Cooperative Steady‑State of Planetary Civilizations Based on the Axiomatic System of Ternary Global‑Domain Unified Dynamics — A Mathematical Characterization Study of a Community with a Shared Future for Mankind《基于三元全域统一动力学公理体系的行星文明协同稳态 —— 人类命运共同体数理刻

Existing studies concerning a community with a shared future for mankind mostly focus on philosophical origins and diplomatic narratives. Mainstream international quantitative models for global governance generally take zero‑sum games as their underlying logic, and indigenous neutral fundamental quantitative analytical tools free from national bias remain scarce. Drawing on the four open‑access archived monographs of Ternary Steady‑State Theory and its 28 companion series papers, this paper takes the three independent and complete meta‑axioms (Closed‑Loop Axiom, Scale‑Matching Axiom and Physical‑Bottom‑Line Axiom) as the sole underlying constraints. An 11‑dimensional hierarchical time‑delay reaction‑diffusion global‑domain dynamical system for closed‑spherical planetary civilisations is constructed, and the dimensionless civilisation‑cooperation phase‑transition criterion is derived. Taking historical time‑series data of Roman, Maya, Chinese and modern global civilisations as independent examples, cross‑civilisation posterior deductions are implemented to verify the mathematical self‑consistency of the model within this axiomatic framework. Based on the objective evolutionary laws of non‑equilibrium dissipative systems, a set‑theoretic formal definition in the ternary global‑domain phase space is provided for the system state corresponding to a community with a shared future for mankind. Necessary and sufficient conditions for global cooperative steady‑state are forward‑deduced; meanwhile, instability mechanisms within this axiomatic framework are reversely deduced for three development modes: camp division, inter‑generational over‑exploitation, and unrestrained resource consumption. All theoretical derivations, equations and historical cases in this paper originate from the archived companion works, with no additional theoretical assumptions introduced throughout, maintaining a neutral analytical perspective for fundamental‑theory research. This paper only completes the construction of underlying axiomatic frameworks and mathematical tools. Fine‑grained parameter calibration and complete regional simulation deductions are left for independent further development by global researchers.