Layered Base & Power Normalization Method: Constructing Multivariate Variable Analysis Paradigm Only Based on Four Arithmetic Operations

In traditional mathematics education, multivariate constrained extremum problems and multi-variable balance analysis are categorized entirely under advanced mathematics, partial differential analysis and numerical optimization. They can only be solved with high-level tools including functions, derivatives, partial derivatives and Lagrange multipliers. This paper proposes a brand-new paradigm that carries high-level thinking with elementary arithmetic: The whole reasoning process only relies on four basic arithmetic operations and real number comparison. No equations, function definitions, completing the square or calculus are adopted. Rigorous structural judgment for constrained extremum of multi-variable expressions can be completed without advanced mathematical tools. The core contribution of this research is not a new elementary problem-solving skill. Instead, it proves that the underlying logic of partial differential analysis, multi-variable restriction, magnitude rate comparison and layered variable optimization can exist independently of advanced mathematical tools, and can be fully established at the arithmetic learning stage. This paradigm completely bridges the cognitive gap between elementary arithmetic and advanced multivariate analysis, and provides a brand-new system for cultivating advanced mathematical thinking at an early learning stage.