Hidden coordinates are not uniquely determined by a language model's input--output function, so representation-derived measurements should be invariant to function-preserving changes of basis. This study shows that column-permutation parallel analysis violates function-preserving reparameterization invariance because its reference distribution and selected component count can change while the model function and observed covariance spectrum remain fixed. More generally, a data-internal reference procedure cannot simultaneously preserve every coordinate marginal, remain orthogonally equivariant, and remove cross-coordinate covariance. Empirically, across five models, three retrieval domains, and 75 transformations, median component-count disagreement is 0.79 and median fixed-threshold decision disagreement is 0.26. A centering-only control isolates the reference-driven effect, with 1,141 of 1,200 component counts changing despite an unchanged observed spectrum, whereas independent parallel analysis seeds change none of the corresponding decisions. By contrast, orthogonally invariant comparator scores remain numerically stable with similar held-out discrimination. Together, these results show that parallel analysis-derived component counts and decisions can reflect hidden-coordinate choice rather than a well-defined property of the model.
This paper considers a multi-environment factor model in which high-dimensional covariates are collected from heterogeneous environments, with auxiliary labels available in a subset of these environments. The joint distribution of the covariates may vary across environments, whereas the latent structure is decomposed into invariant factors with shared loadings and heterogeneous factors with environment-specific loadings. Such a model is motivated by transfer learning and latent factor regression, where one seeks stable low-dimensional representations for both interpretation and robust out-of-sample prediction of the response $Y$. Leveraging the invariance principle, we show that the invariant and heterogeneous factors are disentangled under a minimal structural condition. Based on this, we propose ATLAS, an Auxiliary-label and invariance-guided Transfer via Latent Alignment across heterogeneous environmentS. ATLAS is a unified procedure that leverages the invariance principle to separate aligned invariant and unaligned heterogeneous factors, and further exploits supervision from auxiliary labels to extract prediction-invariant and transferable factors from those unaligned heterogeneous factors. ATLAS yields near-oracle performance for downstream latent factor regression, enables transferable prediction in new environments through the full latent signal when auxiliary labels are available, and reduces to robust invariant-factor-only prediction otherwise. We establish sharp non-asymptotic error bounds for recovering invariant and heterogeneous factors, identifying all the response-invariant factors, and estimating the invariant signal in $Y$.