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.

Smooth functions form a Poisson algebra

Statement

C∞(M), with pointwise multiplication and the symplectic Poisson bracket, is a real Poisson algebra.

Facts & Assumptions

Given: A symplectic manifold (M,ω).

[F1]

The Poisson bracket is bilinear, skew, and a derivation in each entry. The Poisson bracket is bilinear, skew, and a derivation in each entry.

[F2]

It satisfies the Jacobi identity. The Poisson bracket satisfies the Jacobi identity.

Proof

technique · direct
1.1

Pointwise addition and multiplication make C∞(M) a commutative associative real algebra with unit, and [F1] supplies a bilinear skew biderivation.

F1given
2.1

By [F2] that bracket is a Lie bracket. These are exactly the Poisson-algebra axioms, so the claimed structure follows.

F2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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