ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-29

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.

Jensen's formula for a polynomial

Example

For $f (z) = 1 - z$ , Jensen's formula on any disc $∣ z ∣ \leq R$ with $R > 1$ reads

$0 = \frac{1}{2 π} \int_{0}^{2 π} \log ∣ 1 - R e^{i t} ∣ d t - \log R,$

because $f (0) = 1$ and the only zero inside the disc is $a_{1} = 1$ .

Facts & Assumptions

Given: The polynomial $f (z) = 1 - z$ and a radius $R > 1$ .

[F1]

Jensen's formula expresses the boundary mean of $\log ∣ f ∣$ in terms of $f (0)$ and the interior zeros (Jensen's formula on a disc).

Verification

1.1

The function satisfies $f (0) = 1 \neq 0$ , and its unique zero is $a_{1} = 1$ , which lies in $∣ z ∣ < R$ because $R > 1$ .

givenalgebra

2.1

Applying [F1] with this single zero gives $0 = \frac{1}{2 π} \int_{0}^{2 π} \log ∣ 1 - R e^{i t} ∣ d t - \log R$ , equivalently $\frac{1}{2 π} \int_{0}^{2 π} \log ∣ 1 - R e^{i t} ∣ d t = \log R$ .

F1step 1.1algebra∎

Depends on

Jensen's formula on a disc

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

Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 5 §1 (standard reference, not scraped)