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.

Local sections of a distribution are freely generated by a local frame

Statement

Let $D$ be a rank- $k$ smooth distribution on $M$ .

$Γ (D)$ is a $C^{\infty} (M)$ -submodule of the module of smooth vector fields on $M$ .
If $U \subseteq M$ carries a local frame $X_{1}, \dots, X_{k}$ of $D$ , then every $Y \in Γ (D ∣_{U})$ has a unique expression $Y = \sum_{i = 1}^{k} f_{i} X_{i}$ with smooth functions $f_{i}$ on $U$ .

Facts & Assumptions

Given: A rank- $k$ smooth distribution $D$ on $M$ .

[A1]

Fix an open set $U$ on which $D$ has a local frame $X_{1}, \dots, X_{k}$ .

Proof

technique · direct

1.1

If $Y, Z \in Γ (D)$ and $f, g \in C^{\infty} (M)$ , then [given] $(f Y + g Z)_{p} = f (p) Y_{p} + g (p) Z_{p}$ lies in the linear subspace $D_{p}$ for every $p$ . Hence $Γ (D)$ is closed under addition and smooth scalar multiplication.

given

1.2

On $U$ , the vectors $X_{1} (q), \dots, X_{k} (q)$ form a basis of $D_{q}$ , [given] so each $Y_{q} \in D_{q}$ has unique coefficients $f_{i} (q)$ with $Y_{q} = \sum_{i} f_{i} (q) X_{i} (q)$ . Because $Y$ and the frame fields are smooth, those coefficients are smooth on $U$ .

given

1.3

Therefore the assignment $U \mapsto Γ (D ∣_{U})$ is locally free [given] of rank $k$ : on every frame domain its section module is freely generated by that frame. This is a statement about the sheaf of local sections; it does not assert that the global module $Γ (D)$ is free over $C^{\infty} (M)$ .

given∎

Depends on

Used by

Dependency tree · two levels

11 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

Keith Conrad, Local and global Frobenius theorems (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)