Unified nonlinear transformation framework for spectral distributions (GOE, GUE, GSE), convergence on unimodal laws toward the limiting distribution with cv=1 attractor and a multimodal distribution generator

In this work, we highlight two remarkable phenomena related to the action of a single family of Non-Iterative Functions (FFNI). The first result concerns the fundamental spectral distributions of Random Matrix Theory: GUE, GOE, GSE. We show that a simple Nonlinear Transformation (NLT), controlled by an exponent, establishes a direct bridge between these distributions. Each spectral distribution can be transformed into another and then retrieved by applying inverse transformation. All six possible conversions: GUE ↔ GOE, GUE ↔ GSE, GOE ↔ GSE, are obtained within the same framework. The second result concerns the convergence of numerous unimodal distributions toward a Limiting Distribution LD. By applying the same NLT, we observe that diverse distributions, bell-shaped, triangular, exponential, trapezoidal, spectral, and many synthetic types, all converge to a LD characterized by: σ/μ=1 Both phenomena, spectral distribution conversion and unimodal distribution convergence toward LD, are based on the same NIF family, revealing an unexpected functional unification between seemingly independent domains: random matrices, classical distributions, synthetic laws, and asymptotic behaviors. This forms a simple, accessible, and powerful framework for generating, transforming, and connecting numerous probability distributions. In addition, the same functional structure can analytically generate a wide variety of multimodal distributions (with multiple peaks), revealing that complex shapes can emerge without iteration, solely through parameter variation. Taken together, these results suggest the existence of a unified mathematical architecture linking: spectral distributions, the LD attractor, and a structural generator of multimodal forms.