DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Closed sublevel and level set of a smooth function

Definition

Let $f : M \to R$ be smooth on a boundaryless smooth $n$ -manifold. Write $M^{a} = f^{- 1} ((- \infty, a])$ , $M_{a} = f^{- 1} ({a})$ , and $f^{- 1} ([a, b])$ for the closed band. Both endpoints are included. A regular value may have empty fiber. The smooth-manifold convention is Smooth manifolds and their smooth charts.

Depends on

Smooth manifolds and their smooth charts

Closed sublevel and level set of a smooth function

Definition

Depends on

Dependency tree · 2 levels

Sources