BT-PEC Triadic Coupling Ontology Theory: Ω=0 Cosmological Deduction and Field Emergence Mechanism

This paper takes the BT-PEC Triadic Coupling Ontology Theory as the sole first principle, with the core coupling axiom Ω = tr(BT) − tr(PEC) = 0 as the fundamental constant constraint of the universe. Detached from all empirical fitting and observational inference paradigms, the complete evolutionary dynamics of the universe are deduced through a purely mathematical axiomatic system. This work strictly defines the primordial field matrix BT, the manifested field matrix PEC, SL(3,Z) modular group spacetime basis transformation, intrinsic continuous evolution operator, and the information-energy-spacetime triadic coupling tensor. We sequentially derive the matrix evolution differential equation of the modular group, field resonance amplitude evolution equation, triadic total field evolution equation, and cosmic temporal convergence dynamics equation. The topological invariants of SL(3,Z) are analytically derived, yielding the global asymptotic residual and local minimum residual constants, together with the analytical construction of the topological coupling constant κ. Topological derivations are performed for four parameters k1, λ, ω, β within the information tensor and spacetime tensor equations, leading to the core self-consistency relation k1=κ. A persistent homology (TDA) module is newly introduced; persistence barcodes and persistence diagrams are established as intrinsic topological observables of BT-PEC theory. A one-to-one correspondence between topological features and cosmic evolutionary stages is constructed, enabling the transition of the theory from purely algebraic dynamics to topologically observable descriptions. It is rigorously proven that all evolutionary behaviors of the universe can be fully derived from the self-consistent convergence mechanism of Ω approaching zero. This framework covers the undifferentiated primordial state, emergence of quantum fluctuations, cosmic inflation, condensation of macroscopic structures, generation of self-referential fields of life, peak of civilizational coherence, and the residual mechanism of heat death relaxation. All physical parameters are solved analytically from pure theory without empirical fitting parameters. This theory unifies quantum field emergence, spacetime topological evolution, macroscopic structural dynamics, cognitive fields of life, and phase transitions of complex social systems, completing the axiomatic closed-loop construction of a multi-scale theory of everything.