finite_rademacher_weighted_first_moment_le_even_moment_root

This is a formal bridge for the finite Rademacher power-mean step used in the Rudelson/Lust-Picquard noncommutative-Khintchine route.

Source: Candes--Recht, Exact Matrix Completion via Convex Optimization, PDF p. 19, Section 4.2, Theorem 4.2, equation (4.9), where the tangent-sampling expectation is controlled by Rudelson's selection estimate; Rudelson, Random vectors in the isotropic position, JFA 164 (1999), Theorem 1, proof Steps 1--2; and Hardy--Littlewood--Polya, Inequalities, Section 2.10, for power means.

Mathematical statement. Let $iota$ be a finite Rademacher index set and let $F(eps) >= 0$ be a nonnegative statistic on sign subsets $eps : Finset iota$. For every integer $q >= 1$,

Variables and notation. This node uses the auxiliary Rademacher probability model: each sign pattern is represented by a subset $eps subset iota$ and has weight $2^(-|iota|)$. In the downstream Candes--Recht application, $n = max(n_1,n_2)$, $p = m/(n_1 n_2)$ is the Bernoulli sampling rate, $Omega$ is a fixed Bernoulli sample realization, and $F(eps) = ||sum_c eps_c H_c||$ for Hermitian matrices $H_c$.

Formalization note. This is a formal bridge. It does not appear verbatim in Candes--Recht. It bridges the proved source-backed power-mean node rademacher_expectation_power_mean (9a9634f8) and the source-backed child route through rademacher_matrix_operator_norm_first_moment_log_window_from_2p (136263d8) toward inner_sign_average_khintchine_variance_proxy_bound_of_two_le_max (f4806ebd).