ZEBTS: A Unified Operator‑Geometric and Topological Field Theory of Coherence, Quantum Structure, and Extreme Regimes

Abstract This work develops the mathematical foundation of ZEBTS (Zero‑Energy Boundary Topological Structure), a unified operator‑geometric and topological field theory describing coherence, topology, stochasticity, fractal geometry, and quantum behavior across physical, biological, and cosmological systems. The Ω‑field is introduced as a quantum‑topological coherence field governing the formation of Γ‑boundaries, the evolution of topological invariants such as the Euler characteristic χ and Betti numbers b_k, the stochastic synchronization of the P‑modus, and the fractal scaling of coherence. A variational principle is constructed, incorporating local geometric terms, nonlocal kernels, topological contributions, and stochastic corrections. The theory provides a rigorous description of Ω‑flashes, topological transitions, decoherence, and emergent time. Operator definitions for F^, I^, and K^Ω unify curvature, gauge fields, and informational structure. The framework is extended to Planck‑scale physics, non‑Riemannian and fractal geometries, singularities, and topological defects. Strict theorems of existence, uniqueness, stability, topological transitions, coherence invariants, and stochastic thresholds are established. Dimensional analysis and invariances are derived. A set of falsifiable predictions is formulated, including logistic Ω‑flashes, topological jumps, coherence collapse thresholds, fractal dimension bounds, and monotonicity of emergent time. Experimental schemes for measuring Ω, κ, χ, and flytron solitons are proposed. ZEBTS emerges as a mathematically complete, internally consistent, and experimentally testable theory.