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.

The zero section and cotangent fibres as Lagrangians

Example

Assume ${A C}_{ω}$ . In $(T^{*} Q, ω_{c a n})$ , both the zero section and every cotangent fibre $T_{q}^{*} Q$ are Lagrangian.

Facts & Assumptions

Given: A smooth $n$ -manifold $Q$ and its canonical cotangent form.

[F1]

In cotangent coordinates, $ω_{c a n} = \sum_{i} d q^{i} \land d p_{i}$ . The canonical cotangent two-form is symplectic.

[F2]

In a $2 n$ -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

On the zero section every $p_{i}$ is constant zero, so the pullback of [F1] vanishes. On the fibre over fixed $q$ , every $q^{i}$ is constant, so the restriction again vanishes. Both submanifolds are therefore isotropic.

F1given

2.1

Each has dimension $n$ , while $T^{*} Q$ has dimension $2 n$ . By [F2], both are Lagrangian, including the rank-zero case.

F2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

Ana Cannas da Silva, Lectures on Symplectic Geometry (standard reference, not scraped)