Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 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.

Elliptic and hyperbolic automorphisms on the line

Example

On the bi-infinite line, reflection r(n)=−n is elliptic and translation τ2(n)=n+2 is hyperbolic.

Facts & Assumptions

Given: The reflection r and translation τ2 of the bi-infinite line.

[L1]

Tree automorphisms without inversions are elliptic exactly when they fix a vertex, and hyperbolic exactly when they preserve a translation axis with positive translation length. (Tree automorphisms without inversions are either elliptic or hyperbolic)

Verification

technique · direct
1.1

The reflection r fixes the vertex 0, so [L1] places it in the elliptic case.

L1given
2.1

The translation τ2 has no fixed vertex and preserves the whole line while moving every vertex two steps, so [L1] places it in the hyperbolic case.

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