PropositionStatement: 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.

A Hamiltonian is conserved along its own flow

Statement

$H$ is constant along every integral curve of $X_{H}$ .

Facts & Assumptions

Given: A Hamiltonian $H$ and an integral curve $γ$ of $X_{H}$ .

[F1]

The convention is $d H = ι_{X_{H}} ω$ . Hamiltonian vector field and Hamiltonian function.

Proof

technique · direct

1.1

Along $γ$ , $\frac{d}{d t} H (γ (t)) = d H (X_{H}) = ω (X_{H}, X_{H}) = 0$ by alternation.

F1given

2.1

Hence $H \circ γ$ is constant on every connected time interval in the maximal domain. This proves conservation without assuming completeness.

step 1.1∎

Depends on

Hamiltonian vector field and Hamiltonian function

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)