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.

The double of a smooth manifold with boundary

Definition

The double $D M$ is the quotient of the labelled disjoint union $M_{+} ⊔ M_{-}$ by $(p, +) \sim (p, -)$ precisely for $p \in \partial M$ . The labels remain part of the construction.

The double of a smooth manifold with boundary

Definition

Depends on

Dependency tree · 2 levels

Sources