ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-31

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.

Assuming countable choice, the normal bundle of the sphere is trivial

Example

Assume ${A C}_{ω}$ . For the unit sphere $S^{n} \subseteq R^{n + 1}$ , the normal bundle is a trivial line bundle.

Facts & Assumptions

Given: The axiom ${A C}_{ω}$ and the unit sphere $S^{n} \subseteq R^{n + 1}$ with the Euclidean metric.

[L1]

The normal bundle is a smooth vector bundle and Euclidean orthogonality identifies it with the orthogonal normal line bundle (Assuming countable choice, normal and conormal bundles are smooth vector bundles, Assuming countable choice, an ambient metric identifies the two normal bundles).

[L2]

A vector bundle is trivial exactly when it has a global frame (A vector bundle is trivial if and only if it has a global frame).

[L3]

The tangent space of a regular level set is the kernel of the defining map's differential (The tangent space of a regular level set is the kernel).

Verification

technique · direct

1.1

Write $S^{n}$ as the regular level set of $F (z) = ⟨ z, z ⟩$ . Since $d F_{x} (v) = 2 ⟨ x, v ⟩$ , [L3] gives $T_{x} S^{n} = x^{⊥}$ . Its Euclidean orthogonal complement is therefore the one-dimensional space $R x$ , so [L1] identifies the normal bundle with a line bundle spanned by the radial vector.

L1L3givenalgebra

2.1

The smooth section $x \mapsto x$ is nowhere zero and spans that line at every point, so it is a global frame. Therefore [L2] implies that the normal bundle of $S^{n}$ is trivial.

L2step 1.1∎

Depends on

Used by

The hairy-ball theorem for even-dimensional spheres Remark

Dependency tree · two levels

26 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 (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)