Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 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: an unsolvable word problem means no individual word can be decided

Statement

If a finitely presented group has unsolvable word problem, then no individual word in that group can ever be proved trivial or nontrivial.

Facts & Assumptions

Given: A finitely presented group with unsolvable word problem.

‡ [L1]

Some finitely presented group has unsolvable word problem. (Novikov-Boone: some finitely presented group has unsolvable word problem ‡)

Refutation

technique · direct
1.1

An unsolvable word problem means that no single algorithm decides triviality for all input words in that fixed group.

‡ L1given
2.1

That does not prevent a particular word from being settled by an ad hoc calculation, a special normal form, or a direct proof. So failure of one uniform algorithm is not the claim that every individual instance is forever inaccessible.

step 1.1algebra
3.1

Therefore the statement is false.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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