Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-09-12
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: recognizable means total halting

Statement

Whenever a machine recognizes a language, it halts on every input.

Facts & Assumptions

Given: The language {0}⊆{0,1}∗.

[L1]

By Decidable and recognizable languages, a recognizer only has to halt and accept on the members of its language.

Refutation

technique · direct
1.1

Let M be the machine that accepts immediately on input 0 and otherwise moves right forever without entering any halting state. Then M accepts exactly the one-word language {0}.

L1given
2.1

On every nonmember of {0}, the machine from step 1.1 diverges. So M recognizes {0} but does not halt on every input.

step 1.1
3.1

Therefore recognizability does not force total halting, and the statement is false.

step 2.1∎

Depends on

Used by

Dependency tree · two levels

3 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