Chebyshev Moment Regularization (CMR): Condition-Number Control with Moment Shaping

We introduce Chebyshev Moment Regularization (CMR), a simple, architecture-agnostic loss that directly optimizes layer spectra. CMR jointly controls spectral edges via a log-condition proxy and shapes the interior via Chebyshev moments, with a decoupled, capped mixing rule that preserves task gradients. We prove strictly monotone descent for the condition proxy, bounded moment gradients, and orthogonal invariance. In an adversarial $\kappa$-stress setting (MNIST, 15-layer MLP), compared to vanilla training, CMR reduces mean layer condition numbers by $\sim10^3$ (from $\approx3.9\times10^3$ to $\approx3.4$ in 5 epochs), increases average gradient magnitude, and restores test accuracy ( $\approx10\to\approx86$ ). These results support optimization-driven spectral preconditioning: directly steering models toward well-conditioned regimes for stable, accurate learning.