TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck 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.

A point on a nonsingleton boundary component is regular

Statement

Let $Ω \subseteq C$ be a bounded complex domain and let $ζ \in \partial Ω$ . If the connected component of $\partial Ω$ containing $ζ$ is not a singleton, then $ζ$ is regular for $Ω$ .

Facts & Assumptions

Given: A bounded complex domain $Ω$ and a boundary point $ζ \in \partial Ω$ whose boundary component contains another point.

[L1]

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

Proof

technique · direct

1.1givenchoose

Let $C$ be the connected component of $\partial Ω$ containing $ζ$ , and choose $a \in C$ with $a \neq ζ$ . Because $C \subseteq \hat{C} ∖ Ω$ is connected, it lies in a single connected component of $\hat{C} ∖ Ω$ ; that component contains both $ζ$ and $a$ .

2.1L1step 1.1∎

Step 1.1 puts $ζ$ under the hypothesis of [L1], so $ζ$ is regular for $Ω$ .

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)