DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-6-luna)

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.

A smooth map transverse to an embedded submanifold

Definition

Let $F : M \to N$ be smooth, and let $Z \subseteq N$ be an embedded submanifold. Then $F$ is transverse to $Z$ , written $F ⋔ Z$ , when for every $p \in F^{- 1} (Z)$ ,

$d F_{p} (T_{p} M) + T_{F (p)} Z = T_{F (p)} N .$

If $F (M) \cap Z = \emptyset$ , this condition is vacuous, so $F ⋔ Z$ still holds.

A smooth map transverse to an embedded submanifold

Definition

Depends on

Dependency tree · 2 levels

Sources