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.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Every symplectic vector field has a global Hamiltonian function

Statement refuted

Every symplectic vector field has a global Hamiltonian function.

Facts & Assumptions

Given: The proposed universal claim.

[F1]

The obstruction quotient is HdR1(M). Symplectic vector fields modulo Hamiltonian vector fields are first de Rham cohomology.

Refutation

technique · direct
1.1

On T2=(R/Z)2 with ω=dθ1∧dθ2, the field X=∂θ1 satisfies ιXω=dθ2, a closed form, so it is symplectic.

F1algebra
2.1

The form dθ2 integrates to one around the second coordinate circle, so it is not exact. Hence [F1] says X is not Hamiltonian, refuting the claim.

F1step 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