Spectral of Spectrals: Second-Order Mode Decomposition for Complex Systems

Spectral decomposition extracts modes from data: eigenvectors of a covariance or kernel operator that capture maximal variance per component. We introduce a second-order spectral layer: the eigenanalysis of the spectral features themselves. Given a system where each point \(x\) has \(d\) first-order spectral diagnostics \(\phi(x) = (\phi_1, \ldots, \phi_d)\), we construct the feature covariance operator and extract its eigenmodes — the meta-modes. We prove that the meta-spectrum (1) is a sufficient statistic for second-order feature interactions, (2) achieves the optimal low-rank approximation of the feature correlation structure (Eckart-Young), and (3) provides a built-in significance test via comparison to the Marchenko-Pastur null distribution. On the Mandelbrot iteration, where first-level features include Lyapunov exponents, spectral radius, and derivative phase coherence, the meta-spectrum reveals that just two meta-modes capture 92.6% of cross-feature variance, providing a single decision surface that cleanly separates dynamical regimes. We describe extensions to kernel meta-spectra (nonlinear second-order analysis) and time-varying meta-spectra (regime tracking), and outline applications to agent routing, memory retrieval, proof search, and financial risk.