CounterexampleConstruction: 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.

A nondegenerate nonclosed two-form in dimension at least four

Counterexample

On $R^{4}$ , set $η = d x_{1} \land d y_{1} + e^{x_{1}} d x_{2} \land d y_{2}$ .

Facts & Assumptions

Given: The displayed two-form.

[F1]

Symplecticity requires both nondegeneracy and closedness. Symplectic form and symplectic manifold.

Verification

technique · direct

1.1

Its square is $η^{2} = 2 e^{x_{1}} d x_{1} \land d y_{1} \land d x_{2} \land d y_{2}$ , a nowhere-zero top form, so $η$ is nondegenerate.

givenalgebra

2.1

Exterior differentiation gives $d η = e^{x_{1}} d x_{1} \land d x_{2} \land d y_{2} \neq 0$ . Hence [F1] shows that $η$ is not symplectic. Products with standard symplectic factors give the same phenomenon in every even dimension at least four.

F1step 1.1algebra∎

Depends on

Symplectic form and symplectic manifold

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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)