The Structural Bias of ${\ell }_2$-Regularized Cross-Entropy Heads in Class-Incremental Learning: Characterization, Limits, and a Gauge-Anchoring Fix

Class-incremental learning systems trained with ${\ell }_2$-regularized cross-entropy and within-task softmax exhibit a structural bias on cross-task discriminators that no amount of data or regularization tuning can remove. We characterize this bias in closed form: at the population optimum, the cross-task discriminator under sequential within-task training differs from the joint-optimal by exactly ${\Sigma }^{-1}({\bar{\mu }}^{(j)}-{\bar{\mu }}^{(i)})$, an offset arising from the per-task gauge fix imposed by within-task softmax, identical for all $C^2$ class pairs in a task block—this invariance, not the closed form alone, is what makes the diagnosis falsifiable and the correction one-line. Under a separation condition ($\kappa >1$, cross-class distances exceeding inter-task distances) we prove a margin-based bound on task confusion at rate $\exp (-(\kappa -1{)}^2{\gamma }_{ij}/8)$. By explicit construction we show this bound cannot be uniform: when separation fails, cross-task misclassification remains $\Omega (1)$ regardless of inter-task margin, and on configurations with extreme within-task class-norm asymmetry it converges to $1$. We propose post-hoc de-centering, an evaluation-time correction that adds the per-task offset ${\Sigma }^{-1}{\bar{\mu }}^{(t)}$ to each weight and a corresponding scalar to each threshold; it recovers the Bayes-optimal cross-task discriminator at the population exactly and yields a bound on task confusion that does not require the separation condition. The same operation appears in shared-covariance LDA-style heads (SLDA, FeCAM, NCM); our contribution is the gauge-theoretic derivation identifying it as the unique per-task affine correction to the structural bias of within-task cross-entropy training, the falsifiable empirical diagnostic across five published CIL methods, and the finite-sample analysis. Experiments on CIFAR-100, ImageNet-100, ImageNet-1K, ImageNet-R, and DomainNet across five frozen backbones show post-hoc de-centering recovers $82$–$94\%$ of the joint–sequential gap (mean $90.5\%$).