Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-29
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 tree action is free

Statement

Every action of a group on a tree is free.

Facts & Assumptions

Given: The Bass-Serre action for a graph of groups.

[L1]

In the Bass-Serre action, vertex stabilizers are conjugates of the chosen vertex groups. (The fundamental group acts without inversions on its Bass-Serre tree)

Refutation

technique · direct
1.1

If a graph of groups has a nontrivial vertex group Gv, then [L1] says that Gv fixes the corresponding vertex of the Bass-Serre tree.

L1given
2.1

So this tree action has a nonidentity stabilizer and is not free. Therefore the statement is false.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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