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 $λ_x+λλ_y=0$ or the self-dilatation $μ_{\bar z}=μμ_z$ -- we show that the auxiliary Beltrami equation in the classical Vekua pipeline is unnecessary. The canonical coordinate $ξ=y-λx$, computed by arithmetic from the spectral parameter $λ$, reduces every rigid variable-algebra Vekua equation to a standard Vekua equation in $ξ$ on any open set where the characteristic Jacobian $Φ=\barξ_x+λ\barξ_y$ does not vanish, with global reduction on domains where $ξ$ is injective. No PDE is solved at any stage.