Derivation, Steady-State Analysis and Mathematical Self-Consistency of the Essence-Qi-Spirit Dynamic Equations《精气神方程组的推导、稳态分析与数理自洽性 —— 三元生命数理论核心数学框架》

Abstract

Based on the three fundamental axioms of the ternary steady-state theory—closed-loop, scale, and bottom-boundary axioms—this study constructs a coupled dynamic system of essence, qi, and spirit to realize the standardized mathematical reconstruction of the traditional Chinese life theory. Three dimensionless state variables corresponding to life substance structure, energy metabolism, and information regulation are defined, and a one-to-one hierarchical mapping framework is established between the three core life dimensions and the ternary axiom parameters. Traditional Chinese medicine logical mechanisms including "essence generates qi, qi generates spirit, and spirit governs essence and qi" are strictly transformed into a first-order coupled ordinary differential system with a bottom-boundary step operator. Under healthy and safe boundary constraints, the unique steady-state analytical solutions of the system are completely derived. A three-dimensional Jacobian matrix is constructed for linearization, and the necessary and sufficient conditions for global asymptotic stability of the life steady state are proposed through eigenvalue analysis. By algebraic degradation deduction under different working conditions, this study verifies the complete mathematical self-consistency between the dynamic equations and the ternary axiom system, proving that the simultaneous establishment of the three axioms serves as the necessary and sufficient condition for the long-term ordered survival of life systems. Furthermore, the phase-space state deviation trajectories are mapped to typical TCM syndrome types such as essence deficiency, qi stagnation, qi counterflow, and spirit disorder, establishing a unified quantitative analysis paradigm connecting traditional life cognition with modern non-equilibrium thermodynamics and dynamic system theory. All derivation processes in this paper are parameter-free, reversible, and reproducible. The proposed life-specific dynamic equations can be degenerated from the universal ternary reaction-diffusion equations under life-specific boundary constraints, achieving multi-scale theoretical unification spanning microcosmic, life, macroscopic, and cosmic systems.