An Elementary Asymptotic Approach to the Golden Ratio on Markov Cubic Surfaces

Announcement The target journal requires submissions to contain no abstract. The following abstract is provided solely for the preprint version. Abstract: The following short abstract is included only for the preprint version. This note gives an elementary asymptotic derivation of the golden‑ratio scaling on the Markov cubic surface. By projecting the Markov equation to the affine ratio locus and expanding the discriminant \(5Z^{2}-4\) for large \(Z\), the upper branch \(X_{+}(Z)\) is shown to satisfy \(X_{+}(Z)/Z = \varphi^{2} + O(Z^{-2})\). This provides a self‑contained route to the eigenvalue \(\varphi^{2}\) and its curvature‑driven error decay, without invoking modular or hyperbolic machinery.