rudelson_selection_expected_tangent_deviation_from_coordinate_radius_bound_dense_proviso

Corrected Rudelson selection estimate (expected tangent-sampling deviation) WITH the Candès–Recht Theorem 4.2 “right-hand side $\le 1$” proviso. There is an absolute constant $C_{sel}>0$ such that for every $\beta>2$, all sizes with $m\le n_1n_2$, every reduced SVD $S$ of $M$, and every radius $R\ge 0$: if the sampling is dense, $m\ge\beta\,(\max n)\,r\,\log(\max n)$, if the desymmetrization parameter is bounded, $\sqrt{\log(\max n)/p}\cdot R\le 1$ (with $p=m/(n_1n_2)$), and if every tangent-coordinate projection has Frobenius norm at most $R$, then the expected tangent-sampling deviation $\mathbb{E}_\Omega\,p^{-1}\|P_TP_\Omega P_T-pP_T\|$ is at most $C_{sel}\,\sqrt{\log(\max n)/p}\,R$. The added proviso $\sqrt{\log/p}\cdot R\le 1$ is exactly Candès–Recht 2009, Theorem 4.2 part 1 (eq. (4.9), p.18): “provided the right-hand side is smaller than 1.” It comes from the self-bounding desymmetrization $\mathbb{E}Z\le A+A\sqrt{\mathbb{E}Z}$ with $A=\sqrt{\log/p}\,R$, which yields the linear bound only when $A\le 1$; without it the stated linear bound can fail (the true bound is $\sim A^2$ when $A\ge 1$). This corrects the radius_dense node which omitted the proviso (R a free parameter).