Sharp Gaussian constants for row deletion in unit-row matrices

For fixed $0 < \theta < 1$ and a real $m \times n$ matrix $A$ with unit rows,
let $s_\theta(A)$ be the smallest singular value left after the worst choice of
$\lfloor \theta m \rfloor$ of its rows. For independent rows uniform on
$S^{n-1}$, the manuscript proves that $(n/m)\, s_\theta(A)^2$ and
$s_\theta(A)^2/\|A\|_2^2$ converge in probability to the truncated Gaussian
second moment $h_\theta$ along every sequence with $n \to \infty$ and
$m/n \to \infty$, with explicit failure probabilities and error bounds, by
trimmed-mean concentration in a fixed direction, a net argument and a coupling
with Gaussian vectors. For every matrix with unit rows it proves
$(n/m)\, s_\theta(A)^2 \le h_\theta + O_\theta(n^{-1/6})$ for all $m$ once $n$ is
large, by projecting away the leading covariance directions and applying the
Gaussian Poincaré inequality. Together, the two results make $\sqrt{h_\theta}$
the sharp asymptotic constant.