Universal Unified Mathematical Tool Manual for the Fourth Stage of W≡0 Global Topological Unified Field Theory

Project: Construction System of W≡0 Global Topological Unified Field TheoryResearch Stage: Fourth Stage · Pre-mathematical Layer Abstract This paper serves as a standardized universal mathematical reference for the whole W≡0 theoretical system, which systematically standardizes all basic specifications of the exclusive topological mathematical framework of the theory. Its contents cover the definition of global unified symbol systems, rigorous mathematical expressions of self-developed core topological operators, lists of permitted and prohibited mathematical tools, allocation schemes of exclusive mathematical tools for Entity-State-Application three-line deduction, and unified criteria for formula citation, approximation and error analysis. All contents are strictly derived from established axioms and theorems formed in Stage 0 to Stage 3, without external hypotheses, unproven self-defined mathematical tools, physical deductions or numerical calculations. As the unified criterion followed by mathematical deductions in the 4.0 series, derivations of physical constants in the 4.1+ series and all subsequent academic researches, this manual fundamentally avoids mixed use of tools, symbolic ambiguity and logical conflicts, and ensures the uniqueness, self-consistency and reproducibility of all system deductions. Keywords W≡0 theory; mathematical tool manual; topological operator; mathematical tool list; Entity-State-Application three-line tool allocation; topological dimensional system