Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28
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: HNN normal form is canonical without choosing transversals

Statement

The HNN normal form of an element is canonical even before any transversal choice is made.

Facts & Assumptions

Given: An HNN extension with chosen associated subgroup 2Z≤Z and identity isomorphism.

[L1]

HNN normal form is defined relative to explicit transversal data. (The transversal data used for HNN normal forms)

[L2]

Uniqueness holds only after those transversals are fixed. (Normal forms in an HNN extension are unique relative to chosen transversals)

Refutation

technique · direct
1.1

Let A=⟨a⟩≅Z, let C−=C+=⟨a2⟩, and let ϕ=id⁡. With transversal S={e,a}, the element ta is already in normal form. With transversal S′={e,a−1} for the same odd coset, write a=a2a−1 and use the relation ta2=a2t to obtain the different normal form a2ta−1.

L1givenalgebra
2.1

Both written words represent the same group element, but they are distinct as normal forms until the transversal choice is fixed. So the uniqueness clause of [L2] is relative, not canonical without data, and the statement is false.

L1L2step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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