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 function 1 / (1 - z1 z2) extends holomorphically from a Hartogs figure

Example

The function

f(z1,z2)=11−z1z2

is holomorphic on the full bidisc {∣z1∣<1, ∣z2∣<1} and therefore, a fortiori, on every Hartogs figure inside that bidisc.

Facts & Assumptions

Given: The function f(z1,z2)=1/(1−z1z2) on the unit bidisc.

[L1]

Every holomorphic function on a Hartogs figure extends uniquely to the full bidisc hull (A holomorphic function on a Hartogs figure extends to the full bidisc).

Verification

technique · direct
1.1

On the unit bidisc one has ∣z1z2∣<1, so 1−z1z2≠0. Hence the reciprocal f(z1,z2)=1/(1−z1z2) is holomorphic there.

givenalgebra
2.1

Restricting f to any Hartogs figure produces a concrete instance of [L1], and the extension theorem recovers the same global formula on the whole bidisc.

step 1.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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