CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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.

Every bounded simply connected proper plane domain is regular

Statement

Let $Ω \subseteq C$ be a bounded proper complex domain whose complement in the Riemann sphere $\hat{C}$ is connected. Then every boundary point of $Ω$ is regular. In the planar working convention of the batch sources, this says that every bounded simply connected proper plane domain is regular.

Facts & Assumptions

Given: A bounded proper complex domain $Ω$ with connected complement in $\hat{C}$ .

[L1]

If the complementary component of $\hat{C} ∖ Ω$ containing a boundary point also contains another point, then that boundary point is regular (A boundary point whose complementary component contains another point is regular).

Proof

technique · direct

1.1

Fix $ζ \in \partial Ω$ . The complement $\hat{C} ∖ Ω$ is connected by hypothesis, contains $ζ$ , and also contains $\infty$ because $Ω$ is bounded and proper. Therefore the complementary component containing $ζ$ contains a second point.

given

2.1

Applying [L1] to the boundary point $ζ$ shows that $ζ$ is regular. Since $ζ$ was arbitrary, every boundary point is regular.

L1step 1.1∎

Depends on

A boundary point whose complementary component contains another point is regular

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

Boris Khoruzhenko, Potential Theory lecture notes (standard reference, not scraped)
Harold P. Boas, Class Notes Math 618: Complex Variables II, Spring 2016 (standard reference, not scraped)