Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-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.

A cubic image curve winds three times around the origin

Example

Let γ(t)=2eit for 0≤t≤2π and let f(z)=z3−1. Then the image contour f∘γ winds three times around the origin:

n(f∘γ,0)=3.

Facts & Assumptions

Given: The circle γ(t)=2eit and the cubic polynomial f(z)=z3−1.

[L1]

The logarithmic-derivative integral equals the winding number of the image contour, and on a null-homologous contour it also equals the zero-minus-pole count (The argument-principle integral is the winding number of the image cycle).

Verification

technique · direct
1.1givenalgebra

The three zeros of f are the cube roots of unity, so they all lie in ∣z∣<2. The function has no poles.

2.1step 1.1L1∎

Apply [L1] to the circle γ. Since γ is null-homologous in C and encloses all three zeros of f, the argument-principle count is 3. Therefore n(f∘γ,0)=3.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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