Alphabeta Math
PropositionStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27
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.

Finite groups and free groups are hyperbolic

Statement

Every finite group and every finitely generated free group is hyperbolic.

Facts & Assumptions

Given: Either a finite group G with a finite generating set S, or a finitely generated free group F(X) with free basis X.

[L1]

Cayley trees are 0-hyperbolic (Cayley trees are 0-hyperbolic).

[L2]

The Cayley graph of a free group with respect to a free basis is a tree (The Cayley graph of a free group with respect to a free basis is a tree).

Proof

technique · direct
1.1

If G is finite, then its Cayley graph has finite diameter. Every geodesic triangle in a finite-diameter space is diam⁡(Γ(G,S))-slim, so G is hyperbolic.

givenalgebra
2.1

If F(X) is free, then [L2] says that its Cayley graph is a tree, and [L1] therefore makes it 0-hyperbolic. Hence F(X) is hyperbolic.

L1L2∎

Depends on

Used by

Dependency tree · two levels

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Sources