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: translation length is always the distance from an arbitrary basepoint

Statement

For a tree automorphism without inversions and for any chosen basepoint v, one always has ℓ(g)=d(v,gv).

Facts & Assumptions

Given: The definition of translation length and the elliptic/hyperbolic dichotomy.

[L1]

Translation length is the minimum of d(v,gv) over all vertices. (The translation length of a tree automorphism without inversions)

[L2]

Elliptic automorphisms fix a vertex. (Tree automorphisms without inversions are either elliptic or hyperbolic)

Refutation

technique · direct
1.1

Reflect the bi-infinite line by r(n)=−n. The vertex 0 is fixed, so [L2] gives ℓ(r)=0 via [L1].

L1L2given
2.1

At the basepoint v=1 one has d(1,r(1))=d(1,−1)=2≠0. So ℓ(r) is not the displacement from an arbitrary basepoint, and the statement is false.

L1step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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