the $d$-dimensional orthogonal group. For $R$, we will require that it belongs to a Borel set ${\mathbb U}\subseteq {\mathcal O}_{d}$ that is large in terms of $\nu_{d}$.

\[t1\] Fix finite constants $D\geq1 $ and $E\geq1$, and positive finite constants $\rho\in (0,1)$, $\lambda$, $L$ and $\gamma$. For each full-rank $d\times p$ matrix $M$, each $d\times d$ variance/covariance matrix $\Sigma>0$ and each $f_{\tilde{z}}\in {\mathcal F}_{d,20}(D,E)$ there exists a Borel set ${\mathbb U} = {\mathbb U}(M,\Sigma,f_{\tilde{z}}) \subseteq
{\mathcal O}_{d}$ so that $$\sup_{\footnotesize \begin{array}{c}
        M
        \end{array}
    }\;
    \sup_{  
        \Sigma
    }\;
    \sup_{f_{\tilde{z}} \in {\mathcal F}_{d,20}(D,E)}\;
    \nu_{d}({\mathbb U}) 
    \quad
    \stackrel[]{\frac{ p }{\log d} \to 0}{\longrightarrow} 
    \quad
    1$$ and so that the following holds: If $\Xi_n$ denotes either the quantity $$\label{t1.1}
\sup_{t\in{{\mathbb R}}} \left|
    {{\mathbb P}}\Big( \hat{F}_n \leq t \Big) - 
    F_{p, n-p-1, n\Delta}(t) 
    \right|$$ or the quantity $$\label{t1.2}
{{\mathbb P}}\Big(
    \hat{F}_n > F^{-1}_{p, n-p-1, 0}(\alpha)
\Big)
-
\Phi\Big(
    - \Phi^{-1}(\alpha)
    + \sqrt{n} \Delta \sqrt{ \frac{1-p/n}{2 p/n}}
\Big)$$ for some fixed $\alpha\in [0,1]$, then $$\sup_{\footnotesize \begin{array}{c}
        M
    \end{array}
}\;
\sup_{  
    \footnotesize \begin{array}{c}  
        \vartheta, \theta, {\mathcal L}(\epsilon), \mu,
    \Sigma\\ {{\mathbb E}}|\epsilon/\sigma|^{8+\lambda}\leq L\\
    \Delta< \gamma/\sqrt{n}
\end{array}}\;
\sup_{f_{\tilde{z}}\in {\mathcal F}_{d,20}(D,E)}\;
\sup_{R \in {\mathbb U}}\;
\;\;\Xi_n
    \quad
    \stackrel[\frac{n^2 }{\log d} \to 0, 
        \frac{p}{n}\to \rho]{ n\to\infty}{\longrightarrow} 
    \quad
    0.$$ This statement continues to 