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: every HNN extension is an ascending HNN extension

Statement

Every HNN extension is the ascending HNN extension of some injective endomorphism of its base group.

Facts & Assumptions

Given: The definitions of a general HNN extension and of an ascending HNN extension.

[L1]

An ascending HNN extension has one associated subgroup equal to the whole base group. (Ascending HNN extensions of injective endomorphisms)

[L2]

A general HNN extension only requires two injectively embedded associated subgroups. (An HNN extension with its stable letter)

Refutation

technique · direct
1.1

Let A=Z and choose associated subgroups 2Z and 3Z with the isomorphism 2n↦3n. By [L2] this defines an HNN extension.

L2givenconstruct
2.1

Neither associated subgroup is all of A, so this example cannot satisfy the condition in [L1]. Therefore it is an HNN extension that is not ascending, and the statement is false.

L1step 1.1algebra∎

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