Etale commutativity of the maximal-order resolution step

The splitting sequence of the pullback. Let $s_0^{\mathrm{\prime }}>\cdots >s_{k^{\mathrm{\prime }}}^{\mathrm{\prime }}$ be the corresponding sequence of maximal divisor counts for the canonical resolution of ${\phi }^{\ast }(\mathcal{J},E,\mu )$; since $\phi$ is flat, the inverse image of $E$ has the same intersection pattern, so $s_j^{\mathrm{\prime }}\le s_i$-values at corresponding stages. We prove by induction on the pairs $(l,l^{\mathrm{\prime }})$ of the two splitting sequences that the induced sequence ${\phi }^{\ast }(X_i)$ agrees with the canonical sequence of the pullback up to extension and that the invariants correspond.

Case 1: the étale image misses the strata. If $s({\phi }^{\ast }(X_{i_l}))<s_{i_l}$, then the centers blown up in $(X_i{)}_{i_l\le i\le i_{l+1}}$ lie in strata that do not meet the image of $\phi$, so their inverse images are empty and the induced morphisms are isomorphisms; the equality of marked ideals and invariants with the pullback at stage $i_l$ is inherited from stage $l$.

Case 2: the strata meet the image. If $s({\phi }^{\ast }(X_{i_l}))=s_{i_l}>0$, the strata of the pullback are the inverse images of the strata of $X$. First apply the contained-stratum Step 1aa: whether a component is contained in the support is detected by its coefficient-ideal restriction support, so this condition pulls back. Such a component is a regular SNC boundary stratum; its blowup and controlled division commute with the étale base change, and both sides assign the same count/infinity primary value and the auxiliary values $\nu =0$, $\rho =\mathrm{\varnothing }$. No induction is applied to its zero restricted ideal. After these contained components are removed, [F2] makes every retained restricted ideal generically nonzero: an identically zero component restriction would have whole marked support and hence would still be in the contained branch. Now ${\phi }^{\ast }(J_{i_l}{\mathrm{\mid }}_{H_{\alpha }^s})=J_{j_{l^{\mathrm{\prime }}}}^{\mathrm{\prime }}{\mathrm{\mid }}_{(H^{\mathrm{\prime }}{)}_{\alpha }^s}$; the resolution process is reduced on both sides to the restrictions, so by the inductive hypothesis in lower dimension (the strata have dimension $<\dim X$) the canonical resolutions correspond and the invariants satisfy $\mathrm{inv}(x^{\mathrm{\prime }})=\mathrm{inv}(\phi (x^{\mathrm{\prime }}))$, as do $\nu$ and $\rho$.

Case 3: the non-boundary case. If $s({\phi }^{\ast }(X_{i_k}))=s_{i_k}=0$, first apply Step 1ba of [F1]. Its codimension-one support components are regular and isolated; étale pullback preserves their codimension and their local equation $J=(u^{\mu })$. Their labelled Cartier-center blowups divide by $u^{\mu }$ and give the unit ideal near them. Thus this branch corresponds on both sides, with an inserted isomorphism only when a center has empty inverse image. The remaining support has codimension at least two. Only now apply Step 1bb: its maximal-contact hypersurface has a generically nonzero restricted ideal, so the lower-dimensional canonical-resolution induction applies. Its inverse image is a hypersurface of maximal contact by [F3], and the induction gives matching centers and invariants; [F3] gives independence of the direction. The codimension-one branch carries the same top encoded invariant on both sides, and all other points receive the values from their later unchanged lifts as in the proposition's successive assignment.

Conclusion. Combining the preceding cases over the finitely many elements of the splitting sequence, ${\phi }^{\ast }(X_i)$ is an extension of the canonical resolution of the pullback and the invariants agree on all supports; this proves (1) and (2).