Arithmetic Uniformization of Rigid Elliptic Structures: From Rigid to Standard Vekua without the Beltrami Equation

Abstract

For the rigid subclass of variable elliptic structures -- characterized equivalently by the inviscid Burgers law $\lambda_x+\lambda\lambda_y=0$ or the self-dilatation $\mu_{\bar z}=\mu\mu_z$ -- we show that the auxiliary Beltrami equation in the classical Vekua pipeline is unnecessary. The canonical coordinate $\xi=y-\lambda x$, computed by arithmetic from the spectral parameter $\lambda$, reduces every rigid variable-algebra Vekua equation to a standard Vekua equation in $\xi$ on any open set where the characteristic Jacobian $\Phi=\bar\xi_x+\lambda\bar\xi_y$ does not vanish, with global reduction on domains where $\xi$ is injective. No PDE is solved at any stage.