CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

The codimension-one Frobenius criterion

Statement

For a nowhere-zero one-form $α$ , the hyperplane distribution $\ker α$ is integrable if and only if $α \land d α = 0$ .

Facts & Assumptions

Given: The manifolds, forms, vector fields, maps, and coordinates explicitly named in the statement.

[F1]

The preceding result states that If $θ^{1}, \dots, θ^{n - k}$ locally frame $D^{\circ}$ , then $D$ is involutive if and only if $d θ^{a} = \sum_{b} η_{b}^{a} \land θ^{b}$ locally for every $a$ ; equivalently, its annihilator ideal is differential. (The Pfaffian Frobenius criterion).

[F2]

A smooth distribution is integrable if and only if it is involutive (Frobenius local coordinate theorem).

Proof

technique · direct

1.1

Extend the nowhere-zero $α$ to a local coframe. The Pfaffian condition is $d α = η \land α$ .

F1given

2.1

In that coframe, $α \land d α = 0$ is equivalent to the absence of every component of $d α$ not containing $α$ , hence to $d α \in (α)$ . Apply [F1] and then [F2] in both directions.

F1F2step 1.1∎

Depends on

Used by

A contact form on three-space Example
α∧ dα vanishes for every one-form False statement
Closed constant-rank one-forms define integrable hyperplane fields Proposition

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

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)