Toward the Analytic Resolution of the Variance-AM-GM Inequality: The Role of Taylor Expansion and Asymptotic Analysis

 This work provides the direct analytic proof and refinement of the Variance-AM-GM
 Inequality Conjecture De Ceuster [2025b]. The original form was found to fail for moderate
 variance, necessitating the determination of a constant Cn. Through rigorous local analysis
 based on Taylor expansion and a two-value reduction for extreme asymptotic regimes, we
 establish that the best possible universal constant is Cn = 1/2.