ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-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.

Degree of the antipodal map in low dimensions

Example

The antipodal maps on $S^{1}, S^{2}, S^{3}$ have degrees $+ 1, - 1, + 1$ , respectively. Of these three spheres, exactly $S^{1}$ and $S^{3}$ admit a continuous nowhere-zero tangent vector field.

Facts & Assumptions

Given: The objects and hypotheses in the statement above.

[F1]

On $S^{n} \subset R^{n + 1}$ for $n \geq 1$ , the identity, a constant map, any single coordinate reflection, and the antipodal map have degrees $1$ , $0$ , $- 1$ , and $(- 1)^{n + 1}$ respectively. (Degree of identity constant reflection and antipodal sphere maps)

[F2]

For every integer $n \geq 1$ , $S^{n}$ admits a continuous nowhere-zero tangent vector field if and only if $n$ is odd. (A sphere has a nowhere zero tangent vector field iff its dimension is odd)

Verification

1.1F1algebra

In $R^{n + 1}$ , negating all coordinates is the composite of its $n + 1$ coordinate reflections. The antipodal formula in [F1] gives $(- 1)^{2} = + 1$ on $S^{1}$ , $(- 1)^{3} = - 1$ on $S^{2}$ , and $(- 1)^{4} = + 1$ on $S^{3}$ .

2.1F2construct∎

By [F2], the odd dimensions $1$ and $3$ permit such a field and dimension $2$ does not. Concretely, identify $R^{2} = C$ and $R^{4} = C^{2}$ and set $v (z) = i z$ . It has norm $1$ on the sphere and real inner product $Re \sum {\overline{z}}_{j} i z_{j} = 0$ with $z$ , so is tangent and nowhere zero.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Hatcher, Algebraic Topology, Degree property (f) and Theorem 2.28, pp.134–135 (standard reference, not scraped)