CorollaryStatement: 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 manifolds have no local invariants beyond dimension

Statement

Assume ${A C}_{ω}$ . If $(M_{0}, ω_{0})$ and $(M_{1}, ω_{1})$ have the same dimension and $p_{i} \in M_{i}$ , then some neighbourhood of $p_{0}$ is symplectomorphic to some neighbourhood of $p_{1}$ , with $p_{0}$ sent to $p_{1}$ .

Facts & Assumptions

Given: ${A C}_{ω}$ , equal-dimensional symplectic manifolds and chosen points as in the statement.

[F1]

Darboux coordinates identify a neighbourhood of every point with an open neighbourhood of zero carrying the standard form. Darboux theorem.

Proof

technique · direct

1.1

Choose Darboux charts $x_{i} : U_{i} \to V_{i} \subseteq R^{2 n}$ at the two points by [F1]. Their images both contain zero, so restrict them to the inverse images of one common open neighbourhood $V \subseteq V_{0} \cap V_{1}$ of zero.

F1construct

2.1

The map $x_{1}^{- 1} \circ x_{0}$ sends $p_{0}$ to $p_{1}$ and preserves the standard form in the middle, hence pulls $ω_{1}$ back to $ω_{0}$ .

F1step 1.1∎

Depends on

Darboux theorem

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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)