Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27
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.

FALSE: the regularized Perron envelope always attains the prescribed boundary data

Statement refuted

For every bounded plane domain and every continuous boundary datum, the regularized Perron envelope always attains the prescribed boundary values at every boundary point.

Facts & Assumptions

Given: The universal claim in the Statement refuted.

[L1]

On any bounded domain, the regularized Perron envelope is harmonic in the interior (The regularized Perron envelope is harmonic).

[L2]

On the punctured disc, the boundary datum 0 on ∣z∣=1 and 1 at the puncture has no harmonic solution (The punctured disc has an irregular boundary point and a continuous boundary datum with no harmonic solution).

Refutation

technique · direct
1.1

Let H be the regularized Perron envelope for the punctured-disc datum from [L2]. By [L1], H is harmonic on the punctured disc.

L1L2
2.1

If the universal claim were true, then H would also attain the prescribed boundary values at the puncture and on the outer circle, so it would be a harmonic solution of exactly the boundary-value problem ruled out in [L2]. Therefore the claim is false.

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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