Untwisting key lemma for the change-of-variables conjecture

Let $\phi:\mathcal{X}=[M/G]\to X=M/G$ be the proper birational morphism associated with a $G$-variety $M$. Let $\gamma:\mathcal{E}\to\mathcal{X}$ be a twisted arc whose generic point maps into the isomorphism locus of $\phi$, and let $\pi_n$ denote truncation. Define $\phi_n:\pi_n(\mathcal{J}_{\infty}\mathcal{X})\to J_nX$ to be the natural map. Untwisting key lemma. For $n\gg0$,

$$[\phi_n^{-1}(\phi_n(\pi_n(\gamma)))]=\mathbb{L}^{\operatorname{ord}\,\operatorname{Jac}_{\phi}(\gamma)-w_{\mathcal{X}}}.$$

This lemma is proposed as the geometric reduction underlying the main change-of-variables conjecture: fibers of the truncation map have motivic size governed by the Jacobian order corrected by the stack weight. It is not proved in the supplied text.