Motivic graded pieces and spectral sequence

Under [A1], let $F$ be the prime field of $k$. Composing structure morphisms gives a functor from qcqs $k$-schemes to qcqs $F$-schemes: underlying schemes are unchanged, and every $k$-morphism is an $F$-morphism. By the local definitions, both the tower and motivic complexes are restrictions along this functor. Restrict the actual natural equivalence ${\gamma }_r$ and the stage and quotient maps of [F4]. Cofibers of functor diagrams are evaluated objectwise, so the canonical quotient followed by ${\gamma }_r$ gives the asserted triangle. For $r<0$, the constant nonpositive stages make the transition an equivalence, including at $r=-1$, hence its cofiber is zero. This proves the negative-weight zero convention without a new geometric hypothesis.

Apply the generic tower construction in [F1] to these actual diagrams. Its initial page is ${\pi }_{-a-b}{\mathrm{gr}}^{-b}$, and its cochain shift calculation is ${\pi }_n(C[2r])=H^{2r-n}(C)$. Substitution of $r=-b$ and $n=-a-b$ gives $2r-n=a-b$. Thus the page is the stated motivic group, with zero terms for negative weights. The tower reindexing gives the displayed differential; its target has total homotopy degree $-(a+s)-(b-s+1)=-a-b-1$.

Suppose $X$ has finite valuative dimension. Forgetting its $k$-structure changes neither its affine opens nor their rings, so the dimension condition in [F3] holds for its prime-field scheme. Apply the stage bound in [F4] to this very tower and choose its integer $B=B(X)\ge 0$. For any integer $n$, put $U_n=\max (1,B+n+1)$ and $L_n=0$. If $r\ge U_n$, then $n<r-B$, so ${\pi }_n{F}^r=0$; if $r\le L_n$, the map to $K(X)$ is an equivalence. The generic tower interface in [F1] therefore gives finite convergence and its associated-graded identification, with the actual image filtration. This argument also applies to the empty scheme under the same source contract.

Restrict the coherent filtered products of [F4] along the same functor. On the associated graded, their transport along the restricted ${\gamma }_r$ is precisely the source motivic product. The source and local exact couples use the same restricted stage/cofiber maps and connecting homomorphisms. The identity on stages and the actual graded comparisons therefore commute with all exact-couple arrows. Deriving couples takes images and kernel quotients functorially, so induction identifies all pages and differentials, as in the construction recorded in [F2]. Transport the source pairings through these page identifications. Products add bidegrees by step 2.1. The derivation sign is the parity of homotopy degree $-a-b$, hence $(-1{)}^{a+b}$; the shifts by $2r$ are even. This establishes the claimed products and Leibniz rule using actual source pairings.

Repeat the zero-limit calculation of [F3] on these restricted diagrams. The sequential limit is the fiber of $1-\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}$ on ${\prod }_{r\ge 0}{F}^r$. In each degree $n$, both ${\pi }_n{F}^r$ and ${\pi }_{n+1}{F}^r$ are eventually zero by step 2.2. Each of the resulting products of groups has only finitely many possibly nonzero factors; the transition shift is nilpotent, so $1-\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}$ is invertible by a finite geometric sum. The fiber exact sequence makes every homotopy group of the limit zero, hence the limit spectrum is zero. The image filtration is full for $r\le 0$ and zero for $r\ge U_n$. The transported source abutment pairings in step 3.1 identify its products with those in $K_{\ast }(X)$. This proves completeness and multiplicative convergence with the full stated scope.