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.

Circlines and their reflections on the Riemann sphere

Definition

For distinct points a,b,c∈C^, define C(a,b,c):={a,b,c}∪{z∈C^∖{a,b,c}:[a,b;c,z]∈R}. A circline is a subset of C^ of the form C(a,b,c) for some ordered triple of distinct points.

The standard circline is R^:=R∪{∞}, whose reflection is complex conjugation σR^(z)=z‾,σR^(∞)=∞. Whenever a Möbius transformation M carries a circline C to R^, put σC,M:=M−1∘σR^∘M. The well-definedness result Möbius transformations preserve circlines and conjugate their reflections ↗ proves that such normalizing maps exist and that σC,M has the same value for every choice of M. That common map is the reflection in C and is denoted σC.

Depends on

Used by

Dependency tree · two levels

4 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