Alphabeta Math
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.

Time-dependent Hamiltonian evolution is symplectic

Statement

Assume ACω. Every time slice Φt,s of a time-dependent Hamiltonian evolution is a local symplectomorphism wherever it is defined: Φt,s∗ω=ω.

Facts & Assumptions

Given: A time-dependent Hamiltonian and its local evolution.

[F1]

ιXHtω=dHt. Time-dependent Hamiltonian vector field and flow.

[F2]

Along a time-dependent evolution, ∂tΦt,s∗ω=Φt,s∗(LXHtω). Differentiation of a pulled-back form along a time-dependent flow.

Proof

technique · direct
1.1

Cartan's formula and [F1] give LXHtω=d(dHt)+ιXHtdω=0.

F1given
2.1

By [F2], ∂tΦt,s∗ω=0. At t=s the pullback is ω, so it remains ω on the evolution domain.

F2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources