Dynamics of Ω‑Flashes, Topological Transitions, Stochasticity, Fractals, and Quantum Integration in ZEBTS

Abstract This article develops the mathematical foundation of ZEBTS (Zeta‑Extended Boundary Theory of States), a multilayer ontological framework that unifies coherence dynamics, topology, stochastic processes, fractal geometry, quantum integration, and operator‑geometric structures. The Ω‑field is introduced as the primary dynamical variable governing the emergence of Γ‑boundaries, topological invariants such as the Euler characteristic χ and Betti numbers bk, stochastic synchronization of the P‑modus, and the fractal scaling properties of coherence. A nonlinear operator F(Ω) describes Ω‑flashes as coherence phase transitions, while the stochastic term σ(Ω) generates noise‑driven regimes captured by the Fokker–Planck equation. The theory incorporates a variational principle that yields Euler–Lagrange dynamics with local, nonlocal, topological, and stochastic contributions. Quantum behavior is encoded through the Γ‑modus, the wave function ψ(x)=Ω(x)eiθ(x), and Ω‑driven decoherence. The framework is extended to extreme regimes, including Planck‑scale quantization, non‑Riemannian and fractal geometries, singularities, and topological defects. Experimental and computational methods are proposed for measuring Ω, κ, χ, and topological transitions in ultracold atomic systems and biological media. Together, these results form a unified operator‑geometric and topological field theory, establishing ZEBTS as a mathematically coherent and experimentally testable model of complex physical, biological, and cosmological dynamics.