PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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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.

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Level-set distributions are involutive

Statement

Let $F : M \to R^{r}$ be smooth, and let $D_{p} : = \ker d F_{p}$ for each $p \in M$ . Then every smooth vector field tangent to $D$ is closed under Lie bracket; in particular, when $D$ has constant rank it is involutive.

Facts & Assumptions

Given: A smooth map $F : M \to R^{r}$ .

[A1]

Let $X$ and $Y$ be smooth vector fields with values in $\ker d F$ .

Proof

technique · direct

1.1

Write $F = (f^{1}, \dots, f^{r})$ . The tangency assumption says [given] $X (f^{a}) = 0 = Y (f^{a})$ for every component $f^{a}$ .

given

1.2

Therefore [given] $[X, Y] (f^{a}) = X (Y (f^{a})) - Y (X (f^{a})) = 0$ for every $a$ . This means $d F ([X, Y]) = 0$ , so $[X, Y]$ is again tangent to the kernel family.

givenalgebra

1.3

Thus kernel distributions are closed under brackets whenever they are [given] defined as smooth distributions.

given∎

Depends on

Used by

The kernel of a submersion as an integrable distribution Example

Dependency tree · two levels

12 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)