Actual exact-sequence Whitney and filtered-root formula on singular qcqs schemes

Take the source base to be $S=X$, regarded as a classical derived scheme, and take $Y=X$ with the identity smooth map. Smooth derived schemes over a classical base are classical, so the EM restriction used in the model is defined on this smooth site. No assertion that $X$ is smooth over $k$ is needed. The comparison applies because $S=X$ is qcqs over $k$ and $Y=X$ is smooth qcqs over $S$. The model supplier in [F2] gives the homotopy-commutative motivic ring spectrum, and its orientation supplier gives the linear orientation and actual PBF. These are precisely the hypotheses needed to apply [F1]; the separate relation comparison in [F2] then identifies its Chern coefficients with the supplied classes.

If either outer bundle has rank zero, it is the zero sheaf, and exactness identifies the middle bundle with the other outer bundle. Whitney then follows directly from $c_t(0)=1$ in [F2]. Otherwise all three ranks are positive. The sequence is locally split since its quotient is locally free, so its sheaf-complex triangle is a fiber sequence of vector bundles as required by [F1]. Apply part (iii) and transport the equality through [F2]. This proves Whitney in every stated rank.

For a nonempty line-quotient filtration apply part (ii) of [F1] and transport through [F2] to obtain the displayed root factorization. For the empty filtration the bundle is zero, and the empty product and its locally prescribed Chern series are both $1$. The substantial geometric splitting argument remains supplied by [F1]. This proves the filtered formula with the same rank-zero convention as in step 2.1.