Corollary: Möbius-weight stability

For nonempty $J$,

A nonzero weight larger than this bound remains nonzero. An ideal zero weight becomes at most this bound.

Proof. The change in $Z_J$ is a signed sum of $2^{|J|}$ pinched copies of $Y$. Each has trace norm at most $\Vert Y{\Vert }_1$. Apply the triangle inequality, $\Lambda$, and the prefactor $1/2$.