CounterexampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 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.

Two noncompatible atlases on the real line

Statement refuted

Any two atlases on the same topological manifold are smoothly compatible.

Facts & Assumptions

Given: The real line with the two singleton atlases $A = {(R, id)}$ and $B = {(R, ψ)}$ , where $ψ (x) = x^{3}$ .

[F1]

The real line is a smooth manifold, so the two displayed charts are charts on one and the same manifold (Euclidean spaces and Euclidean open subsets as smooth manifolds).

[F2]

A smooth atlas is a covering family of pairwise smoothly compatible charts (Smooth atlases).

[F3]

Smooth compatibility requires both transition directions to be smooth (Smoothly compatible charts and the smoothness of Euclidean transition maps).

Counterexample

technique · direct

1.1

Each singleton family $A$ and $B$ covers $R$ , so [F1, F2] by [F2] each is an atlas provided its single chart is legitimate, and [F1] supplies that legitimacy.

F1F2

2.1

The transition $ψ \circ {id}^{- 1} (x) = x^{3}$ is smooth, but the [F3, step 1.1] reverse transition $id \circ ψ^{- 1} (x) = x^{1 / 3}$ is not differentiable at $0$ . Hence [F3] says the two charts are not compatible.

F3step 1.1

3.1

Therefore $A$ and $B$ are atlases on the same manifold [step 2.1] that are not compatible, which refutes the statement.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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, §2.2 (standard reference, not scraped)
Rob van der Vorst, Introduction to differentiable manifolds, §2 (standard reference, not scraped)