured as a simple subtraction of two state-vectors, so the system doesn't need a special operation to quantify feature transitions. This shortcut only works if the vectors carry meaningful length information, not just direction, which constrains how representations must be built.

Dependents

These beliefs depend on this one:

IN directionality-as-universal-geometric-property — Asymmetry (A→B ≠ B→A) appears independently in ROME's separate-edit constraint, Park's signed binary contrasts (ℓ̄_{w0⇒w1} = ℓ̄_{w1} − ℓ̄_{w0}), and SBERT's tied-weight siamese architecture, establishing directionality as an invariant geometric property of the superposed space rather than a per-implementation artifact.