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 Hartogs figure in (|z1|, |z2|) coordinates

Example

For 0<r,s<1, the Hartogs figure H(r,s) is the union of a thin inner cylinder and a thick outer shell:

H(r,s)={∣z1∣<1, ∣z2∣<s}∪{r<∣z1∣<1, ∣z2∣<1}.

In the (∣z1∣,∣z2∣)-plane this is exactly the region obtained by taking the rectangle [0,1)×[0,s) together with the vertical strip (r,1)×[0,1).

Facts & Assumptions

Given: Real numbers 0<r,s<1.

[L1]

The Hartogs figure and its bidisc hull are defined by the displayed modulus conditions in the two coordinates (The Hartogs figure H(r,s) and its bidisc hull).

Verification

technique · direct
1.1

By [L1], membership in H(r,s) depends only on the two moduli ∣z1∣ and ∣z2∣, and the two defining pieces are exactly the inequalities listed in the Example.

L1
2.1

So the picture in modulus coordinates is the union of the thin horizontal rectangle [0,1)×[0,s) with the outer vertical strip (r,1)×[0,1), while the missing core is [0,r]×[s,1).

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