ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passaudited 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.

Boundary orientation of the unit sphere by the outward normal

Example

For $n \geq 1$ , the boundary orientation of $S^{n - 1} = \partial B^{n}$ is the standard hypersurface orientation for which the radial outward normal is first.

Facts & Assumptions

Given: An integer $n \geq 1$ , the closed unit ball $B^{n} \subset R^{n}$ with the orientation induced by the standard ordered basis of $R^{n}$ , and its boundary $S^{n - 1}$ .

[L1]

The induced boundary orientation is outward-normal-first (Induced boundary orientation).

[L2]

The boundary of $B^{n}$ is $S^{n - 1}$ (The closed ball and its sphere boundary).

[L3]

The standard oriented interval satisfies $\partial [- 1, 1] = {1} - {- 1}$ (Boundary orientation is independent of the outward vector field).

Verification

technique · direct

1.1

At $u \in S^{n - 1}$ , the radial vector $u$ points outward from $B^{n}$ .

givenL2

2.1

If $n = 1$ , then $B^{1} = [- 1, 1]$ and [L3] gives the positive determinant ray at $1$ and the negative determinant ray at $- 1$ , exactly as the outward vectors $1$ and $- 1$ require under [L1]. If $n \geq 2$ , a tangent basis $(v_{1}, \dots, v_{n - 1})$ is positive precisely when $(u, v_{1}, \dots, v_{n - 1})$ is positive in the standard orientation of $R^{n}$ . Thus in every case the boundary orientation is the standard hypersurface orientation cooriented by the radial outward normal.

givenL1L3step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Ioan Mărcuț, Manifolds (2017 lecture notes), §§14.5, 15.1 (standard reference, not scraped)
Will Merry, Differential Geometry (2021), Lecture 24 (standard reference, not scraped)