ExampleConstruction: 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.

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A linear shear makes $z_{1} z_{2}$ regular in $z_{2}$

Example

Let $T (z_{1}, z_{2}) = (z_{1} + z_{2}, z_{2})$ . Then

$(z_{1} z_{2}) \circ T = (z_{1} + z_{2}) z_{2} = z_{1} z_{2} + z_{2}^{2},$

and along the slice $z_{1} = 0$ this becomes $z_{2}^{2}$ . So the sheared germ is regular in $z_{2}$ of order $2$ .

Facts & Assumptions

Given: The shear $T (z_{1}, z_{2}) = (z_{1} + z_{2}, z_{2})$ .

[L1]

Regularity in the last variable is the exact-order condition of Regular holomorphic germs in the last variable.

[L2]

Every nonzero germ can be made regular after an invertible complex-linear change of coordinates (After a linear coordinate change, every nonzero germ is regular in the last variable).

Verification

technique · direct

1.1

Direct substitution gives $(z_{1} z_{2}) \circ T = z_{1} z_{2} + z_{2}^{2} .$

givenalgebra

2.1

Setting $z_{1} = 0$ in step 1.1 yields the slice $z_{2}^{2}$ , which has exact order $2$ at the origin. So [L1] makes the transformed germ regular in $z_{2}$ of order $2$ , exhibiting the coordinate-change mechanism promised by [L2].

step 1.1L1L2∎

Depends on

Used by

FALSE: every nonzero germ is regular in the last variable without a coordinate change False statement

Dependency tree · two levels

5 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)

A linear shear makes z1z2z1​z2​ regular in z2z2​