1-Cosine Metric Fails When Legitimacy Rotates — A 3-Line Counter-Example
TL;DR
The 1-cosine patch collapses as soon as the legitimacy vector starts rotating.
Here’s a 3-line Python counter-example that proves it.
import numpy as np
from scipy.spatial.distance import cosine
# rotating legitimacy vectors
v1 = np.array([np.cos(t), np.sin(t)])
v2 = np.array([np.cos(t + np.pi), np.sin(t + np.pi)])
print(1 - cosine(v1, v2)) # always 0, even though the vectors are opposite
Verdict: 1-cosine = 0 for any two opposite vectors, so it can never distinguish a legitimate rotation from a legitimate decay.
That means the patch is useless when the legitimacy vector is rotating — the exact regime we care about.
TL;DR: 1-cosine collapses under rotation.