The identified support fiber is the classical relative Hom before Gysin

In the derived category of Nisnevich sheaves with transfers, let $f:{\mathbb{Z}}_{\mathrm{t}\mathrm{r}}(V\setminus x)\to {\mathbb{Z}}_{\mathrm{t}\mathrm{r}}(V)$ be the actual map and let $D=\mathrm{cone}(f)$. Resolve the A¹-local coefficient by the derived-Hom model in [F1]. Yoneda evaluation on that resolution makes the two Hom complexes exactly the classical values on $V$ and $V\setminus x$, with the actual restriction between them. Applying Hom to the ordinary cone identifies $R\mathrm{Hom}(D,\mathbb{Z}(j))$ with the fiber of that arrow. By [F1], this is the already identified EM support fiber. Taking cohomology, A¹-locality of the target identifies these derived Hom groups with the displayed groups in $DM$, by the localization property in [F1]. No unrelated objectwise equivalences have entered this construction.

In $DM$, the image of $D$ is a cofiber of $M(V\setminus x)\to M(V)$, the same first arrow as in [F2]. Its Gysin cofiber is therefore compatibly isomorphic to $M(\mathrm{Spec}L)(d)[2d]$; no uniqueness of cone is asserted from the triangulated axioms alone. Substitute this isomorphism in step 1.1. For $j\ge d$, apply [F3] to obtain exactly the stated residue-field group. This proves the Gysin/coefficient route for the actual support fiber, with its full scope checked before use.