Mathematical Formalization of Eigenvergence in Recursive Expansive Dynamics

We establish a rigorous mathematical formalization of eigenvergence, a novel convergence phenomenon in recursive expansive dynamical systems characterized by golden ratio scaling (phi^-n) and fractal self-similarity. This theoretical framework unifies concepts from spectral operator theory, algebraic deformation theory, and differential geometric topology within the context of Recursive Expansive Dynamics (REDS). The eigenvergence phenomenon emerges naturally from operators R: X → X on complete metric spaces satisfying three fundamental conditions: (i) local expansive properties with sensitivity parameter delta > 0; (ii) recursive structure with golden ratio spectral modulation (spectral radius rho(R) = phi^-1); and (iii) self-similarity transformations maintaining fractal consistency. We prove that sequences {x_n} generated by such operators converge to unique limit points x* with optimal rate |x_n - x*| ≤ C * phi^-n, establishing eigenvergence as a fundamental convergence mode distinct from classical fixed-point theory. The theoretical scaffolding encompasses four major components. First, Gromov-Hausdorff convergence in metric space sequences with rate d_GH(X_n, X_infty) ≤ C * phi^-n. Second, recursive Lie algebra deformations with structure constants C^k_{ij}(n) = C^k_{ij}(0) + sum_{m=1}^n phi^-m * delta C^k_{ij}(m), satisfying cohomological constraints H^2(g, g) = 0. Third, Cykloid Influence spacetime geometries with curvature recursion R^{(n+1)}{mu nu rho sigma} = R^{(n)}{mu nu rho sigma} + phi^-n * K^{(n)}{mu nu rho sigma}, yielding fractal dimensions D_H approximately 3.48. Fourth, eigenvergent hypergeometric functions of the form 2F1_eig(a, b; c; z) = sum{n=0}^infty [(a)_n (b)_n / (c)_n n!] * phi^-n * z^n, with convergence radius R = phi. Physical applications generate testable predictions, including gravitational wave echoes with frequency scaling f_n = f_0 * phi^-n and cosmic microwave background angular correlations exhibiting golden ratio modulation. The mathematical formalization extends beyond classical convergence paradigms to encompass multi-scale recursive dynamics, non-local memory effects, and fractal attractor structures across algebraic, geometric, and physical domains.