arXiv:2609.23468v2 Announce Type: replace Abstract: We study exact sign recovery for PCA connectivity analysis in mixture models, using the projection onto the first $(K-1)$ principal components. For any fixed number of components with affinely independent means, we derive the oracle signal projection explicitly. Under anisotropic sub-Gaussian noise, with a random number of observations in each component, a rowwise eigenspace bound yields a sufficient condition, governed by the weakest signal direction, for recovering all connectivity signs without requiring a bounded condition number of the nonzero signal spectrum. Under isotropic Gaussian noise, when the configuration of the component means is fixed up to scaling and orthogonal transformations, we identify an asymptotically sharp threshold for the signal strength: exact sign recovery succeeds above the threshold and fails below the threshold, with probability tending to one when the signal strength stays a fixed relative distance from the threshold. The threshold has an explicit formula when the mixing proportions are equal and the means form a regular simplex.