Alphabeta Math
TheoremStatement: 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.

The Hamiltonian vector-field map is a Lie antihomomorphism

Statement

With ιXHω=dH and {F,G}=ω(XF,XG),

[XF,XG]=−X{F,G}.

Facts & Assumptions

Given: Smooth functions F,G on (M,ω).

[F1]

XF is symplectic, so LXFω=0. A vector field is symplectic iff ιXω is closed.

[F2]

Cartan calculus gives ι[X,Y]=LXιY−ιYLX. Cartan commutator identities.

[F3]

XF(G)=−{F,G} in the library convention. Poisson bracket on a symplectic manifold.

Proof

technique · direct
1.1

Apply [F2] to ω: ι[XF,XG]ω=LXF(dG)−ιXG(LXFω)=d(XFG)=−d{F,G}.

F1F2F3given
2.1

The right side is ι−X{F,G}ω. Nondegeneracy makes contraction injective, so the vector fields are equal.

step 1.1∎

Depends on

Used by

Dependency tree · two levels

9 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