Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28
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  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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FALSE: every Möbius self-map of the Riemann sphere restricts to an entire biholomorphism of the complex plane

Statement

Every Möbius self-map of the Riemann sphere restricts to an entire biholomorphism C→C.

Facts & Assumptions

Given: The Möbius map M(z)=1/z.

[L1]

Every Möbius transformation is a sphere biholomorphism (Every Möbius transformation is a biholomorphism of the Riemann sphere).

Refutation

technique · direct
1.1

Fact [L1] makes M(z)=1/z a biholomorphic self-map of the sphere.

L1given
2.1

As a sphere map, M is defined at 0 and satisfies M(0)=∞. Therefore its restriction to the finite plane does not even take values in C, so it is not an entire map C→C, let alone an entire biholomorphism.

given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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