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 tree automorphism fixes a vertex

Statement

Every automorphism of a simplicial tree fixes a vertex.

Facts & Assumptions

Given: The elliptic/hyperbolic dichotomy.

[L1]

A tree automorphism without inversions is either elliptic with a fixed vertex or hyperbolic with a translation axis and positive translation length. (Tree automorphisms without inversions are either elliptic or hyperbolic)

Refutation

technique · direct
1.1

Translation by n↦n+1 on the bi-infinite line is hyperbolic: it preserves the whole line and moves every vertex a positive distance. This is the second case of [L1].

L1given
2.1

A hyperbolic automorphism has no fixed vertex, so the translation from step 1.1 is a counterexample. Hence the statement is false.

L1step 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