The Burgers Transform: From Holomorphic Functions to Rigid Elliptic Structures

We introduce the Burgers transform $\mathcal{B}$, a nonlinear bijection between holomorphic functions $f\text{\hspace{0.17em}⁣}:U\to {\mathbb{C}}^{+}$ 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 $\mathrm{Hol}(U,{\mathbb{C}}^{+})$ as the maximal seed space compatible with rigidity. The obstruction formula $H{\mathrm{\mid }}_{x=0}=2i(\mathrm{Im}f)f_{\bar{w}}$ 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.