DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

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.

Dual and Hom vector bundles

Let $E \to M$ and $F \to M$ be smooth vector bundles over the same base.

The dual bundle $E^{*} \to M$ has fibre $E_{p}^{*}$ over $p \in M$ .

The Hom bundle $Hom (E, F) \to M$ has fibre $Hom (E_{p}, F_{p})$ over $p \in M$ .

The next theorem equips these fibrewise constructions with smooth vector-bundle structures.

Depends on