Alphabeta Math
ExampleConstruction: AI-adaptedVerification: 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.

Product and opposite symplectic manifolds

Example

For every symplectic manifold (M,ω), the diagonal ΔM⊂M−×M is Lagrangian for the product form −pr⁡1∗ω+pr⁡2∗ω.

Facts & Assumptions

Given: A symplectic manifold (M,ω).

[F1]

Opposites and products carry the stated symplectic forms. Products and opposites of symplectic manifolds.

[F2]

In a 2n-dimensional symplectic vector space a subspace is Lagrangian exactly when it is isotropic and of dimension n, and a submanifold is Lagrangian exactly when its tangent spaces are Lagrangian subspaces. Equivalent characterizations of Lagrangian subspaces, Isotropic, coisotropic, symplectic, and Lagrangian submanifolds.

Verification

technique · direct
1.1

A tangent vector to the diagonal is (v,v). On two such vectors, the product form gives −ω(v,w)+ω(v,w)=0, so the diagonal is isotropic.

F1givenalgebra
2.1

If dim⁡M=2n, then dim⁡ΔM=2n and dim⁡(M−×M)=4n. Thus [F2] upgrades isotropy to Lagrangianity, including n=0.

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