The actual supported third edge sends the positive point class to its residue object

The finite-type schemes and their open complements are noetherian and qcqs, so [F4] applies to the actual restricted tower on $X$. For $j=0,1,2$ and every $n\in \mathbb{Z}$, it gives ${\pi }_n{\mathrm{gr}}_x^{j}K(X)=H_x^{2j-n,j}(X,\mathbb{Z})=0$. A spectrum with all homotopy groups zero is zero. Thus the three supported graded triangles make $F_x^{3}K(X)\to F_x^{2}K(X)\to F_x^{1}K(X)\to F_x^{0}K(X)$ equivalences. Stage zero is $K_x(X)$ by the actual stage-zero identifications on $X$ and its complement. Their composite is exactly $a_{3,x}$, so this augmentation is an equivalence on the possibly singular ambient X.

Choose the smooth affine pure-three-dimensional finite-type neighbourhood $V$ of $x$ from [F2]. The same argument as step 1.1 gives the actual stage-three augmentation equivalence on $V$. The local ring at $x$ is regular of dimension three, so its point support is integral of codimension three; $V$ is quasi-projective over the perfect field. Apply [F1] at $q=3$ and transport through the actual augmented comparison in [F2]. Its resulting inverse-augmentation leading map ${\psi }_{x,V}:K_{0,x}(V)\to H_x^{6,3}(V)$ is an isomorphism taking the positive residue class to the positive point cycle. Its domain, projection and sign are the specified actual maps.

Restriction from $X$ to $V$ commutes with augmentations and stage-to-layer maps by the actual natural triangles and support construction in [F4]. The K-support and motivic-support restrictions are equivalences by [F2]. Because both stage-three augmentations are equivalences by steps 1.1 and 2.1, their commuting square also makes the stage-three support restriction an equivalence. Consequently the leading map ${\psi }_{x,X}=p_{3,x}({\pi }_0{a}_{3,x}{)}^{-1}$ is identified by the actual support restrictions with ${\psi }_{x,V}$. It is therefore an isomorphism taking $u_x$ to ${\delta }_x$, with the generators transported by [F2]. This establishes the global supported normalization without inferring stage excision merely from K-theory excision.

Since $a_{3,x}$ is an equivalence and ${\psi }_{x,X}$ is an isomorphism, the equality $p_{3,x}={\psi }_{x,X}{\pi }_0{a}_{3,x}$ makes $p_{3,x}$ an isomorphism. Inverting this equality gives ${ϵ}_{3,x}={\pi }_0{a}_{3,x}{p}_{3,x}^{-1}={\psi }_{x,X}^{-1}$ and hence ${ϵ}_{3,x}({\delta }_x)=u_x$. These are the actual augmented tower maps, consistent with the supported version of the global edge notation in [F3].