DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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 degree of a based circle loop

Definition

Let $α : I \to R / Z$ be a based loop at $[0]$ . Since $p : R \to R / Z$ is a covering map ( $p : R \to R / Z$ is a covering map with translated interval sheets), path lifting (Existence and uniqueness of path lifts through a covering map) gives a unique lift $\tilde{α} : I \to R$ with $\tilde{α} (0) = 0$ and $p \circ \tilde{α} = α$ ; this is a lift in the sense of Lifts of maps, paths, and homotopies through a covering map.

Because $α (1) = [0]$ , its terminal value satisfies $\tilde{α} (1) \in p^{- 1} ([0]) = Z$ by The circle as $S^{1} = R / Z$ with basepoint $[0]$ . Define $\deg (α) = \tilde{α} (1)$ .

This defines degree on based loops. Its independence from a representative of a path-homotopy class is proved separately before degree is used on $π_{1} (S^{1}, [0])$ .

The degree of a based circle loop

Definition

Depends on

Dependency tree · 2 levels

Sources