BT-PEC Fundamental Theory: Axiomatic Construction of Tripartite Coupling Zero-State Equation and Its Topological Dynamical Deduction

This paper establishes the BT-PEC Fundamental Theory starting from three minimal independent existence axioms: distinguishability, state evolution, and intrinsic constraint. Without presupposing spacetime, matter or physical constants, we construct the ontological base operator B, transformation operator T, and constraint-encoding operator PEC. The core tripartite coupling zero-state equation Ω = tr(B·T) − tr(PEC) = 0 is rigorously derived as the necessary global condition for stable observable systems. We analyze the topological properties of solution manifold M_Ω. Under compactness, 3-dimensionality and fundamental group π₁=ℤ³, M_Ω is topologically equivalent to 3-torus T³ or its finite quotient space. The SL(3,ℤ) modular group acts as the automorphism group of T³. Combined with persistent homology, we encode topological information and define intrinsic projected gradient flows on the torus, which yields the emergent programming bootstrapping principle under discrete sampling. The framework supports cross-domain mapping: physical, informational, computational and observational systems are different coordinate representations on T³. The theory contains 100 falsifiable scientific predictions at characteristic bifurcation points on the constraint surface. We strictly separate proven theorems and pending conjectures. In proper asymptotic limits, Ω=0 reduces to Newtonian mechanics, relativity and quantum mechanics. The whole construction relies only on logical consistency, functional analysis and algebraic topology, independent of experimental input.