[proofplan] We first record the precise external asymptotic result being used: the rank-one Gaussian spiked covariance eigenvector limit of [Paul's spiked covariance eigenstructure theorem](https://doi.org/10.1214/009053607000000839), stated for covariance $I_{p_n}+\theta v_nv_n^\top$, proportional dimension $p_n/n\to\gamma$, and a unit top sample eigenvector $\hat v_{1,n}$. We then verify that the theorem statement has exactly this Gaussian column model, covariance normalization, varying unit spike direction, and proportional regime. The cited theorem gives a non-zero squared-overlap limit when $\theta>\sqrt\gamma$ and zero overlap at and below the threshold, and the two alternatives exhaust all fixed spike strengths $\theta>0$. [/proofplan] [step:State and verify the spiked covariance eigenvector theorem in the supercritical regime] Assume first that $\theta > \sqrt{\gamma}$. We use the rank-one Gaussian spiked covariance eigenvector limit of [Paul's spiked covariance eigenstructure theorem](https://doi.org/10.1214/009053607000000839), specialized as follows: if $p_n/n\to\gamma\in(0,\infty)$, if $v_n\in\mathbb{R}^{p_n}$ satisfies $|v_n|=1$, if the columns of $X_n$ are independent Gaussian random vectors in $\mathbb{R}^{p_n}$ with covariance $I_{p_n}+\theta v_nv_n^\top$, and if $S_n=n^{-1}X_nX_n^\top$ has unit top eigenvector $\hat v_{1,n}$, then for $\theta>\sqrt\gamma$, \begin{align*} |\langle \hat{v}_{1,n},v_n\rangle|^2 \to \frac{1-\gamma/\theta^2}{1+\gamma/\theta} \end{align*} in probability. The theorem statement supplies each hypothesis: the data columns are Gaussian, the covariance is normalized as $I_{p_n}+\theta v_nv_n^\top$, the spike direction is allowed to vary with $n$ but is normalized by $|v_n|=1$, the spike rank is one, the spike strength $\theta$ is fixed, the proportional regime is $p_n/n\to\gamma$, and this step assumes the supercritical separation condition $\theta>\sqrt\gamma$. Therefore the cited theorem applies to the indexed pair $(\hat v_{1,n},v_n)$ and gives the displayed supercritical limit. [guided] We first isolate the exact external result, because the proof depends on matching its model conventions. The rank-one Gaussian spiked covariance eigenvector limit in [Paul's spiked covariance eigenstructure theorem](https://doi.org/10.1214/009053607000000839) applies to a sequence of sample covariance matrices $S_n=n^{-1}X_nX_n^\top$ whose columns are independent Gaussian random vectors in $\mathbb{R}^{p_n}$ with covariance matrix $I_{p_n}+\theta v_nv_n^\top$. It permits the spike direction $v_n$ to vary with $n$, provided $|v_n|=1$, and assumes the proportional asymptotic regime $p_n/n\to\gamma\in(0,\infty)$ with fixed spike strength $\theta>0$. Under the supercritical condition $\theta>\sqrt\gamma$, the theorem states that the squared projection of a unit eigenvector $\hat v_{1,n}$ associated with the largest eigenvalue of $S_n$ onto the population spike direction $v_n$ converges in probability according to \begin{align*} |\langle \hat{v}_{1,n},v_n\rangle|^2 \to \frac{1-\gamma/\theta^2}{1+\gamma/\theta}. \end{align*} We now verify the hypotheses one by one. The theorem statement defines the sample covariance matrix with the normalization $S_n=n^{-1}X_nX_n^\top$, so the covariance normalization agrees with the cited result. The population covariance is $I_{p_n}+\theta v_nv_n^\top$, so the perturbation has rank one and spike strength $\theta$. The vectors $v_n$ are unit vectors, so the overlap $|\langle\hat v_{1,n},v_n\rangle|^2$ is exactly the squared projection appearing in the theorem. Finally, $p_n/n\to\gamma\in(0,\infty)$ and the present step assumes $\theta>\sqrt\gamma$. Hence every hypothesis of the cited eigenvector theorem is satisfied, and the displayed non-zero limit follows in probability. [/guided] [/step] [step:Apply the same theorem at and below the phase transition] Assume now that $0<\theta\le \sqrt{\gamma}$. The subcritical and critical part of [Paul's spiked covariance eigenstructure theorem](https://doi.org/10.1214/009053607000000839), in the same rank-one Gaussian covariance normalization $I_{p_n}+\theta v_nv_n^\top$, states that when the spike strength is at or below the threshold, the top sample eigenvector is asymptotically orthogonal to the population spike direction: \begin{align*} |\langle \hat{v}_{1,n},v_n\rangle|^2 \to 0 \end{align*} in probability. The Gaussian column assumption, the normalization $S_n=n^{-1}X_nX_n^\top$, the proportional limit $p_n/n\to\gamma$, the unit normalization $|v_n|=1$, and the rank-one covariance perturbation have already been verified for the cited theorem; replacing the supercritical inequality by $0<\theta\le\sqrt\gamma$ gives exactly its subcritical and critical conclusion. [guided] The second regime uses the same external theorem, but on the other side of the phase transition. The cited rank-one Gaussian spiked covariance eigenvector theorem is formulated for the same objects: independent Gaussian columns with covariance $I_{p_n}+\theta v_nv_n^\top$, the normalized sample covariance matrix $S_n=n^{-1}X_nX_n^\top$, a unit top eigenvector $\hat v_{1,n}$, a unit population spike direction $v_n$, and a proportional limit $p_n/n\to\gamma\in(0,\infty)$. What changes is only the threshold condition. If $0<\theta\le\sqrt\gamma$, the spike is not separated above the spectral bulk in the sense required to produce a non-zero limiting projection. The subcritical and critical conclusion of the cited theorem is therefore \begin{align*} |\langle \hat{v}_{1,n},v_n\rangle|^2 \to 0 \end{align*} in probability. The hypotheses are satisfied for the same reasons as in the supercritical case: the model is Gaussian, the sample covariance has the $n^{-1}$ normalization, the covariance perturbation is rank one, $v_n$ is normalized, and the dimensions obey $p_n/n\to\gamma$. Since this step assumes precisely $0<\theta\le\sqrt\gamma$, the theorem gives asymptotic orthogonality of the top empirical eigenvector to the spike direction. [/guided] [/step] [step:Combine the two regimes] The two preceding steps cover all fixed spike strengths $\theta>0$: either $\theta>\sqrt{\gamma}$ or $0<\theta\le\sqrt{\gamma}$. In the first case, the cited supercritical eigenvector limit gives \begin{align*} |\langle \hat{v}_{1,n},v_n\rangle|^2 \to \frac{1-\gamma/\theta^2}{1+\gamma/\theta} \end{align*} in probability. In the second case, the cited subcritical and critical orthogonality limit gives \begin{align*} |\langle \hat{v}_{1,n},v_n\rangle|^2 \to 0 \end{align*} in probability. This proves the two asserted alternatives. [guided] There are only two possibilities for a fixed positive spike strength $\theta$: either it lies above the threshold $\sqrt\gamma$, or it lies at or below that threshold. The first step handled $\theta>\sqrt\gamma$ and proved \begin{align*} |\langle \hat{v}_{1,n},v_n\rangle|^2 \to \frac{1-\gamma/\theta^2}{1+\gamma/\theta} \end{align*} in probability. The second step handled $0<\theta\le\sqrt\gamma$ and proved \begin{align*} |\langle \hat{v}_{1,n},v_n\rangle|^2 \to 0 \end{align*} in probability. These two regimes partition the range $\theta>0$, so the theorem follows exactly as stated. [/guided] [/step]