a0_implies_tangent_coordinate_frobenius_bound

Role. It is a reusable node in the Candes-Recht decomposition, phrased as a standalone theorem so that downstream sketches can import it directly.

Problem and notation. Exact matrix completion asks when an unknown low-rank real matrix can be recovered from a random subset of its entries. Here $M\in\mathbb R^{n_1\times n_2}$ has rank $r$, $m$ entries are observed, and $n=\max(n_1,n_2)$. Recovery means nuclear-norm minimization: minimize $\|X\|_*$ among matrices $X$ agreeing with $M$ on the observed entries. Probability notation. $\operatorname{successProb}(m,M)$ is the fixed-cardinality success probability: $\Omega$ is chosen uniformly among all subsets of $n_1n_2$ entries with $|\Omega|=m$, and the event is that the convex program uniquely returns $M$. In Bernoulli nodes, $\mathbb P_p(E)$ or $\operatorname{bernoulliEventProb}(p,E)$ means each entry is sampled independently with probability $p$, usually $p=m/(n_1n_2)$. Coherence notation. The object $S$ records SVD/singular-vector data for $M$. The hypotheses $A0(S,\mu_0)$ and $A1(S,\mu_1)$ are the Candes-Recht incoherence assumptions: $\mu_0$ measures how spread out the singular vector spaces are, and $\mu_1$ measures the largest entry of the sign matrix $UV^\top$. The parameter $\beta>2$ controls polynomial failure probabilities such as $n^{-\beta}$. For tangent-space nodes, the main event is that the sampled tangent operator is well conditioned; this prevents a nonzero tangent perturbation from agreeing with M on the sampled entries.

Claim. Candes-Recht coordinate estimate used before Theorem 4.2: under A0, each projected coordinate matrix has Frobenius norm squared at most $2 \mu_{0} r/n$, where $n = \max n_{1} n_{2}$.

Lecture-note formulation:

Decomposition status. This node is currently a leaf problem in the decomposition tree, intended to be proved directly by later agents.