Green-Function and Information-Geometric Correspondences Between Inverse Eigenvalue Loci of Generalized Lucas Sequences and the Mandelbrot Set

Abstract

We investigate geometric, potential-theoretic, and information-theoretic correspondences between the inverse eigenvalue loci of companion matrices associated with generalized Lucas sequences and the boundary of the Mandelbrot set. Through systematic numerical experiments, we show that these algebraic spectral loci exhibit a striking low-distortion geometric correspondence with the Mandelbrot boundary at macroscopic scales, together with a coherent organization within its external potential field, characterized by concentration along narrow equipotential annuli of the Mandelbrot Green function. This correspondence is quantified using a suite of complementary diagnostics, including optimal transport matching, Procrustes alignment, local distortion measures, fractal and spectral statistics, Green-function-based potential comparisons, and convex simplex update analyses. Taken together, these results indicate that the observed similarity extends beyond visual resemblance, reflecting shared structural organization across geometric, harmonic, and statistical levels. While the present work is entirely numerical in nature, it establishes a robust multi-scale framework for comparing algebraic spectral constructions with nonlinear dynamical fractals, and it highlights several avenues for future analytical investigation.

Figures (14)

• Figure 1: Optimal transport matching between inverse eigenvalue loci points and the sampled Mandelbrot boundary. Color-coded lines connect each $\lambda_i$ to its assigned $m_{\pi(i)}$, illustrating the global one-to-one correspondence. Axes show $\Re(z)$ (horizontal) and $\Im(z)$ (vertical).
• Figure 2: Final pointwise correspondence after Procrustes alignment. Matched pairs $(\lambda_i,m_{\pi(i)})$ are indicated by line segments. The aligned inverse eigenvalue loci closely trace the Mandelbrot boundary, including the main cardioid and primary bulbs. Axes show $\Re(z)$ (horizontal) and $\Im(z)$ (vertical).
• Figure 3: Histogram of pointwise matching distances $d_i$. Most pairs cluster in a narrow interval, with only rare outliers corresponding to the finest Mandelbrot filaments.
• Figure 4: Turned–angle curvature distributions. Both boundaries exhibit strong concentration near $\kappa \approx 0$, but the Mandelbrot boundary shows a broader and heavier tail of large $\kappa$, indicating more frequent curvature spikes and fine–scale boundary irregularity. The inverse eigenvalue loci show a tighter distribution with reduced tail weight, suggesting smoother local geometry.
• Figure 5: Coarse scan of reflection symmetry preservation as a function of the reflection axis angle $\theta$. The plotted score corresponds to the preservation fraction defined in Section \ref{['sec:symmetry_methods']}, evaluated jointly on the aligned inverse eigenvalue loci and the Mandelbrot boundary. A pronounced maximum occurs near $\theta \approx 0.6^\circ$, corresponding to reflection across an axis close to the real axis. The vertical dashed line indicates the best-fit axis. Legacy labeling in the vertical axis (“Construct + Mandel”) reflects earlier internal nomenclature and denotes the combined symmetry score of both systems.

Theorems & Definitions (11)

• Remark 2.1: Yoneda-type viewpoint
• Definition 2.2: Local Distortion Diagnostics
• Proposition 5.1: KL-monotone interpolation contraction
• proof
• Lemma 5.2: Pinsker inequality
• Proposition 5.3: Numerical verification of Appendix A assumptions
• Lemma 5.4: Histogram TV--control implies weak convergence to a common limit
• proof
• Theorem 5.5: Inverse Eigenvalue Loci--Mandelbrot Correspondence
• proof