CorollaryStatement: AI-adaptedProof: 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.

Hamiltonian flow has zero divergence with respect to symplectic volume

Statement

Every Hamiltonian vector field $X_{H}$ has zero divergence with respect to the symplectic volume $Ω = ω^{n} / n!$ .

Facts & Assumptions

Given: A Hamiltonian vector field on a symplectic manifold.

[F1]

Its local flow preserves $Ω$ . Liouville volume preservation.

[F2]

Divergence relative to $Ω$ is defined by $L_{X} Ω = ({div}_{Ω} X) Ω$ . Divergence relative to a volume form.

Proof

technique · direct

1.1

Differentiate the identity $ϕ_{t}^{*} Ω = Ω$ from [F1] at $t = 0$ to obtain $L_{X_{H}} Ω = 0$ .

F1given

2.1

By [F2], $({div}_{Ω} X_{H}) Ω = 0$ . Since $Ω$ is nowhere zero, the divergence function vanishes.

F2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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)