Alphabeta Math
ExampleConstruction: AI-adaptedVerification: 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.

A Möbius transformation is recovered from three point correspondences

Example

The unique Möbius transformation carrying (1,i,−1) to (∞,0,1) is M(z)=(1−i)(z−i)z−1.

Facts & Assumptions

Given: The source triple (1,i,−1) and the target triple (∞,0,1).

[L1]

A unique Möbius transformation carries one ordered triple of distinct sphere points to another (A unique Möbius transformation carries any ordered triple of distinct sphere points to any other).

Verification

technique · direct
1.1givenalgebra

The formula M(z)=(1−i)(z−i)/(z−1) satisfies M(1)=∞, M(i)=0, and M(−1)=1 by direct substitution.

2.1L1given∎

Fact [L1] makes a Möbius transformation with those three prescribed values unique, so this formula is exactly the desired map.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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