CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-02

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.

The complement of the halting problem is not recognizable

Statement

The complement of $H A L T_{T M}$ is not recognizable.

Facts & Assumptions

Given: The halting language $H A L T_{T M}$ .

[L1]

The language $H A L T_{T M}$ is recognizable and undecidable, by The halting problem is recognizable and undecidable.

[L2]

A language is decidable if and only if both it and its complement are recognizable, by A language is decidable if and only if it and its complement are recognizable.

Proof

technique · direct

1.1

Suppose for contradiction that the complement of $H A L T_{T M}$ were recognizable. Then both $H A L T_{T M}$ and its complement would be recognizable, by [L1].

assume-contraL1

2.1

Applying [L2] to step 1.1 would make $H A L T_{T M}$ decidable. That contradicts the undecidability clause of [L1].

L1L2step 1.1contradiction

3.1

Therefore the complement of $H A L T_{T M}$ is not recognizable.

step 2.1discharge-contradiction: simultaneous recognizability would decide $HALT_{TM}$∎

Depends on

Used by

Halting on every input is not recognizable Theorem
There exists a language that is not recognizable Theorem

Dependency tree · two levels

10 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

Stan Nelson, Computability: Undecidable Problems (standard reference, not scraped)
John Watrous, Introduction to the Theory of Computing, Lecture 17: More undecidable languages; reductions (standard reference, not scraped)