DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05

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.

Involutive distributions

Definition

A smooth distribution $D$ on $M$ is involutive when

$X, Y \in Γ (D) ⟹ [X, Y] \in Γ (D) .$

That is, the smooth vector fields tangent to $D$ are closed under the Lie bracket.

Depends on

Used by

A bracket-closed variable-rank family outside regular Frobenius Counterexample
The standard contact plane field is not integrable Counterexample
The ambient value of a Lie bracket is determined by the two pointwise vector values False statement
Integrable distributions are involutive Proposition
Involutivity can be checked on a local frame Proposition
Level-set distributions are involutive Proposition
Frobenius local coordinate theorem Theorem

Dependency tree · two levels

5 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

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Keith Conrad, Local and global Frobenius theorems (standard reference, not scraped)