A Pure Logical Proof of the Zhu-Liang Exponential Function and Euler's Formula Recursive Element Theorem

Key Points

• To provide a logical proof demonstrating that the exponential function and Euler's formula are fundamental recursive elements in mathematics.
• Defined four axioms for recursive elements: existence, encoding invariance, metabolic conservation, and generativity.
• Used complex analysis to prove existence and generativity properties.
• Applied model theory and homotopy type theory to demonstrate encoding invariance.
• Utilized functional analysis to establish metabolic conservation.
• The proof validates the Zhu-Liang exponential function and Euler's formula as core recursive elements.
• It confirms these functions play a crucial role in modern mathematics fields like Fourier analysis and differential equations.