ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Affine functions and mixed quadratic monomials are harmonic

Example

Let $n \geq 1$ , $a \in R$ , and $b \in R^{n}$ . Every affine function $x \mapsto a + b \cdot x$ is harmonic on $R^{n}$ . If $n \geq 2$ and $i, j \in {0, \dots, n - 1}$ are distinct, then $x \mapsto x_{i} x_{j}$ is harmonic on $R^{n}$ .

Verification

Given: the displayed dimension, coefficients, coordinate indices, and the classical Laplacian The Laplacian of a $C^{2}$ function and of a $C^{2}$ vector field.

1.1

All second partial derivatives of $a + b \cdot x$ vanish [given].

2.1

For $i \neq j$ , each diagonal second derivative of $x_{i} x_{j}$ vanishes [given]. ∎

Depends on

The Laplacian of a $C^2$ function and of a $C^2$ vector field

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

PDE source treatment (standard reference, not scraped)