Spectral Matrix Evolution of the Mandelbrot Iteration: Jacobian Products, Coherence, and Meta-Modes

The Mandelbrot set \(\mathcal{M} = \{c \in \mathbb{C} : z_n = z_{n-1}^2 + c \text{ stays bounded}\}\) is traditionally visualized by escape-time coloring. We introduce a spectral matrix-evolution analysis: at each parameter \(c\), we track the product of local Jacobian matrices along the orbit, extract the spectral radius, finite-time Lyapunov exponent, condition number, and a novel derivative phase-coherence score. The phase-coherence metric measures the consistency of the local derivative's rotation direction along the orbit — a diagnostic invisible to scalar escape-time analysis. A second-order "spectral of spectrals" meta-layer then reveals that the full four-feature dynamical landscape is governed by just two meta-modes capturing 92.6% of cross-feature variance. The meta-resonance map provides a cleaner regime separator than any single diagnostic: interior points have meta-resonance \(0.90\), escaped points \(0.57\), and boundary points \(0.45\)-\(0.55\). The method generalizes to any iterated holomorphic or real map and provides a template for spectral analysis of nonlinear dynamical systems.