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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.

Poincaré recurrence for finite-volume Hamiltonian invariant regions

Statement

Let R be a measurable invariant region of finite symplectic volume for a Hamiltonian flow, and fix a nonzero time τ for which the time map and all its iterates are defined on R. For every measurable E⊆R, almost every x∈E returns to E under ϕnτ for infinitely many positive integers n.

Facts & Assumptions

Given: The invariant finite-volume region and time map in the statement.

[F1]

Hamiltonian time maps preserve symplectic volume. Liouville volume preservation.

[F2]

In a finite measure-preserving system, almost every point of each measurable set returns infinitely often. Poincare recurrence for finite measure-preserving systems.

Proof

technique · direct
1.1

Restrict T=ϕτ and the symplectic volume measure to R. Invariance keeps T on R, [F1] makes it measure preserving, and the hypothesis gives finite total measure.

F1given
2.1

Apply [F2] to (R,T). Since Tn=ϕnτ wherever the iterates are defined, its conclusion is exactly the stated recurrence. No assertion is made for an incomplete time map.

F2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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