DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedaudited 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.

Smooth charts, atlases, and structures with boundary

Definition

A boundary chart is a homeomorphism $φ : U \to V \subseteq H^{n}$ , where $U \subseteq M$ is open and $V$ is relatively open. Two charts are compatible if each transition map is smooth in the local-extension sense. A smooth atlas is a compatible covering atlas; its smooth structure is its maximal compatible atlas.

Smooth charts, atlases, and structures with boundary

Definition

Depends on

Dependency tree · 2 levels

Sources