= Solution
Because both vectors are linear transformations of the same <multivariate normal> response, $(\widehat Y,e)$ is jointly <multivariate normal>. Its cross-<covariance matrix> is
$$
\operatorname{Cov}(\widehat Y,e)=P(\sigma^2I_n)(I_n-P)^T
=\sigma^2(P-P^2)=0.
$$
Zero cross-<covariance> implies independence for jointly <multivariate normal> vectors, including singular ones. Therefore \b[the <fitted values> and the entire vector of <regression residuals> are independent]. The <fitted-residual orthogonality> identity gives the zero <covariance>; the <normal distribution> assumption is what upgrades it to independence.
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