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.

The polynomial z^5 + 3z + 1 has one zero in the unit disc

Example

The polynomial

$p (z) = z^{5} + 3 z + 1$

has exactly one zero in the unit disc.

Facts & Assumptions

Given: The polynomial $p (z) = z^{5} + 3 z + 1$ and the unit circle $∣ z ∣ = 1$ .

[L1]

Rouché's theorem preserves the zero count when the strict boundary inequality holds (Rouche's theorem in the classical strict-inequality form).

Verification

technique · direct

1.1

On $∣ z ∣ = 1$ , $∣ z^{5} + 1 ∣ \leq ∣ z ∣^{5} + 1 = 2 < 3 = ∣ 3 z ∣ .$

givenalgebra

2.1

Apply [L1] with $f = p$ and $g (z) = 3 z$ . The function $3 z$ has exactly one zero in $∣ z ∣ < 1$ , so $p$ has exactly one zero there as well.

step 1.1L1∎

Depends on

Rouche's theorem in the classical strict-inequality form

Used by

Nothing in the library uses this result yet.

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

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