Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedSession-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.

The bidisc minus the origin is not a domain of holomorphy

Example

The punctured bidisc

Ω={(z1,z2):∣z1∣<1, ∣z2∣<1}∖{(0,0)}

is not a domain of holomorphy.

Facts & Assumptions

Given: The punctured bidisc Ω.

[L1]

A holomorphic function on a punctured several-variable domain extends across the missing point (An isolated puncture is removable in complex dimension at least two).

[L2]

A domain of holomorphy is one for which no single larger overlap works for every holomorphic function (Holomorphic extension and domains of holomorphy in several variables).

Verification

technique · direct
1.1

Let f∈O(Ω). By [L1], f extends holomorphically across the missing origin to the full bidisc. So the same larger domain works for every holomorphic function on Ω.

L1
2.1

The fixed overlap can be taken to be any small bidisc around a point of Ω and the larger domain is the whole bidisc. By [L2], this shows that Ω is not a domain of holomorphy.

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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