Alphabeta Math
DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge 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.

The cross-ratio of an ordered quadruple of sphere points

Definition

Let z1,z2,z3,z4∈C^ be pairwise distinct. Their cross-ratio is [z1,z2;z3,z4]:=(z1−z3)(z2−z4)(z1−z4)(z2−z3) when all four points are finite, and when exactly one point is ∞ we use the limiting conventions [∞,z2;z3,z4]=z2−z4z2−z3,[z1,∞;z3,z4]=z1−z3z1−z4, [z1,z2;∞,z4]=z2−z4z1−z4,[z1,z2;z3,∞]=z1−z3z2−z3.

Because the four points are distinct, at most one of them is ∞, so these cases are exhaustive. The ordering matters: permuting the four entries changes the value by the usual fractional-linear transformations on C∖{0,1}.

Depends on

Used by

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