The second-stage projection is minus the actual second Chern map on its homotopy fiber

On the smooth big Zariski site ${\mathrm{S}\mathrm{m}}_X$, evaluate the functorial stages and maps of [F4]. Construct $r$ by the fiber-product inverse of $T\times B\to S$, $(t,b)\mapsto i(t)+\kappa (b)$, using the specified homotopy in [F1]. Together with the actual $Q$ and the functorial universal $C_2$, this defines $D_{\mathrm{b}\mathrm{i}\mathrm{g}}=(Q+C_2{i}_K)r:S\to A$. Its restriction to small-site objects is the specified difference $D$, because every constituent, including the determinant-fiber path, is the same natural construction. Smooth qcqs objects over the noetherian finite-dimensional $X$ have finite presentation and remain noetherian finite-dimensional. Thus all bundle calculations used on this site meet [F1]'s hypotheses.

The rank-zero test is the pointed zero class. For positive ranks the reduced bundle-class construction defines $j_0:BGL={\mathrm{colim}}_{n}BG{L}_n\to S$ on this big site, with $BG{L}_n={\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_n$ after Zariski descent. The composite $D_{\mathrm{b}\mathrm{i}\mathrm{g}}{j}_0$ on ${\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_n$ is a map into the actual spectrum level in [F4], so the adjunction identifies its class with a weight-two cohomology class of that stack. This applies to this particular constructed map; it is no claim that arbitrary small-site maps extend to motivic stacks. For pointed classes, forgetting the basepoint is injective here: the chosen trivial-bundle basepoint splits the coefficient-valued suspension spectrum of ${\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_{n,+}$ into its unit and reduced summands, as also recorded in Asakura–Sato Proposition 5.2. Thus a zero class in the displayed unreduced cohomology also gives a zero pointed map class.

At a Nisnevich local ring $R$, the small-site map $j_0$ is the actual rank-zero plus map $BGL(R)\to BGL(R{)}^{+}$. Projectives over $R$ are free, and the rank target is discrete by [F4]. The component-zero local comparison in [F2] therefore makes $j_0$ an integral-chain equivalence. This uses the actual plus model of this rank fiber; it does not claim that an arbitrary homotopy fiber preserves a homology equivalence.

Pull this class back along $G(n,N)\to {\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_n$, where $G(n,N)$ carries the actual universal rank-$n$ quotient. By derived Yoneda this pullback is the value on that universal bundle as an element of its full cohomology group. Its image under $r{j}_0$ has underlying class $\delta (V)$, and [F1] gives $Q(r([V]-n))+C_2(\delta (V))=0$. These Grassmannians are smooth qcqs over $X$ and remain noetherian finite-dimensional. Their cohomology classes therefore vanish by the bundle calculation, including the bundle descent data.

Apply [F3] on the big site over the qcqs base $X$. Its actual model is homotopy commutative, linearly oriented and satisfies PBF; use the unit coring in Corollary 4.4.7. In mapping homotopy degree $u=0,1$, a Chern monomial of weight $w$ has coefficient ${\mathcal{E}}_X^{4-2w-u,2-w}(X)$, which vanishes for $w>2$ by the actual negative-coefficient lemma. The finite basis on $G(n,N)$ is indexed by $\alpha$ with $\mathrm{\mid }\alpha \mathrm{\mid }={\sum }_j{\alpha }_j\le N-n$ and weight $w={\sum }_{j}j{\alpha }_j$. Every potentially nonzero monomial has $\mathrm{\mid }\alpha \mathrm{\mid }\le 2$, so ambient ranges $N-n\ge 2$ include them all, and the transition maps preserve them. This holds for both mapping degrees. Hence the finite-approximation tower has zero ${\lim }^1{\pi }_1$; the map-induced Grass/Vect comparison detects the class in step 2.1 from the zero restrictions in step 3.1. It follows that $D_{\mathrm{b}\mathrm{i}\mathrm{g}}{j}_0$ is zero on each ${\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_n$.

Stabilization $V\mapsto V\oplus \mathcal{O}$ preserves $[V]-n$, its determinant, and the specified universal Chern maps with their natural identifications. Thus these restrictions are restrictions of a single stable map. In the coefficient formula of step 4.1, variables newly introduced after rank two have weight greater than two. Both mapping ${\pi }_0$ and ${\pi }_1$ therefore stabilize for $n\ge 2$. The Milnor exact sequence for the mapping-spectrum inverse limit has ${\lim }^1{\pi }_1=0$. Its map to $\lim {\pi }_0$ is consequently injective, and the zero rank restrictions imply $D_{\mathrm{b}\mathrm{i}\mathrm{g}}{j}_0\simeq 0$ on stable $BGL$. This separately controls the rank limit, rather than choosing incompatible nullhomotopies at each rank.

Represent the homotopy in step 5.1 in the local simplicial-sheaf model with fibrant derived coefficient target. It is then a homotopy of presheaf maps. Restrict it along the inclusion of qcqs étale $X$-schemes into ${\mathrm{S}\mathrm{m}}_X$, and apply the supplied derived Nisnevich sheafification. At finite rank the restricted bundle stack is the small-site ${\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_n$: both assign to an étale test scheme its groupoid of rank-$n$ bundles and their isomorphisms. Use the canonical map from the presheaf colimit of these restricted finite-rank stacks to the restriction of the big-site stable $BGL$, and precompose the restricted homotopy with it. After Nisnevich sheafification its source is exactly the small-site stable $BGL$, since sheafification preserves this colimit. The target remains the actual $A$ by its derived Nisnevich descent in [F4]; the rank and determinant fibers, retraction, augmentation and stage map are the same natural constructions by [F1] and [F4]. On each finite-rank stack the precomposed map is exactly the reduced bundle-class map followed by $D$, since its constituents were restricted from step 1.1. The resulting homotopy is therefore precisely $D{j}_0\simeq 0$. This uses restriction of the particular constructed big-site homotopy; it requires neither an equivalence of sites nor commutation of restriction with a sheaf colimit.

The derived free-chain adjunction of [F2] makes precomposition with this $j_0$ injective on homotopy classes of maps into $A$. Steps 6.1 and 2.2 imply $D\simeq 0$ on $S$. Restrict along $i:T\to S$ and use $ri\simeq \mathrm{i}\mathrm{d}$ from [F1] to obtain $Q+C_2{i}_K\simeq 0$. Evaluation on derived sections proves the asserted equality, retaining all gluing morphisms. No global bundle-difference presentation of a perfect class is used.