ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-29

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 $n$ -sphere with its standard smooth atlas

Example

For $n \geq 1$ , let

$S^{n} : = {(x_{1}, \dots, x_{n + 1}) \in R^{n + 1} : x_{1}^{2} + \dots + x_{n + 1}^{2} = 1} .$

With north and south poles $N = (0, \dots, 0, 1)$ and $S = (0, \dots, 0, - 1)$ , the stereographic charts

$σ_{N} (x) = \frac{(x_{1}, \dots, x_{n})}{1 - x_{n + 1}}, σ_{S} (x) = \frac{(x_{1}, \dots, x_{n})}{1 + x_{n + 1}}$

define a smooth atlas on $S^{n}$ . Their overlap transition is $u \mapsto u / ∥ u ∥^{2}$ on $R^{n} ∖ {0}$ .

Facts & Assumptions

Given: The sphere $S^{n}$ and the two stereographic maps $σ_{N}, σ_{S}$ .

[F1]

A smooth atlas is a covering family of pairwise smoothly compatible charts (Smooth atlases).

[F2]

Smooth compatibility means that both transition maps are smooth on the overlap (Smoothly compatible charts and the smoothness of Euclidean transition maps).

[F3]

A smooth manifold is a topological manifold equipped with a smooth structure (Smooth manifolds and their smooth charts).

Verification

technique · direct

1.1

The domains $S^{n} ∖ {N}$ and $S^{n} ∖ {S}$ are open and cover [given] $S^{n}$ . The inverse formulas $σ_{N}^{- 1} (u) = (\frac{2 u}{1 + ∥ u ∥^{2}}, \frac{∥ u ∥^{2} - 1}{1 + ∥ u ∥^{2}}), σ_{S}^{- 1} (u) = (\frac{2 u}{1 + ∥ u ∥^{2}}, \frac{1 - ∥ u ∥^{2}}{1 + ∥ u ∥^{2}})$ show that both maps are homeomorphisms onto $R^{n}$ .

given

2.1

On the overlap the transition maps are $σ_{S} \circ σ_{N}^{- 1} (u) = \frac{u}{∥ u ∥^{2}}, σ_{N} \circ σ_{S}^{- 1} (u) = \frac{u}{∥ u ∥^{2}},$ defined on $R^{n} ∖ {0}$ , so they are smooth rational maps there. Hence the two stereographic charts are smoothly compatible by [F2], and [F1] makes them a smooth atlas on $S^{n}$ .

F1F2step 1.1

3.1

Therefore $S^{n}$ is a smooth manifold by [F3].

F3step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Rob van der Vorst, Introduction to differentiable manifolds, §1, Example 1.8 (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds, §2.3 (standard reference, not scraped)

The nn-sphere with its standard smooth atlas