PropositionStatement: 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.

Symplectic vector spaces have even dimension

Statement

Every symplectic vector space $(V, ω)$ has dimension $2 n$ for a unique $n \geq 0$ . It has a basis $e_{1}, \dots, e_{n}, f_{1}, \dots, f_{n}$ in which $ω (e_{i}, f_{j}) = δ_{i j}$ and all $e$ -- $e$ and $f$ -- $f$ pairings vanish.

Facts & Assumptions

Given: A symplectic vector space $(V, ω)$ .

[F1]

Symplectic means that the radical of $ω$ is zero. Symplectic vector space.

[F2]

An alternating form has a basis of symplectic pairs followed by a basis of its radical. Every alternating form on a finite-dimensional space has a basis of symplectic pairs followed by a basis of its radical; in particular its rank is even.

Proof

technique · direct

1.1

Apply [F2] to $ω$ . Its normal-form basis consists of $r$ pairs $e_{i}, f_{i}$ and $s$ radical vectors, with $\dim V = 2 r + s$ .

2.1

By [F1] the radical is zero, so $s = 0$ . Taking $n = r$ gives the asserted even dimension and the displayed standard symplectic basis. This includes $V = 0$ , where $n = 0$ and the basis is empty.

F1F2step 1.1∎

Depends on

Used by

Symplectic manifolds can have odd dimension False statement

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

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