quadratic_neumann_middle_index_distinct_mean_as_coefficient_sum

Role. It belongs to the golfing/Neumann-series certificate branch, where the certificate is decomposed into linear and quadratic sampling terms.

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 certificate nodes, $T$ is the tangent space at $M$, $P_T$ and $P_{T^\perp}$ are the tangent and normal projections, and $P_\Omega$ keeps only observed entries. The Neumann-series estimates control the dual certificate used to prove uniqueness of nuclear-norm recovery.

Claim. The mean part of the $\omega_{1} = \omega_{3} \ne \omega_{2}$ quadratic term is the coefficient sum $p^{-1}(1-p) \sum E_{\omega_{1}} H_{\omega_{1}} F_{\omega_{1}}$ used in the proof of Lemma 4.6.

Lecture-note formulation:

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