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.

Isotropic, coisotropic, symplectic, and Lagrangian coordinate subspaces

Example

In standard $R^{4}$ with symplectic basis $(e_{1}, e_{2}, f_{1}, f_{2})$ , coordinate subspaces realize each of the four subspace types.

Facts & Assumptions

Given: $ω (e_{i}, f_{j}) = δ_{i j}$ , $ω (f_{j}, e_{i}) = - δ_{i j}$ , and the pairings among two $e$ -vectors or among two $f$ -vectors are zero.

[F1]

The four types are determined by $W$ , $W^{ω}$ , their inclusion, and their intersection. Isotropic, coisotropic, symplectic, and Lagrangian subspaces.

Verification

technique · direct

1.1

Direct pairing gives $⟨ e_{1} ⟩^{ω} = ⟨ e_{1}, e_{2}, f_{2} ⟩$ , so $⟨ e_{1} ⟩$ is isotropic but not Lagrangian. Taking orthogonals reverses this equality, so $⟨ e_{1}, e_{2}, f_{2} ⟩$ is coisotropic but not Lagrangian.

F1givenalgebra

1.2

Also $⟨ e_{1}, f_{1} ⟩^{ω} = ⟨ e_{2}, f_{2} ⟩$ , making $⟨ e_{1}, f_{1} ⟩$ symplectic, while $⟨ e_{1}, e_{2} ⟩^{ω} = ⟨ e_{1}, e_{2} ⟩$ , making the latter Lagrangian.

F1givenalgebra

2.1

The four displayed coordinate subspaces therefore exhibit, respectively, a proper isotropic, a proper coisotropic, a proper symplectic, and a Lagrangian subspace.

F1step 1.1step 1.2∎

Depends on

Isotropic, coisotropic, symplectic, and Lagrangian subspaces

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

Eckhard Meinrenken, Symplectic Geometry (standard reference, not scraped)