ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

Spheres as SO(n+1)/SO(n)

Example

Assume ${A C}_{ω}$ . For every $n \geq 1$ , the standard action gives a canonical $S O (n + 1)$ -equivariant diffeomorphism

$S O (n + 1) / S O (n) ≅ S^{n} .$

Here $S O (n)$ is embedded as $diag (A, 1)$ .

Facts & Assumptions

Given: ${A C}_{ω}$ , $n \geq 1$ , and the standard linear action of $S O (n + 1)$ on the unit sphere $S^{n} \subseteq R^{n + 1}$ .

[A1]

A transitive smooth action identifies the manifold equivariantly with the quotient by a point stabilizer. The Axiom of Countable Choice ( ${A C}_{ω}$ ), Transitive smooth actions identify M with G/H.

Verification

technique · prove transitivity and compute the stabilizer

1.1

The action is smooth and preserves $S^{n}$ . Given $u, v \in S^{n}$ , extend each to an oriented orthonormal basis whose last vector is respectively $u$ and $v$ ; when necessary, changing the sign of one of the first $n$ vectors corrects the orientation. The linear map carrying the first oriented basis to the second is in $S O (n + 1)$ and sends $u$ to $v$ . Thus the action is transitive.

givenalgebra

1.2

A matrix in $S O (n + 1)$ fixes $e_{n + 1}$ exactly when it preserves $e_{n + 1}^{⊥}$ and has block form $diag (A, 1)$ . Orthogonality and determinant one then say precisely $A \in S O (n)$ . Hence the stabilizer is the displayed copy of $S O (n)$ .

givenalgebra

2.1

Apply [A1] at $e_{n + 1}$ . The map $g S O (n) \mapsto g e_{n + 1}$ is the asserted equivariant diffeomorphism. For $n = 1$ , $S O (1) = {1}$ and the quotient is $S O (2) ≅ S^{1}$ . The excluded value $n = 0$ also has the analogous point quotient if one adopts $S O (0) = S O (1) = {1}$ , but it is not needed for the stated family. Countable choice is used through [A1].

A1step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)
Pavel Etingof, MIT 18.745 Lie Groups and Lie Algebras I (standard reference, not scraped)