PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passjudge 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.

Involutivity can be checked on a local frame

Statement

Let $D$ be a smooth distribution. Then $D$ is involutive if and only if every point has a neighborhood $U$ with a local frame $X_{1}, \dots, X_{k}$ of $D ∣_{U}$ such that

$[X_{i}, X_{j}] \in Γ (D ∣_{U}) for all i, j .$

Facts & Assumptions

Given: A smooth distribution $D$ .

[A1]

Fix a neighborhood $U$ with local frame $X_{1}, \dots, X_{k}$ of $D ∣_{U}$ .

Proof

technique · direct

1.1

If $D$ is involutive, then every bracket of tangent vector fields [given] is tangent, so in particular every bracket $[X_{i}, X_{j}]$ is tangent on $U$ .

given

1.2

Conversely, assume all frame brackets are tangent on $U$ . Any tangent [given] fields on $U$ have the form $X = \sum_{i} f_{i} X_{i}$ and $Y = \sum_{j} g_{j} X_{j}$ with smooth coefficients. Expanding $[X, Y]$ with the Leibniz rule expresses the bracket as a sum of terms involving the tangent fields $[X_{i}, X_{j}]$ and the frame fields $X_{i}$ , hence again as a tangent field.

givenalgebra

1.3

Since this holds on a neighborhood of every point, $D$ is [given] involutive exactly when one may check bracket closure on a local frame.

given∎

Depends on

Used by

Dependency tree · two levels

9 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

Will J. Merry, Differential Geometry (standard reference, not scraped)