External non-A¹ projective-relation filtered and Whitney theorem

PDF p. 8: Actual Chern and Whitney identities; PDF p. 11: Total virtual Chern class

Let $\mathcal{E}$ be a homotopy-commutative linearly oriented motivic ring spectrum over a qcqs derived scheme $S$, satisfying its actual class-power PBF. For a bundle $V$ of positive constant rank $r$ on $Y\in {\mathrm{S}\mathrm{m}}_S$, define ${\tilde{c}}_i(V)$ by the quotient-projective relation ${\sum }_{j=0}^r(-1{)}^j{\xi }^j{p}^{\ast }{\tilde{c}}_{r-j}(V)=0,{\tilde{c}}_0=1.$ Then a subbundle filtration with line quotients $L_s$ satisfies ${\tilde{c}}_t(V)={\prod }_s(1+c_1(L_s)t)$, and every fiber sequence of vector bundles $V^{\mathrm{\prime }}\to V\to V^{\mathrm{\prime }\mathrm{\prime }}$ satisfies ${\tilde{c}}_t(V)={\tilde{c}}_t(V^{\mathrm{\prime }}){\tilde{c}}_t(V^{\mathrm{\prime }\mathrm{\prime }})$.

For the zero bundle adopt ${\tilde{c}}_t(0)=1$, and read coefficients above the degree of any Chern polynomial as zero. These are explicit local coefficient conventions extending the positive-rank definition. With them both identities also hold when a bundle has rank zero: a zero end term in a fiber sequence makes the other map an isomorphism, and the rank-zero filtered formula is the empty product $1$.

The positive-rank input is Definition 4.4.1 and Lemma 4.4.3(ii)–(iii) of the April 14, 2025 author PDF at printed pages 30–31. Its quotient convention and smooth-qcqs relative site are as in section 2.1. The rank-zero extension and the polynomial coefficient convention are local, rather than assertions that Definition 4.4.1 defines a zero-bundle class. For a zero end term, the isomorphism just noted preserves the positive-rank coefficients because their projective relation and orientation are preserved; if the middle bundle is zero, all three bundles are zero.

The local chain verifies the model, orientation, product and PBF hypotheses and applies the input with base $S=X$, evaluated at the smooth identity $X\to X$. Thus smooth-relative scope imposes no smoothness of $X\mathrm{/}k$. The substantial geometric splitting and vanishing argument is imported through the stated source result; its proof and upstream machinery are not reconstructed here.