For $g\sim\mathcal{N}(0,I_k)$ and fixed full-row-rank $H$, $Hg$ is independent of the squared residual $\|(I-P_H)g\|^2$

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For $g\sim\mathcal{N}(0,I_k)$ and fixed full-row-rank $H$, $Hg$ is independent of the squared residual $\|(I-P_H)g\|^2$ · Prove2Me