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 and cotangent bundles are smooth vector bundles

Example

Assume ${A C}_{ω}$ . For a smooth manifold $M$ , the tangent bundle $T M \to M$ and the cotangent bundle $T^{*} M \to M$ are smooth vector bundles of rank $\dim M$ .

Facts & Assumptions

Given: The axiom ${A C}_{ω}$ and a smooth manifold $M$ .

[L1]

The tangent bundle has its canonical smooth $2 n$ -manifold structure (Assuming countable choice, the tangent bundle has a canonical smooth 2n-manifold structure).

[L2]

The cotangent bundle has its canonical smooth $2 n$ -manifold structure (Assuming countable choice, the cotangent bundle has a canonical smooth 2n-manifold structure).

Verification

technique · direct

1.1

The earlier tangent-bundle construction gives local coordinates of the form $(p, v) \in U \times R^{n}$ , and the fibre over $p$ is the vector space $T_{p} M$ . Thus $T M \to M$ is a smooth rank- $n$ vector bundle.

L1given

2.1

The cotangent-bundle construction gives the same kind of local description with fibres $T_{p}^{*} M$ , so $T^{*} M \to M$ is likewise a smooth rank- $n$ vector bundle.

L2step 1.1∎

Depends on

Used by

Assuming countable choice, the tangent bundle of the circle is trivial Example

Dependency tree · two levels

15 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)