Algebraic Completeness of Ternary Total Integrals: Topological Conversion from Calculus, Explicit-Implicit Separation, and the ∅-State Operator

Traditional calculus, built upon binary logic, is restricted to processing the linear ornonlinear evolution of explicit physical variables, leading to inherent compute gaps whenfacing complex systems with ∅(Null) states. This paper formally proposes the "TernaryTotal Integral," a mathematical tool aimed at unifying explicit (M+, M−) and implicit (∅)variables within physical systems. Research demonstrates that classical calculus is merely alow-dimensional approximation of the Total Integral when the energy escape constant κ→0.By introducing the ∅-state topological operator, this paper provides the conversion equationsbetween the two frameworks and proves the algebraic completeness of the Total Integral inprocessing deterministic deviations within non-symmetric operation logs.