TheoremStatement: Literature-sourcedProof: Literature-sourced

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

External non-A¹ projective-relation filtered and Whitney theorem

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

Statement

Let $E$ be a homotopy-commutative linearly oriented motivic ring spectrum over a qcqs derived scheme $S$ , satisfying its actual class-power PBF. For a rank- $r$ bundle $V$ on $Y \in {S m}_{S}$ , define ${\tilde{c}}_{i} (V)$ by the quotient-projective relation $\sum_{j = 0}^{r} (- 1)^{j} ξ^{j} p^{*} {\tilde{c}}_{r - j} (V) = 0, {\tilde{c}}_{0} = 1.$ 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 $V^{'} \to V \to V^{''}$ satisfies ${\tilde{c}}_{t} (V) = {\tilde{c}}_{t} (V^{'}) {\tilde{c}}_{t} (V^{''})$ .

External input

This is Definition4.4.1 and Lemma4.4.3(ii)–(iii). The local chain verifies its model, orientation, product and PBF hypotheses and applies it with base $S = X$ , evaluated at the smooth identity $X \to X$ . Thus the theorem’s smooth-relative scope does not impose smoothness of $X / k$ . The substantial geometric splitting/vanishing argument is imported; no unsupplied A¹-invariance or affine-torsor detection is used.

Depends on

Used by · 2

Dependency tree · 2 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.

External dependencies

Annala–Iwasa, Motivic spectra and universality of K-theory (author PDF,14April2025) · Definition4.4.1 and Lemma4.4.3(ii)–(iii), full proof pp30–31; §2.1.5 quotient convention and Lemma3.2.3

Classes are coefficients of Σ_(j=0)^r (−1)^j ξ^j p*c_(r−j)(V)=0, c0=1. A line-quotient filtration gives c(V)=Π(1+c1(L_s)t); any fiber sequence of vector bundles gives c(V)=c(V′)c(V″).

Hypotheses: Homotopy-commutative linearly oriented motivic ring spectrum over any qcqs derived base S, satisfying the source PBF; bundle on Y∈Sm_S. No A¹-invariance or finite quasi-smooth-transfer hypothesis.

Reading limits: Read Definition4.4.1 and full Lemma4.4.3 proof across pp30–31, including its use of the splitting trick, sections on the quotient projective bundle, empty common zero locus and formal group law. Those geometric steps are imported as an established theorem, not claimed reconstructed from a flag filtration alone. Read Lemma3.1.8 fully and the relevant projective conventions. The prior AI22 proof gap mentioned by the source is historical and corrected in this version; no new independent defect diagnosis is made.

Sources

Annala–Iwasa, Motivic spectra and universality of K-theory (author PDF,14April2025) · Definition4.4.1 and Lemma4.4.3(ii)–(iii), full proof pp30–31; §2.1.5 quotient convention and Lemma3.2.3