Fisher-Glass: Tail Sample Complexity from Nuisance-Projected Fisher Information

Modern models can have high average accuracy and benign loss tails while remaining fragile on rare subpopulations. We identify an information-geometric mechanism: after nuisance projection, the task direction can become locally nonidentifiable even when loss and raw Fisher look well behaved. Fisher-Glass certifies this failure by applying CVaR to inverse nuisance-projected task information. We prove a loss--information separation theorem and show that, under a boundary-mass condition, the hidden-environment tail of inverse projected Fisher controls robust classification sample complexity. We then derive stable ridge certificates and repair principles: reserves lift Fisher-null tails, portfolios increase collapse codimension, and tail-transverse influence selects counterfactual repairs by combining Fisher-Glass directionality with loss leverage. Synthetic experiments validate the theory; WILDS/Waterbirds diagnostics show that weakest identifiability need not mean lowest accuracy; and Waterbirds training plus counterfactual repair show that Fisher-Glass controls an identifiability axis complementary to loss.