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 tangent bundle of the circle is trivial

Example

Assume ${A C}_{ω}$ . The tangent bundle of $S^{1}$ is a trivial line bundle.

Facts & Assumptions

Given: The axiom ${A C}_{ω}$ and the circle $S^{1} \subseteq R^{2}$ .

[L1]

The tangent bundle of the circle is a smooth rank-one vector bundle (Assuming countable choice, the tangent and cotangent bundles are smooth vector bundles).

[L2]

A smooth 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).

Verification

technique · direct

1.1

Define $X : S^{1} \to T S^{1}$ by $X (x, y) = (- y, x)$ . This vector is tangent to $S^{1}$ at $(x, y)$ because it is orthogonal to the radial vector $(x, y)$ , and it is never zero on the circle.

L1givenconstruct

2.1

Because $T S^{1}$ has rank $1$ , the nowhere-zero tangent field $X$ is a global frame. Hence [L2] shows that $T S^{1}$ is trivial.

L2step 1.1∎

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

John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)