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.

Smooth families of maps and their evaluation maps

Let $M$ , $S$ , and $N$ be smooth manifolds. A smooth family of maps from $M$ to $N$ parametrized by $S$ is a smooth map

$F : M \times S \to N .$

For each $s \in S$ , its slice map is

$F_{s} : M \to N, F_{s} (p) : = F (p, s) .$

The map $F$ itself is the evaluation map of the family.

Depends on