Exact order-three ambiguity of the Einstein-Maxwell-dilaton coupling $a^2$ in metric jets, and its fourth-order collapse

Is the EMD coupling square $a^2$ a function of the metric three-jet on an explicit active, non-null, simple-spectrum family of truncated Einstein-Maxwell-dilaton data, and can one more derivative recover it? Proved: no function of the common metric three-jet returns $a^2$ - the order-three ambiguity is exactly a free affine shear orbit ($\mathbb{R}$) mixing $B=a\sin 2\theta$ with the phase gradient - while the fourth-order quotient recovers $a^2=A^2+B^2$, with equality fiber exactly $a=\pm b$ ($\mathbb{Z}_2$). In particular Kaluza's $a=\sqrt{3}$ and the control $a=1$ collide through metric order three.

Finite-jet theorems about compiled truncated EMD equation certificates: the exact shear-orbit fiber classification of the complete first seed channels, an explicit collision family with one metric three-jet realized by an actual cubic metric germ (genuine Frechet Ricci value and first derivative), the compiled impossibility theorem, and the fourth-order recovery with equality fiber $a=\pm b$. Not settled here: promotion to analytic EMD solution germs (separate written argument pending specialist audit, in the parent repository), chart covariance beyond the fixed presentation, density of the active locus, degenerate branches, and any sufficiency of $a^2=3$ for a Kaluza uplift.