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.

Pullback of the tautological line bundle along the antipodal cover

Example

Let $q : S^{n} \to {R P}^{n}$ be the antipodal quotient map. The pullback of the tautological line bundle $q^{*} γ^{1} \to S^{n}$ is trivial.

Facts & Assumptions

Given: The antipodal quotient map $q : S^{n} \to {R P}^{n}$ .

[L1]

The tautological bundle $γ^{1} \to {R P}^{n}$ is the line of scalar multiples of the represented vector (The tautological line bundle over real projective space).

[L2]

A rank-one vector bundle is trivial once it has a nowhere-zero global frame (A vector bundle is trivial if and only if it has a global frame).

Verification

technique · direct

1.1

A point of $q^{*} γ^{1}$ is a pair $(x, ([x], v))$ with $v \in R x$ . Define a section $s : S^{n} \to q^{*} γ^{1}$ by $s (x) = (x, ([x], x))$ . This is well defined because $x$ lies in the line represented by $[x]$ .

L1givenconstruct

2.1

The section $s$ is nowhere zero. Since $q^{*} γ^{1}$ has rank $1$ , it is a global frame, so [L2] implies that the pulled-back tautological bundle is trivial.

L2step 1.1∎

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 (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Rob van der Vorst, Introduction to differentiable manifolds (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)