BT-PEC Origin Theory: Ternary Coupled Matrix Driven Bootstrap Emergent Virtual Machine, Interpreter, Compiler and Syntax Paradigm

This paper proposes the BT-PEC (Berry-Tensor / Pseudo-Euclidean Coupling) Origin Theory, establishing a formal bootstrapping system based on the coupled constraints of ternary matrices B, T, PEC. The core conservation constraint of the system is defined asΩ = tr(BT) − tr(PEC) = 0Under this global trace balance constraint, the high-dimensional matrix dynamical system can spontaneously emerge a complete formal grammar, machine instruction set, compilation rules and virtual machine execution architecture. Under the continuous limit condition, the generalized Einstein field equations of general relativity can be rigorously derived. This paper fully constructs the BT-PEC axiomatic system, matrix evolution dynamics, eight sets of core mathematical theorems, emergent formal grammar pedigree, AST compilation architecture and bytecode virtual machine model, and completes the full reference engineering implementation and standardized test verification. Numerical experimental results show that at the steady fixed point where Ω→0, the system automatically completes structural differentiation: the Berry tensor matrix B emerges the topological structure of program non-terminal grammar, the state transition matrix T emerges grammar derivation and control-flow jump rules, and the pseudo-Euclidean coupling matrix PEC emerges the underlying executable terminal instruction set, forming a fully self-consistent emergent programming language system without artificial presupposition. At the level of physical unification, this paper derives the Einstein-Hilbert action independently from the same set of underlying axioms based on the information bottleneck complexity penalty continuation scheme, and obtains the Einstein field equations strictly through metric variation, realizing the deep unification of discrete computational dynamics and continuous spacetime geometry. This research connects matrix algebra, formal language theory, compiler architecture, information geometry and general relativity, providing a brand-new fundamental paradigm for bootstrap emergent computation and information-gravity unified theory.