Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 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.

Four distinct sphere points lie on one circline exactly when their cross-ratio is real

Statement

Four distinct sphere points lie on one circline if and only if their cross-ratio is real, in the sense that z1,z2,z3,z4 lie on one circline ⟺[z1,z2;z3,z4]∈R.

Facts & Assumptions

Given: Four distinct sphere points z1,z2,z3,z4.

[L1]

A circline is exactly a set of the form C(a,b,c)={a,b,c}∪{z∉{a,b,c}:[a,b;c,z]∈R} (Circlines and their reflections on the Riemann sphere).

Proof

technique · direct
1.1

If the four points lie on one circline, then taking that circline to be C(z1,z2,z3) forces z4∈C(z1,z2,z3). Because the four points are distinct, z4∉{z1,z2,z3}, so [L1] gives [z1,z2;z3,z4]∈R.

L1given
2.1

Conversely, if [z1,z2;z3,z4]∈R, then z4∉{z1,z2,z3} and [L1] says exactly that z4∈C(z1,z2,z3). Hence the four points lie on a single circline.

L1given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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