Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-29
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One smooth transition direction does not guarantee chart compatibility

Statement

False claim: if one transition map between two charts is smooth, then the charts are smoothly compatible.

Facts & Assumptions

Given: The two charts on R with coordinate maps φ(x)=x and ψ(x)=x3.

[F1]

Two charts are smoothly compatible only when both transition maps on the overlap are smooth (Smoothly compatible charts and the smoothness of Euclidean transition maps).

Refutation

technique · direct
1.1

The transition ψ∘φ−1(x)=x3 is smooth on R. [given] However the reverse transition is φ∘ψ−1(x)=x1/3, which is not differentiable at 0.

given
2.1

By [F1], the failure of the reverse transition means these charts are not smoothly compatible. So one smooth direction is not enough.

F1step 1.1∎

Depends on

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