Comprehensive Verification, Steady‑State Analysis and Standard Definition of Global Application Boundaries for the Ternary Global‑Domain Unified Mathematical System《三元全域统一数理体系综合核验、稳态分析与全域应用边界标准化界定》

This paper is the final concluding manuscript (Series No. 27) of the complete Yang‑type mathematical trilogy and represents the core achievement for final system validation and formalisation within the third phase (V23‑V27) of the theoretical framework. The full Yang‑type mathematical system is constructed in three progressive stages: V1‑V6 establishes the core framework of ternary foundational axioms; V7‑V22 realises interdisciplinary extended deductions of the ternary axioms; V23‑V27 focuses on system verification, normative unification and closed‑loop finalisation. Based on the theoretical accumulation of V1‑V26 and outcomes of the core series papers, this paper takes the 11‑dimensional global‑domain unified dynamical equations and the three‑layer hierarchical vector theoretical framework as its main research carrier and completes three key research tasks:(1) Conduct full‑system mathematical self‑consistency re‑verification for the 11‑dimensional global‑domain state vector, global‑domain coupling matrix and degraded equation sets at all hierarchical levels. Supplement the previously missing complete proof of global Lyapunov stability for the 11‑dimensional global‑domain system. Systematically classify and deduce precise mathematical criteria for multi‑steady‑state attraction basins, critical bifurcation points and irreversible collapse absorbing states within the global‑domain phase space.(2) Systematically sort out the full theoretical inheritance chain from foundational axioms and extended deductions to global‑domain unification. Fully calibrate dimension‑mapping rules, parameter standards and the complete mathematical‑symbol system across series papers, and establish a unified, standardised academic paradigm for the whole Ternary Steady‑State Theory.(3) Precisely and standard‑define rigid boundaries between pure fundamental‑theory research and applied research including engineering simulation, traditional Chinese‑medicine clinical practice and social empirical studies, and build a comprehensive and implementable normative framework for the extension and application of the global‑domain theory. Relying on the complete 11‑dimensional global‑domain state‑vector system, this paper further deepens the underlying mathematical interpretation of a community with a shared future for mankind within planetary‑scale giant dissipative systems, constructs a complete steady‑state‑discrimination indicator system adapted to global‑system evolution, and strictly demarcates research boundaries between pure axiomatic deduction and real‑world data‑driven fitting. All mathematical derivations in this manuscript strictly follow indigenous original‑research paradigms and adopt pure axiomatic‑deductive logic without empirical fitting parameters or ad‑hoc simplifying assumptions. Through three core efforts: full‑dimensional system‑stability verification, cross‑series standard unification, and normalisation of theory‑application boundaries, this paper thoroughly remedies mathematical, normative and boundary‑definition deficiencies of the full ternary mathematical system. It ultimately delivers an indigenous Chinese interdisciplinary mathematical‑theoretical system characterised by logical self‑consistency, thorough deduction, reproducibility and extensibility.