Long-Horizon Meta-Objectives Induce Spectral Flattening

Muon accelerates matrix optimization by flattening the singular spectrum of gradient updates. But why should spectral flattening be a good optimization strategy, and can it arise without being explicitly prescribed? We approach these questions from a meta-optimization perspective, where update transformations are evaluated by their consequences for future learning rather than immediate descent alone. In controlled matrix-factorization problems, we find that extending the optimization horizon systematically shifts the preferred update toward flatter spectra. This shift reflects a long-term trade-off: spectrum-aware updates may sacrifice short-term progress along dominant directions while improving the optimization of modes that gradient descent leaves behind. We connect this behavior to the evolution of error modes and show how spectral flattening creates more balanced learning dynamics. Our results interpret Muon-like flattening as an emergent strategy for improving the geometry of subsequent optimization. This perspective suggests a path toward learning spectrum-aware optimizers directly from long-horizon objectives.