Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30
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: Runge's theorem gives polynomial approximation on every compact set

Statement

False claim: Every compact set in C admits uniform polynomial approximation for every function holomorphic on a neighbourhood of that set.

Facts & Assumptions

Given: The unit circle and the function 1/z on it.

[L1]

The function 1/z on the unit circle is not uniformly approximable there by polynomials (1/z is not uniformly approximable by polynomials on the unit circle).

Refutation

technique · direct
1.1

The unit circle is compact, and 1/z is holomorphic on a neighbourhood of it.

given
2.1

If the displayed claim were true, then 1/z would be uniformly approximable on that compact set by polynomials. This contradicts [L1].

step 1.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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