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

We introduce $\mathbf{\text{Chebyshev}}\mathbf{\text{Moment}}\mathbf{\text{Regularization}}\mathbf{\text{(CMR)}}$, 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), \emph{compared to vanilla training}, CMR reduces mean layer condition numbers by $\sim \text{\hspace{0.17em}⁣}{10}^3$ (from $\approx 3.9\text{\hspace{0.17em}⁣}\times \text{\hspace{0.17em}⁣}{10}^3$ to $\approx 3.4$ in 5 epochs), increases average gradient magnitude, and restores test accuracy ( $\approx 10\mathrm{\%}\text{\hspace{0.17em}⁣}\to \text{\hspace{0.17em}⁣}\approx 86\mathrm{\%}$ ). These results support $\mathbf{\text{optimization-driven}}\mathbf{\text{spectral}}\mathbf{\text{preconditioning}}$: directly steering models toward well-conditioned regimes for stable, accurate learning.