CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-29

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.

Complex conjugation preserves angle magnitudes but is not conformal

Statement refuted

Every map that preserves angle magnitudes is conformal.

Facts & Assumptions

Given: The map $c : C \to C$ , $c (z) = \overline{z}$ .

[F1]

This page's conformal convention is orientation-preserving: biholomorphisms preserve both angle magnitude and orientation, while complex conjugation is the standard orientation-reversing exclusion (Biholomorphisms are conformal and have holomorphic inverse).

Counterexample

1.1

On tangent vectors at $0$ , $c$ sends $1 \mapsto 1$ and $i \mapsto - i$ , so the unoriented angle still has magnitude $π / 2$ but the oriented angle changes from $+ π / 2$ to $- π / 2$ .

F1given

2.1

The complex difference quotient at $0$ is $(c (h) - c (0)) / h = \overline{h} / h$ ; along real $h \to 0$ this equals $1$ , while along purely imaginary $h \to 0$ it equals $- 1$ , so the limit does not exist, $c$ is not holomorphic, and [F1] therefore excludes it from being conformal in the library's sense.

F1step 1.1algebra∎

Depends on

Biholomorphisms are conformal and have holomorphic inverse

Used by

Nothing in the library uses this result yet.

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

Lars V. Ahlfors, Complex Analysis, 3rd ed., Ch. 3 §2.3 Conformal Mapping (standard reference, not scraped)