Alphabeta Math
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.

Distributional harmonicity removes an apparent interior corner

Example

Let n≥1 and define v:Rn→R by v(x)=∣x1∣. This apparent corner is not distributionally harmonic on Rn; indeed Δv=2δ{x1=0}. Thus a distributionally harmonic locally integrable function has a unique smooth harmonic representative; in particular, the actual corner ∣x1∣ cannot be distributionally harmonic on any open set meeting the hyperplane {x1=0}.

Verification

Given: n≥1, the domain Rn, and the distributional derivative convention Distributional harmonicity and Poisson's equation on an open subset of Rn.

1.1

In one variable, integrating by parts twice gives (∣t∣)′′=2δ0; tensoring with the remaining variables gives the stated hyperplane term [given].

2.1

Conversely Weyl's lemma for the Laplacian gives every distributionally harmonic distribution a smooth representative [step 1.1]. ∎

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Sources