Clopen rank extension and genuine representative independence

Each bundle has a locally constant rank function, so the common rank loci for a finite bundle family are pairwise disjoint open-and-closed subschemes covering $X$. Quasi-compactness gives a finite subcover, which contains every nonempty locus by disjointness. Each locus is quasi-compact because it is closed in $X$, and quasi-separated because it is open in $X$. Thus this is a finite clopen partition by qcqs $k$-schemes. For empty $X$ use the empty partition.

Let $F$ be the prime subfield of $k$. Forgetting to $F$ changes neither the underlying qcqs schemes nor the clopen cover and its fiber products. For every $j\ge 0$, [F2] therefore supplies the actual derived Nisnevich sheaf and applies finite-clopen descent to exactly this cover. It gives $\mathbb{Z}(j{)}_{\mathrm{m}\mathrm{o}\mathrm{t}}(X)\simeq {\prod }_{\alpha }\mathbb{Z}(j{)}_{\mathrm{m}\mathrm{o}\mathrm{t}}(X_{\alpha })$ by restriction. Taking cohomology and using finite-product compatibility in [F2] gives $H^{a,j}(X)\cong {\prod }_{\alpha }{H}^{a,j}(X_{\alpha })$ for every integer $a$. The same holds for every finite clopen refinement. Empty covers give the zero complex and zero groups. This argument imposes no algebraic-closure assumption.

For each $i\ge 1$, use the isomorphism of step 2.1 in bidegree $(2i,i)$ to define $c_i$ as the unique class restricting to the constant-rank values from [F1]. Refinement leaves this class unchanged by uniqueness and naturality of those values. Use the unit for $c_0$. For nonempty $X$ the rank has a finite maximum, so coefficients above that maximum vanish by injectivity of restriction; on empty $X$ all coefficient groups are zero and the series is the prescribed unit. Thus the actual series is a polynomial in every case.

Pullback refines the rank partition and preserves the constant-rank classes, hence preserves the glued classes by step 2.1. For an exact sequence, use a common rank partition. On each component apply actual Whitney [F1]; joint injectivity from step 2.1 gives the global identity coefficient by coefficient. The same argument preserves isomorphism invariance.

All hypotheses of [F3] now hold. Its construction kills every exact-sequence relation in the presentation of $K_{\mathrm{v}\mathrm{b}}^0(X)$, so descends to a unique homomorphism. Its value on any difference is the stated quotient, with the inverse defined recursively in formal series, not by convergence. Two differences representing the same class have the same image because this is a map on the presented group. Its naturality also follows from [F3]. Thus this is genuine representative independence.

The degree-zero series group is commutative because all coefficient first degrees are even. The product and inverse formulas through degree3 in [F3] therefore apply. In particular higher coefficients need not be additive, whereas $c_1$ is additive on virtual sums and $c_2(x+y)=c_2(x)+c_1(x)c_1(y)+c_2(y)$.