ExampleConstruction: 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.

The maps z+1, 2z, and 1/z realize the parabolic, hyperbolic, and elliptic branches of the classification

Example

The three maps $z \mapsto z + 1, z \mapsto 2 z, z \mapsto 1 / z$ realize the parabolic, hyperbolic, and elliptic branches of the Möbius classification.

Facts & Assumptions

Given: The Möbius maps $z \mapsto z + 1$ , $z \mapsto 2 z$ , and $z \mapsto 1 / z$ .

[L1]

Nonidentity Möbius transformations are parabolic or conjugate to a dilation, with the elliptic/hyperbolic/loxodromic convention recorded on the classification theorem (Nonidentity Möbius transformations are parabolic or conjugate to a dilation, with the projective trace invariant).

Verification

technique · direct

1.1

The map $z \mapsto z + 1$ fixes only $\infty$ , so [L1] classifies it as parabolic; the map $z \mapsto 2 z$ fixes $0$ and $\infty$ and is already the dilation normal form with multiplier $2$ , so [L1] classifies it as hyperbolic.

L1given

2.1

The map $z \mapsto 1 / z$ fixes $1$ and $- 1$ , and conjugating by $(z - 1) / (z + 1)$ turns it into $w \mapsto - w$ . Since that multiplier has modulus $1$ , [L1] places it in the elliptic branch.

L1givenalgebra∎

Depends on

Nonidentity Möbius transformations are parabolic or conjugate to a dilation, with the projective trace invariant

Used by

Nothing in the library uses this result yet.

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

Lars V. Ahlfors, Complex Analysis, 3rd ed., Ch. 3 §§2.2-3.5 (standard reference, not scraped)
Matthias Weber, Complex Analysis, Ch. 1 §§1.3-1.4 (standard reference, not scraped)
Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 8 §§1-2 (standard reference, not scraped)