ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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.

An exact polynomial bound from the boundary maximum principle

Example

For $p (z) = z^{2} - 2 z + 2$ on the closed unit disc, $\max_{∣ z ∣ \leq 1} ∣ p (z) ∣ = 5,$ and equality is attained at the boundary point $z = - 1$ .

Facts & Assumptions

Given: The polynomial $p (z) = z^{2} - 2 z + 2$ and the triangle inequality and multiplicative law for complex modulus (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[L1]

If $Ω$ is a bounded complex domain and $f$ is continuous on $\overline{Ω}$ and holomorphic on $Ω$ , then $∣ f ∣$ attains its maximum on $\partial Ω$ (Boundary maximum modulus principle on a bounded domain).

[L2]

Every complex polynomial is entire (Complex polynomials are entire with the power-rule derivative, and rational functions are holomorphic wherever their denominator is nonzero).

Verification

technique · direct

1.1givenalgebra

If $∣ z ∣ \leq 1$ , then $∣ p (z) ∣ \leq ∣ z ∣^{2} + 2 ∣ z ∣ + 2 \leq 1 + 2 + 2 = 5$ .

1.2L1L2

By [L2], $p$ is entire, so [L1] applies to the open unit disc and confirms that its maximum on the closed disc occurs on the unit circle.

2.1step 1.1step 1.2algebra∎

The point $- 1$ lies on that circle and $p (- 1) = 1 + 2 + 2 = 5$ . Together with step 1.1, this proves that the exact maximum is $5$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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

J. Lebl, Guide to Cultivating Complex Analysis, §3.3 (standard reference, not scraped)