ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

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The cotangent bundle of a circle as a symplectic cylinder

Example

Assume ${A C}_{ω}$ . The cotangent bundle of the circle is the symplectic cylinder

$T^{*} S^{1} ≅ S^{1} \times R, λ = p d θ, ω_{c a n} = d θ \land d p .$

Facts & Assumptions

Given: The standard angular atlas of $S^{1}$ and its induced cotangent coordinates.

[F1]

In cotangent coordinates, $λ = p_{i} d q^{i}$ . The tautological one-form is intrinsic and smooth.

[F2]

In cotangent coordinates, $ω_{c a n} = d q^{i} \land d p_{i}$ . The canonical cotangent two-form is symplectic.

Verification

technique · direct

1.1

On overlaps, angular coordinates differ by a locally constant multiple of $2 π$ , so $d θ^{'} = d θ$ and the fibre coefficient is $p^{'} = p$ . Thus the local products glue to $S^{1} \times R$ , and $p d θ$ is global.

givenalgebra

2.1

Applying [F1] and [F2] gives $λ = p d θ$ and $ω_{c a n} = - d (p d θ) = d θ \land d p$ , the standard area form on the cylinder.

F1F2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Ana Cannas da Silva, Lectures on Symplectic Geometry (standard reference, not scraped)