CounterexampleConstruction: 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 bijective smooth map with nonsmooth inverse

Statement refuted

Every bijective smooth map is a diffeomorphism.

Facts & Assumptions

Given: The bijection $F : R \to R$ , $F (x) = x^{3}$ .

[L1]

The A-page false statement already proves that $F$ is smooth and bijective but that $F^{- 1} (y) = y^{1 / 3}$ is not smooth (A bijective smooth map need not be a diffeomorphism).

[F1]

A diffeomorphism is a bijective smooth map with smooth inverse (Diffeomorphisms and local diffeomorphisms of manifolds).

Counterexample

technique · direct

1.1

By [L1], the map $F$ is smooth and bijective.

1.2

The same cited refutation shows that $F^{- 1}$ is not smooth, so [F1] rules [F1, L1] out $F$ being a diffeomorphism.

F1L1

2.1

Hence $F (x) = x^{3}$ is the desired counterexample.

step 1.1step 1.2∎

Depends on

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

Nigel Hitchin, Differentiable Manifolds, §2.4 (standard reference, not scraped)
Rob van der Vorst, Introduction to differentiable manifolds, §2 (standard reference, not scraped)