False statementConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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: Sigma_n^0 and Pi_n^0 are disjoint

Statement

For every positive $n$ , the classes $Σ_{n}^{0}$ and $Π_{n}^{0}$ are disjoint.

Facts & Assumptions

Given: a positive finite $n$ .

Refutation

technique · direct

1.1

The decidable set $E = {2 k : k \in N}$ is $Δ_{1}^{0}$ , hence is both $Σ_{1}^{0}$ and $Π_{1}^{0}$ .

givenconstruct

2.1

Prefix padding puts both descriptions of $E$ at every positive level $n$ . Thus $E \in Σ_{n}^{0} \cap Π_{n}^{0}$ , refuting disjointness.

step 1.1∎

Depends on

Used by

A set lying in both Sigmaₙ⁰ and Piₙ⁰ Counterexample

Dependency tree · two levels

5 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

Ludovic Patey, Computability Theory, §5.2 (standard reference, not scraped)