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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Graphs of exact and closed one-forms as Lagrangians

Example

Assume ${A C}_{ω}$ . The graph of $d f$ is Lagrangian in $T^{*} Q$ for every smooth $f$ . More generally, the graph of every closed one-form is Lagrangian, including closed forms that are not exact.

Facts & Assumptions

Given: A smooth manifold $Q$ and the canonical cotangent convention.

[F1]

A one-form has Lagrangian graph exactly when it is closed. A graph of a one-form is Lagrangian exactly when the form is closed.

Verification

technique · direct

1.1

Since $d (d f) = 0$ , [F1] makes $graph (d f)$ Lagrangian. The same argument applies to any closed one-form, without asserting it is exact.

F1given

2.1

On $S^{1}$ , the global angular form $d θ$ is closed but not exact because its integral around the positively oriented circle is $2 π$ , whereas an exact form integrates to zero. Its graph is the section $p = 1$ in $T^{*} S^{1}$ , so [F1] supplies the promised closed-nonexact example.

F1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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)