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False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-29‡ rests on unproved material
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.

‡ Rests on 1 statement not proved in this library. Every dependency marked ‡ below is recorded with a citation but is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

FALSE: recursively enumerable trivial words already give a decision algorithm

Statement

If the trivial words of a presentation form a recursively enumerable language, then the word problem for that presentation is solvable.

Facts & Assumptions

Given: A recursively presented group.

[L1]

In a recursively presented group, the trivial words form a recursively enumerable language. (The trivial words of a recursively presented group form a recursively enumerable language)

[L2]

Every finite presentation is recursively presented. (Recursive presentations and finite presentations of groups)

‡ [L3]

The Novikov-Boone theorem gives a finitely presented group with unsolvable word problem. (Novikov-Boone: some finitely presented group has unsolvable word problem ‡)

Refutation

technique · direct
1.1

By [L1], every recursively presented group has a semidecision procedure that halts on trivial words and may run forever on nontrivial words.

L1given
2.1

By [L3], there exists a finitely presented group with unsolvable word problem; by [L2], that group is recursively presented, so step 1.1 applies to it.

L2‡ L3given
3.1

The group from step 2.1 has recursively enumerable trivial words by step 1.1, but its word problem is not solvable. Therefore recursively enumerable positive instances do not force a decision algorithm.

step 1.1step 2.1
4.1

Therefore the statement is false.

step 3.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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