Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14
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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.
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Darboux theorem makes all symplectic manifolds globally symplectomorphic

Statement refuted

Darboux's theorem makes all equidimensional symplectic manifolds globally symplectomorphic.

Facts & Assumptions

Given: ACω and the proposed consequence of Darboux's theorem.

[F1]

Darboux's theorem supplies coordinates only on a neighbourhood of each chosen point. Darboux theorem.

Refutation

technique · direct
1.1

The standard area forms make both S2 and R2 two-dimensional symplectic manifolds, so [F1] does identify small neighbourhoods of their points.

F1
2.1

A global symplectomorphism would in particular be a diffeomorphism, but S2 is compact and R2 is not; homeomorphisms preserve compactness. Hence no global symplectomorphism exists, and the local conclusion cannot be globalized.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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