Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-31
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.

FALSE: every simply connected plane domain is star-shaped

Statement

False claim. Every simply connected plane domain is star-shaped.

Facts & Assumptions

Given: The dumbbell witness from A dumbbell-shaped plane domain can be simply connected without being star-shaped.

[L1]

There is a simply connected dumbbell domain that is not star-shaped (A dumbbell-shaped plane domain can be simply connected without being star-shaped).

Refutation

technique · direct
1.1

Apply [L1]. It gives a plane domain that is simply connected and simultaneously not star-shaped.

givenL1
2.1

Hence the universal claim is false.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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