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 a coordinate reflection on a sphere

Example

For $n \geq 1$ , a coordinate reflection $r : S^{n} \to S^{n}$ has degree $- 1$ . A reflection-invariant triangulation exhibits this sign by sending its oriented fundamental cycle to its negative.

Facts & Assumptions

Given: The objects and hypotheses in the statement above.

[F2]

For an abstract simplicial complex $K$ and an integer $n \geq 0$ , the simplicial chain group $C_{n} (K)$ is the free abelian group generated by the oriented $n$ -simplices of $K$ , subject to the relation $[v_{π (0)}, \dots, v_{π (n)}] = sgn (π) [v_{0}, \dots, v_{n}]$ for every permutation $π$ of the vertices of a simplex. For $n < 0$ , set $C_{n} (K) = 0$ . The boundary operator $\partial_{n} : C_{n} (K) \to C_{n - 1} (K)$ is $\partial_{0} = 0$ in degree $0$ . For $n \geq 1$ , it is defined on an oriented simplex by $\partial_{n} [v_{0}, \dots, v_{n}] = \sum_{i = 0}^{n} (- 1)^{i} [v_{0}, \dots, \hat{v_{i}}, \dots, v_{n}] .$ The well-definedness of this formula with respect to the chosen oriented representative is recorded in lem-simplicial-boundary-is-independent-of-oriented-representative through justified_by. (Simplicial chain groups and the boundary operator)

[F3]

For every simplicial complex $K$ , the natural simplicial-to-singular chain map induces $H_{n}^{s i m p} (K; G) ≅ H_{n} (∣ K ∣; G)$ for all $n$ . (Simplicial and singular homology agree)

Verification

1.1

Take a regular $(n + 1)$ -simplex in $R^{n + 1}$ centered at the origin, with vertices $v_{0}, \dots, v_{n + 1}$ . Radial projection of its boundary to $S^{n}$ is a homeomorphism: each ray meets its convex boundary exactly once, and the continuous bijection has compact source and Hausdorff target. The orthogonal symmetry interchanging $v_{0}, v_{1}$ and fixing the other vertices is reflection across the perpendicular bisector hyperplane. Rotate this configuration so that this hyperplane is the coordinate hyperplane of $r$ ; radial projection then commutes with the reflection.

construct

1.2

With the orientation convention of [F2], put $c = \partial [v_{0}, \dots, v_{n + 1}] = \sum_{j = 0}^{n + 1} (- 1)^{j} [v_{0}, \dots, {\hat{v}}_{j}, \dots, v_{n + 1}]$ . Each codimension-two face occurs twice with opposite signs, so $\partial c = 0$ . Conversely, cancellation along every common facet forces the coefficients of any top cycle to be a common integer multiple of those of $c$ . There are no $(n + 1)$ -simplices in the boundary, so $c$ generates its top simplicial homology and hence its oriented fundamental class under the natural simplicial-to-singular isomorphism [F3].

F2F3algebra

2.1

The vertex transposition $τ$ commutes with the boundary formula. In the filled simplex its action on $[v_{0}, \dots, v_{n + 1}]$ is multiplication by $- 1$ . Thus $τ c = \partial τ [v_{0}, \dots, v_{n + 1}] = - c$ . Radial projection carries this equality to $r_{*} [S^{n}] = - [S^{n}]$ , independently of which sign was chosen for the generator.

F2step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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 (e), p.134 (standard reference, not scraped)