A Field-Coupled Measure-Valued Dynamical System with Spectral Conservation and Branching Graph Evolution

We introduce a measure-valued dynamical graph framework defined over the complex plane ℂ, in which system states are represented as weighted atomic measures supported on evolving node sets. Each node corresponds to a structured object , where is a complex parameter governing internal dynamics, denotes an iterated nonlinear orbit generated by a Mandelbrot-type recurrence , and encodes a spectral state derived from the orbit via Fourier and phase-rotation analysis. The global system is represented by a time-dependent measure , whose support evolves through coupled branching and interaction dynamics. Node evolution is governed by a field-coupled interaction structure defined through a spectral similarity kernel , inducing nonlocal drift in parameter space. Reproduction is modeled as a branching operator acting on the measure, producing discrete offspring with entropy-constrained weight distribution. Spectral content is transported through the system via inheritance rules coupling orbit functionals with interaction-driven perturbations. The dynamics are subject to two global constraints: (i) conservation of total spectral energy, defined as the sum of squared spectral norms across all nodes and enforced via a renormalization operator acting on the measure space, and (ii) bounded reproduction entropy, restricting the admissible branching distributions and ensuring controlled generative complexity. The main result is the construction of a unified nonlinear evolution operator acting on the measure space, decomposable into interaction, branching, and renormalization components. This operator induces a well-defined phase structure over the parameter space , yielding distinct asymptotic regimes characterized by collapse, frozen fractal formation, stable ecosystem dynamics, and turbulent overcritical growth. The framework establishes a coherent mathematical bridge between complex dynamical systems, measure-valued processes, and field-coupled branching graph evolution with embedded spectral structure. We introduce a measure-valued dynamical graph framework defined over the complex plane ℂ, in which system states are represented as weighted atomic measures supported on evolving node sets. Each node corresponds to a structured object , where is a complex parameter governing internal dynamics, denotes an iterated nonlinear orbit generated by a Mandelbrot-type recurrence , and encodes a spectral state derived from the orbit via Fourier and phase-rotation analysis. The global system is represented by a time-dependent measure , whose support evolves through coupled branching and interaction dynamics. Node evolution is governed by a field-coupled interaction structure defined through a spectral similarity kernel , inducing nonlocal drift in parameter space. Reproduction is modeled as a branching operator acting on the measure, producing discrete offspring with entropy-constrained weight distribution. Spectral content is transported through the system via inheritance rules coupling orbit functionals with interaction-driven perturbations. The dynamics are subject to two global constraints: (i) conservation of total spectral energy, defined as the sum of squared spectral norms across all nodes and enforced via a renormalization operator acting on the measure space, and (ii) bounded reproduction entropy, restricting the admissible branching distributions and ensuring controlled generative complexity. The main result is the construction of a unified nonlinear evolution operator acting on the measure space, decomposable into interaction, branching, and renormalization components. This operator induces a well-defined phase structure over the parameter space , yielding distinct asymptotic regimes characterized by collapse, frozen fractal formation, stable ecosystem dynamics, and turbulent overcritical growth. The framework establishes a coherent mathematical bridge between complex dynamical systems, measure-valued processes, and field-coupled branching graph evolution with embedded spectral structure.