BT-PEC Theory: Analysis of Omega=0 Critical Topology and Phase Structure of Open Quantum Systems

Within a trace-class operator framework on a separable Hilbert space and based on three fundamental axioms, this paper derives the Ω = 0 steady-state condition for stable observable systems under the BT-PEC tri-coupling fundamental theory. Five classes of unstable configurations are constructed to cover known physical breakdown modes. Knot topology is adopted for dimensional selection analysis to discuss how 3D spacetime supports irreversible evolution and the emergence of time. A prediction system with strict falsification criteria is established, and inherent limitations of the framework are explicitly stated, yielding a self-consistent, quantitatively verifiable physical framework for cosmic steady-state and dimensional origin.