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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Moser isotopy for area forms on a compact surface

Example

Assume ${A C}_{ω}$ . Let $ω_{0}, ω_{1}$ be positive area forms on a nonempty compact connected oriented surface $Σ$ without boundary. If $\int_{Σ} ω_{0} = \int_{Σ} ω_{1}$ , then they are related by a Moser isotopy.

Facts & Assumptions

Given: The surface and two forms in the statement.

[F1]

Integration is an isomorphism on top compactly supported de Rham cohomology of a connected oriented boundaryless manifold. Integration is an isomorphism on top compactly supported de Rham cohomology.

[F2]

A cohomologous symplectic path on compact $M$ is trivialized by an isotopy. Moser stability theorem.

Verification

technique · direct

1.1

Compactness makes both top forms compactly supported. Their difference has integral zero, so [F1] makes it exact and therefore $[ω_{0}] = [ω_{1}]$ .

F1given

2.1

Relative to any fixed positive area form, write $ω_{i} = f_{i} μ$ with $f_{i} > 0$ . Then $ω_{t} = (1 - t) ω_{0} + t ω_{1} = ((1 - t) f_{0} + t f_{1}) μ$ stays positive and hence symplectic, and step 1.1 makes its class constant.

step 1.1givenalgebra

3.1

Apply [F2] to obtain $ϕ_{t}^{*} ω_{t} = ω_{0}$ ; in particular $ϕ_{1}^{*} ω_{1} = ω_{0}$ . The connected nonempty hypothesis is exactly what [F1] uses.

F2step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

22 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)