ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck 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 $γ (t) = 2 e^{i t}$ for $0 \leq t \leq 2 π$ and let $f (z) = z^{3} - 1$ . Then the image contour $f \circ γ$ winds three times around the origin:

$n (f \circ γ, 0) = 3.$

Facts & Assumptions

Given: The circle $γ (t) = 2 e^{i t}$ and the cubic polynomial $f (z) = z^{3} - 1$ .

[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

1.1

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

givenalgebra

2.1

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 $n (f \circ γ, 0) = 3$ .

step 1.1L1∎

Depends on

The argument-principle integral is the winding number of the image cycle

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

R. W. Howell and J. H. Mathews, Complex Analysis, §8.7 (standard reference, not scraped)