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.

A compatible complex structure on standard symplectic space

Example

On standard $R^{2 n} = R_{q}^{n} \oplus R_{p}^{n}$ , the map $J (q, p) = (- p, q)$ is compatible with $ω_{0} = \sum_{i} d q^{i} \land d p_{i}$ .

Facts & Assumptions

Given: The displayed $J$ and standard form.

[F1]

Compatibility requires $J^{2} = - I$ and positivity and symmetry of $ω_{0} (\cdot, J \cdot)$ . Compatible complex structure on a symplectic vector space.

Verification

technique · direct

1.1givenalgebra

Direct substitution gives $J^{2} (q, p) = (- q, - p)$ . For $u = (a, b)$ and $v = (c, d)$ , $ω_{0} (u, J v) = ω_{0} ((a, b), (- d, c)) = a \cdot c + b \cdot d$ .

2.1F1step 1.1∎

The last expression is the Euclidean inner product, hence symmetric and positive definite. By [F1], $J$ is compatible; the same calculation gives $ω_{0} (J u, J v) = ω_{0} (u, v)$ .

Depends on

Compatible complex structure on a symplectic vector space

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)