ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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 real parts of z^n are harmonic polynomials

Example

For every integer $n \geq 1$ , the real part of $z^{n}$ is a harmonic polynomial on $C$ . For instance,

$Re (z) = x, Re (z^{2}) = x^{2} - y^{2}, Re (z^{3}) = x^{3} - 3 x y^{2} .$

Facts & Assumptions

Given: An integer $n \geq 1$ .

[L1]

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

[L2]

The $C^{2}$ real and imaginary parts of a holomorphic function are harmonic (The $C^{2}$ real and imaginary parts of a holomorphic function satisfy Laplace's equation and form a harmonic-conjugate pair).

Verification

technique · direct

1.1

The polynomial $f (z) = z^{n}$ is entire by [L1], hence holomorphic on all of $C$ .

2.1

Its real and imaginary parts are polynomials in $x$ and $y$ , so they are of class $C^{\infty}$ and in particular $C^{2}$ ; [L2] therefore makes $Re (z^{n})$ harmonic.

step 1.1L2∎

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.