Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28
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.

FALSE: every self-homeomorphism of the Riemann sphere preserves the cross-ratio

Statement

Every self-homeomorphism of the Riemann sphere preserves the cross-ratio.

Facts & Assumptions

Given: Complex conjugation κ(z)=z‾ on the sphere.

[L1]

Complex conjugation is a sphere homeomorphism, and the quadruple (1,−i,0,∞) has cross-ratio i while its conjugate quadruple (1,i,0,∞) has cross-ratio −i (Complex conjugation is a homeomorphism of the Riemann sphere that is not holomorphic, The cross-ratio of an ordered quadruple of sphere points).

Refutation

technique · direct
1.1

Fact [L1] provides a sphere homeomorphism that sends the cross-ratio value i to the different value −i on an explicit quadruple.

L1given
2.1

Therefore not every sphere homeomorphism preserves cross-ratios, so the statement is false.

given∎

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