The actual EM point calculation through an étale chart

By [F1], $x$ is rational and there is an affine neighbourhood with a map to ${\mathbb{A}}_k^3$ that is étale at $x$. Its étale locus is open by reference 0; refine inside it to a principal affine neighbourhood $U$ using [F2]. The resulting $q_0:U\to {\mathbb{A}}_k^3$ is étale everywhere. Its image at $x$ has coordinates $a_1,a_2,a_3\in k$. Replace each coordinate by $t_i-a_i$, whose inverse translation adds $a_i$; this gives an étale map $q$ sending $x$ to $0$. Both translations are inverse polynomial morphisms, so this composition preserves étaleness. The scheme-theoretic zero fibre is affine and quasi-compact by [F2]–[F3]. Reference 1 makes it a discrete disjoint union of finite separable field spectra; quasi-compactness makes the union finite, and algebraic closedness in [F1] makes every field $k$. Thus it is a finite disjoint union of reduced rational points. Its points are closed in $U$, because the fibre is closed and its finite discrete points are closed in that fibre. Delete the other points and use [F2] to choose a principal affine neighbourhood $W$ of $x$ inside the remaining open. Restriction of $q$ stays étale by reference 0, and its entire scheme-theoretic zero fibre is now the single reduced rational point $x$. The point codimension is three: the neighbourhood result in [F1] gives this at $x$, whose local ring is unchanged by open restriction, and the origin has its three coordinate parameters. Before applying [F4], check the schemes in its whole restriction diagram. Every open $O$ of $V$ is quasi-compact by [F2], smooth by the open-source property in [F5], and locally of finite type over $k$ by [F5]. Its structure morphism is quasi-compact: the only opens of $\mathrm{Spec}k$ are the empty and whole subsets, and their inverse images are quasi-compact. Hence $O$ is finite type over $k$ by [F5]. The affine $k$-scheme $W$ is separated over $k$ by [F5], and its open complement $W\setminus \{x\}$ is separated by the open-immersion and composition assertions there. The target ${\mathbb{A}}_k^3$ is the smooth affine standard chart of the projective test in [F3]; the same argument gives finite type and separatedness for it and its punctured complement. Thus both $q$ and its restriction to the complements lie between smooth separated finite-type schemes over the same perfect field, with the fixed size bounds in [A1]. These domain checks and the reduced rational fibre and codimension checks are precisely the hypotheses of [F4].

Write $W=\mathrm{Spec}B$ and $A=k[t_1,t_2,t_3]$. By reference 0 and [F3], $B$ is flat over $A$. The bounded free coordinate Koszul resolution of $k=A\mathrm{/}{\mathfrak{m}}_0$ from [F3], pulled back by the actual perfect functor, is its termwise tensor with $B$. Exactness of this tensor identifies the result with $B{\otimes }_{A}k=B\mathrm{/}{\mathfrak{m}}_{0}B$. Step 1.1 identifies its affine scheme with the reduced point $x$, so as a $B$-module this is exactly its residue sheaf, with its quotient action. Hence actual supported pullback sends $u_0$ to $u_x$, and sends $n{u}_0$ to $n{u}_x$ for every integer $n$. By [F4], $q^{\ast }{\delta }_0={\delta }_x$. Naturality of the actual pointed transformation on the restriction square, followed by functoriality of homotopy fibres, therefore gives $c_{3,x}(n{u}_x)=q^{\ast }{c}_{3,0}(n{u}_0)=q^{\ast }(2n{\delta }_0)=2n{\delta }_x$, using the standard calculation in [F3]. Open excision in [F4] transfers this equality from $W$ to the original $V$, preserving the selected generators. No isomorphism or injectivity of the support-to-global map on $V$ is used, and no independent classical Chern operation is substituted for the actual EM operation.