The Burgers Transform: From Holomorphic Functions to Rigid Elliptic Structures

We introduce the Burgers transform $\mathcal{B}$, a nonlinear bijection between holomorphic functions $f\colon U\to\mathbb{C}^+$ and rigid variable elliptic structures on the plane, defined implicitly by $\lambda = f(y-\lambda x)$. The output automatically satisfies the conservative complex Burgers equation $\lambda_x+\lambda\lambda_y=0$. Our main result is that holomorphicity of the seed $f$ is necessary, not merely sufficient, for rigidity: any $C^1$ function whose implicit solution satisfies $\lambda_x+\lambda\lambda_y=0$ must be holomorphic. This closes a gap in the existing literature and identifies $\operatorname{Hol}(U,\mathbb{C}^+)$ as the maximal seed space compatible with rigidity. The obstruction formula $H|_{x=0} = 2i\,(\operatorname{Im} f)\,f_{\bar{w}}$ quantifies the cost of non-holomorphicity at the level of the initial data. We characterise the domain of $\mathcal{B}$ through shock formation, its interaction with affine automorphisms of $\mathbb{C}^+$, and the infinitesimal structure: the propagator $\mathcal{P}_f = D\mathcal{B}_f$ satisfies a Jacobian-twisted multiplicativity that deforms the seed algebra by the density of characteristics. Four worked examples -- affine, exponential, inverse, and trigonometric seeds -- show that the complexity class of a seed and that of the resulting structure are generically unrelated.