Alphabeta Math
CounterexampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge 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.

Hopf conclusion needs a strict nonconstant extremum

Statement refuted

For n≥2, a boundary maximum and an interior sphere alone do not force a strictly positive outward derivative. On B‾1(0), the harmonic function u≡0 attains its maximum at every boundary point and has outward derivative zero there.

Facts & Assumptions

Given: The objects and hypotheses in the statement refuted.

[F1]

Hopf assumes a strict interior inequality below the boundary maximum as well as an interior tangent ball and the specified continuity and derivative conditions. (Hopf boundary point lemma for the laplacian).

Counterexample

technique · direct
1.1

The function is smooth with zero Laplacian and zero boundary values. At each p∈∂B1, the ball itself is an interior tangent ball and its outward direction is p.

givenalgebra
2.1

For 0<t<2, (u(p)−u(p−tp))/t=0, so the outward derivative is zero. The strict condition u(x)<u(p) inside the domain, required by Hopf, fails; all its other stated conditions are satisfied.

F1step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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