park-2025-theorem-4-magnitude-binary-projection

IN premise — summaries/2026/08/24/park-2024-categorical-hierarchical-concepts-s2-using-this-result-we-show-that-semantic-hierarchy-between-co.md

Created 2026-08-25T02:58:25+00:00

Theorem 4 states that for a binary feature w with linear representation ℓ̄_W, ℓ̄_Wᵀ g(y) = b_w (a positive constant) for all y ∈ Y(w) and ℓ̄_Wᵀ g(y) = 0 for y ∉ Y(w).

Summary

This theorem says that a binary (on/off) feature can be read out of any representation using a single linear probe: the probe gives a fixed positive value when the feature is present and exactly zero when it is not, with no in-between cases. For the system, this means detection of that feature is perfectly reliable and unambiguous, so downstream reasoning can treat the feature as a clean yes/no signal rather than a graded or noisy one.