TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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.

Every exact C1 vector field is closed

Statement

Let $U \subseteq R^{n}$ be open. Every exact $C^{1}$ vector field $F : U \to R^{n}$ is closed.

Facts & Assumptions

Given: The open set and exact $C^{1}$ field in the Statement.

[L1]

Exactness means $F_{i} = \partial_{i} ϕ$ for a $C^{2}$ scalar function $ϕ$ , while closedness means $\partial_{j} F_{i} = \partial_{i} F_{j}$ for all indices $i, j$ (Exact and closed C1 vector fields).

[L2]

Continuous second partials commute: $\partial_{j} \partial_{i} ϕ = \partial_{i} \partial_{j} ϕ$ (Continuous second partials of a scalar potential commute).

Proof

technique · direct

1.1

Choose the $C^{2}$ potential $ϕ$ from [L1]. For every $i, j$ , $\partial_{j} F_{i} = \partial_{j} \partial_{i} ϕ = \partial_{i} \partial_{j} ϕ = \partial_{i} F_{j},$ where the middle equality is [L2].

givenL1L2algebra

2.1

The equalities in step 1.1 are precisely the closedness condition in [L1], so $F$ is closed.

step 1.1L1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 33 results over 15 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J.-B. Campesato, Poincare Lemma, section 1 (standard reference, not scraped)