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 ~!103 (from ≈3.9!×!103 to ≈3.4 in 5 epochs), increases average gradient magnitude, and restores test accuracy ( ≈10%!→!≈86% ). These results support optimization-driven spectral preconditioning: directly steering models toward well-conditioned regimes for stable, accurate learning.