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. By [F3] the model is a homotopy-commutative linearly oriented motivic ring spectrum with the actual PBF. The diagonal identifications preserve its products, and [F2] identifies the projective-relation coefficients with the supplied Chern classes.

The exact sequence is locally split since its quotient is locally free. Its associated sheaf-complex triangle is therefore a fiber sequence of vector bundles as required by [F1]. When both end terms have positive rank, apply part (iii) and transport the equality through [F2]. If an end term has rank zero, it is the zero sheaf and the other map is an isomorphism; isomorphism invariance and the convention $c_t(0)=1$ prove the same identity, including the case in which all terms are zero. This proves Whitney in the required non-A¹ theory.

For a positive-rank filtered bundle, apply part (ii) of [F1] to the filtration to obtain the displayed root factorization, again using [F2]. The geometric splitting trick underlying this implication is imported through that lemma, not inferred from a mere flag or K-zero equality. For the zero bundle the empty filtration has empty product $1=c_t(0)$, so the formula includes rank zero explicitly.