False statementConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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 nonzero germ is regular in the last variable without a coordinate change

Statement

False claim: every nonzero holomorphic germ is already regular in the last variable, without any coordinate change.

Facts & Assumptions

Given: The false claim above.

[L1]

The germ $z_{1} z_{2}$ is not regular in $z_{2}$ at the origin ( $z_{1} z_{2}$ is not regular in $z_{2}$ at the origin).

[L2]

A linear shear can nevertheless make that same germ regular in $z_{2}$ (A linear shear makes $z_{1} z_{2}$ regular in $z_{2}$ ).

Refutation

technique · direct

1.1

Fact [L1] gives a specific nonzero germ that fails the claimed property in the original coordinates.

2.1

Fact [L2] shows that the coordinate-change lemma is doing real work rather than decorating an already-true statement. Therefore the claim is false.

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Jiří Lebl, Tasty Bits of Several Complex Variables, Exercise 6.2.5 (standard reference, not scraped)