False statementConstruction: Literature-sourcedVerification: 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.

Every closed differential form is globally exact

Statement

The assertion that every closed differential form has a global primitive is false.

Facts & Assumptions

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

[F1]

On a chart, if $η = \sum_{I} η_{I} d x^{I}$ , then $d η = \sum_{I} d η_{I} \land d x^{I}$ (The local coordinate formula for the exterior derivative).

Refutation

technique · direct

1.1

For $ω = (- y d x + x d y) / (x^{2} + y^{2})$ , writing $r^{2} = x^{2} + y^{2}$ gives $\partial_{x} (x / r^{2}) - \partial_{y} (- y / r^{2}) = 0$ ; therefore [F1] gives $d ω = 0$ on $R^{2} ∖ {0}$ .

F1given

2.1

If $d F = ω$ globally, then for $γ (t) = (\cos t, \sin t)$ one has $(F \circ γ)^{'} = ω (γ^{'}) = 1$ . The fundamental theorem of calculus would give $0 = F (γ (2 π)) - F (γ (0)) = 2 π$ , a contradiction.

step 1.1∎

Depends on

The local coordinate formula for the exterior derivative

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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)