Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passaudited 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.

z12−z2 prepares to the Weierstrass polynomial z2−z12

Example

For f(z1,z2)=z12−z2, the Weierstrass preparation in the variable z2 is

f=(−1) (z2−z12).

So the prepared polynomial is exactly W(z1,z2)=z2−z12, and the unit is the constant −1.

Facts & Assumptions

Given: The germ f(z1,z2)=z12−z2 at the origin.

[L1]

A regular germ factors as a unit times a Weierstrass polynomial (Weierstrass preparation theorem).

Verification

technique · direct
1.1

The slice f(0,z2)=−z2 has a simple zero at 0, so f is regular in z2 of order 1.

givenalgebra
2.1

The identity f=(−1) (z2−z12) already has the required form: the factor −1 is a unit and z2−z12 is monic in z2 with lower coefficient −z12 vanishing at the origin. Thus [L1] produces exactly this preparation.

step 1.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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