Orientation Can Reverse the Harm of Representation Collapse

Representation collapse is blamed for brittle features, yet neural collapse accompanies good generalisation, and no label-free statistic says which of the two a model is in. We show that collapse magnitude alone does not decide it: at a matched contraction, the orientation of that contraction relative to semantic and nuisance directions changes how much task-relevant information survives, in a construction where every spectral statistic of the Jacobian is identical. We formalise this with a representation's linear semantic Fisher information and test the formal claim where its hypotheses hold exactly, on a frozen encoder: the predicted change in semantic information is reproduced to better than $4\cdot 10^{-10}$, while the companion statement about nuisance contraction has no exact realisation on these encoders, which is consistent with contracting a nuisance response costing information rather than saving it. Holding one regulariser fixed and rotating only the factor it acts on, the representation-level out-of-distribution error is $65.7\%$ for a semantic contraction against $42.4\%$ for a neutral nuisance, and, at effective ranks within $4\%$ of each other, $6.4\%$ for a semantic protection against $38.5\%$ for a neutral one; six of our eight pre-registered predictions nonetheless failed and one of two kill criteria fired. Over unregularised networks swept through two training regimes our index keeps one sign in all thirty strata where each of eight collapse statistics reverses, but an information-matched control shows most of that advantage is the counterfactual pool rather than our summary of it. The one failure the index provably cannot see is a fold, which we exhibit and measure.