Actual perfect Cartan through degree three on all qcqs field-schemes

Fix $j\le 3$. Form the Cartan-difference map from $\mathcal{K}\times \mathcal{K}$ to $A_j$. Its restriction to ${\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_n\times {\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_m$ vanishes on every pair of finite Grassmannian approximations by actual bundle Whitney [F2]. These approximations are smooth qcqs over $X$; no finite-type hypothesis on $X\mathrm{/}k$ is needed. The actual universal coordinate comparison in [F1] detects this vanishing on the product stacks.

Stabilize in both ranks. A monomial of total Chern weight $w>j$ has coefficient in negative weight $j-w$, zero in every cohomological degree by [F2]. Thus the mapping-space ${\pi }_0$ and ${\pi }_1$ systems stabilize once both ranks are at least $j$. The inverse-limit sequence has zero ${\lim }^1{\pi }_1$, and the vanishing passes to $BGL\times BGL$ without phantom terms. This is the precise bounded-weight consequence of [F1], not an assumed finitarity of motivic cohomology.

The actual class maps ignore trivial rank and the free-class comparison matches addition with block sum. Hence the difference pulls back to zero along $j\times j$. Apply [F3] to obtain zero as a derived sheaf-map class. Evaluating on $X$ gives the formula on every perfect class, irrespective of global bundle representability.

Pointedness gives $c_j(0)=0$ for $j>0$. All diagonal first degrees are even, so the target coefficient ring is commutative; truncated series with constant term one have unique recursive inverses. Apply Cartan to $x+(-x)=0$ to obtain formal inversion. The same proof works for each fixed finite $j$, but no claim about a separately chosen E∞ total-series enhancement is required.