CounterexampleConstruction: 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.

A harmonic function can vanish on a line without being zero everywhere

Statement refuted

Refuted claim: if a harmonic function vanishes on a line segment, then it vanishes identically on its domain.

The witness is

$u (x, y) = x .$

It is harmonic on $C$ , vanishes on the whole $y$ -axis, and is not the zero function.

Facts & Assumptions

Given: The function $u (x, y) = x$ .

Counterexample

technique · direct

1.1

One has $u_{x x} = 0$ and $u_{y y} = 0$ , so $u_{x x} + u_{y y} = 0$ and $u$ is harmonic on $C$ by Plane harmonic functions.

algebra

2.1

On the line $x = 0$ , the function $u$ vanishes identically, but $u (1, 0) = 1 \neq 0$ . So vanishing on a line does not force vanishing everywhere.

step 1.1algebra∎

Depends on

Plane harmonic functions

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

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