False statementConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-29

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.

A chart domain need not be a Euclidean open set

Statement

False claim: in a chart $(U, φ)$ , the domain $U$ is an open subset of Euclidean space.

Facts & Assumptions

Given: A chart $(U, φ)$ on a manifold $M$ .

[F1]

A chart has $U \subseteq M$ open in the manifold and $φ : U \to \hat{U} \subseteq R^{n}$ a homeomorphism onto an open Euclidean set; the Euclidean open set is the image $\hat{U} = φ (U)$ (Manifold charts, coordinate domains, and coordinate functions).

Refutation

technique · direct

1.1

By [F1], the set $U$ lives in the manifold $M$ and is open there, while the [F1] Euclidean open set is $\hat{U} = φ (U) \subseteq R^{n}$ . The two sets lie in different ambient spaces and play different roles.

2.1

Therefore the false claim swaps the chart domain with the chart image and is wrong at the level of the definition itself.

step 1.1∎

Depends on

Manifold charts, coordinate domains, and coordinate functions

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Nigel Hitchin, Differentiable Manifolds, §2.1 (standard reference, not scraped)
Rob van der Vorst, Introduction to differentiable manifolds, §1 (standard reference, not scraped)