= Solution
Let $Q=P_1-P_0$ and $R=I-P_1$. For nested subspaces these are orthogonal <projections> onto $\Theta_1\cap\Theta_0^\perp$ and $\Theta_1^\perp$, of ranks $d_1-d_0$ and $d-d_1$. Under the null, both annihilate $\mu$. In an <orthonormal basis> adapted to $\Theta_0$, these two spaces and their complement, the coordinates of $X-\mu$ are <independent> $N(0,\sigma^2)$. Consequently
$$
\boxed{\frac{\|X-P_1X\|^2}{\sigma^2}\sim\chi^2_{d-d_1},\qquad
\frac{\|P_1X-P_0X\|^2}{\sigma^2}\sim\chi^2_{d_1-d_0},}
$$
and the two variables are <independent>. Equivalently the <joint probability density> of the two unscaled sums $a,b>0$ factors into gamma <densities> of respective shapes $(d-d_1)/2,(d_1-d_0)/2$ and common scale $2\sigma^2$.
The nuisance <variance> cancels in
$$
\boxed{F=\frac{\|P_1X-P_0X\|^2/(d_1-d_0)}{\|X-P_1X\|^2/(d-d_1)}
\sim F_{d_1-d_0,d-d_1}\quad\text{under }H_0.}
$$
Reject for a value above the upper $\alpha$ critical quantile to obtain a level-$\alpha$ test. Under a genuine alternative the numerator is noncentral chi-square with noncentrality $\|Q\mu\|^2/\sigma^2$, while the denominator retains its central law, explaining the upper-tail choice.
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