ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.

The Mobius line bundle from a transition function

Example

The Mobius line bundle over $S^{1}$ is obtained by gluing two trivial line bundles over the standard arcs with transition function $1$ on one overlap component and $- 1$ on the other.

Facts & Assumptions

Given: The two-arc cover $U_{0} = S^{1} ∖ {(- 1, 0)}$ and $U_{1} = S^{1} ∖ {(1, 0)}$ of the circle.

[L1]

A smooth cocycle on a countable cover constructs a smooth vector bundle (Construction of a vector bundle from a smooth cocycle).

[L2]

The claim that every vector bundle is globally trivial is false (Every vector bundle is globally trivial).

Verification

technique · direct

1.1

The overlap $U_{0} \cap U_{1}$ has two connected components. Defining the rank-one transition function to be $1$ on the upper component and $- 1$ on the lower one gives a locally constant smooth cocycle, so [L1] constructs a smooth line bundle $L \to S^{1}$ .

L1givenconstruct

2.1

The refutation recorded in [L2] is exactly the argument that this line bundle admits no global frame, so $L$ is not trivial. This smooth nontrivial line bundle is the Mobius bundle.

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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)