Alphabeta Math
CounterexampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28
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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.

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z1z2 is not regular in z2 at the origin

Statement refuted

Refuted claim: the germ z1z2 is regular in the variable z2 at the origin.

Facts & Assumptions

Given: The germ f(z1,z2)=z1z2.

[L1]

Regularity in z2 requires the slice z1=0 to have a finite exact order of vanishing in the remaining variable (Regular holomorphic germs in the last variable).

Counterexample

technique · direct
1.1

On the slice z1=0 one has f(0,z2)=0 for every z2. So the last-variable restriction vanishes identically rather than to a finite exact order.

given
2.1

Step 1.1 contradicts the requirement in [L1]. Therefore the germ is not regular in z2 at the origin.

step 1.1L1∎

Depends on

Used by

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