CounterexampleConstruction: 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.

A set lying in both Sigma_n^0 and Pi_n^0

Statement refuted

The assertion that $Σ_{n}^{0} \cap Π_{n}^{0}$ is empty for positive $n$ .

Facts & Assumptions

Given: a positive finite level $n$ .

Counterexample

technique · direct

1.1givenconstruct

The set $E$ of even natural numbers has the decidable predicate $x = 2 q$ for some $q$ , and its complement has the decidable predicate $x = 2 q + 1$ for some $q$ .

2.1step 1.1∎

Therefore $E$ is $Δ_{1}^{0}$ and is in both $Σ_{n}^{0}$ and $Π_{n}^{0}$ after padding for every $n \geq 1$ . It witnesses overlap, not equality of the two classes.

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

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