Alphabeta Math
CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11
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.

In ⟨a,b∣ab⟩\langle a,b\mid ab\rangle, the trivial word baba is stuck under free cancellation and delete-only relator rewriting

Statement refuted

Consider the following specific syntactic procedure for a displayed finite presentation: at each step, freely cancel an adjacent inverse pair or delete a contiguous occurrence of one displayed relator or its inverse. Never insert a relator and never lengthen the word.

The false claim is that this delete-only procedure reduces every word that represents the identity to the empty word. In ⟨a,b∣ab⟩\langle a,b\mid ab\rangle, the word baba represents the identity but admits no step at all.

Facts & Assumptions

Given: The presentation G=⟨a,b∣ab⟩G=\langle a,b\mid ab\rangle and the delete-only procedure just stated.

[F1]

In a presented group, every displayed relator becomes the identity (Group presentation by generators and relations).

[F2]

In a group, an equation yx=eyx=e determines y=x−1y=x^{-1} (Group and abelian group).

Counterexample

technique · direct
1.1

The relation ab=eab=e gives b=a−1b=a^{-1} by [F2], and therefore ba=a−1a=eba=a^{-1}a=e in GG.

F1F2
1.2

The word baba has no adjacent inverse pair and contains neither the relator abab nor its inverse b−1a−1b^{-1}a^{-1} as a contiguous subword.

given
2.1

Thus baba represents the identity by step 1.1 but is stuck and nonempty under the stated procedure by step 1.2, refuting the claim.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 19 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.