sign_plus_normal_projection_operator_norm_le_one

Sign matrix plus a normal-space contraction is a contraction (Candès–Recht 2009, arXiv:0805.4471, Lemma 3.2 achiever, p.15). Let $E=\operatorname{sign}(M)$ be the tangent sign matrix of a rank-$r$ SVD $S$ of $M$, so $E\in T$ has column space inside the left singular span $U$ and row space inside the right singular span $V$. For any matrix $Z$ with operator norm $\lVert Z\rVert\le 1$, the normal projection $W=P_{T^\perp}Z$ lies in $T^\perp$ (its column space is orthogonal to $U$ and its row space orthogonal to $V$) and satisfies $\lVert W\rVert\le\lVert Z\rVert\le 1$ ($P_{T^\perp}$ is an operator-norm contraction). Because $E$ and $W$ have orthogonal column spaces and orthogonal row spaces, $E+W$ acts blockwise and $\lVert E+W\rVert=\max(\lVert E\rVert,\lVert W\rVert)\le 1$. This is exactly the construction guaranteeing $E+W\in\partial\lVert M\rVert_*$ in the subgradient characterization (3.4).