Tail-Weight Control and Localized Generalization in Nearly Low-Rank Adversarial Classification

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We study norm-constrained linear classification under Eu clidean adversarial perturbations in a Gaussian model with a low-dimen sional informative subspace and an independent noise tail. For bounded ramp loss, we prove that a principal-space witness with risk below one half forces every near-optimal predictor to have small tail weight. A path-specific density bound yields constants without requiring positive tail variance. Under isotropic principal covariance, we establish a unique population minimizer and joint local growth. Boundary normalization then removes the common attack penalty from centered margins, giving localized finite-sample guarantees governed by principal dimension and total tail energy. Globalized growth removes the entrance condition at weaker constants; a model-aware comparison retains local guarantees. Experiments with twenty paired repetitions show decreasing excess risk and tail use with sample size, and nearly unchanged behavior when tail dimension grows at fixed total energy. Pure-noise controls and optimizer diagnostics clarify the scope and limitations of these conclusions.