Alphabeta Math
PropositionStatement: Literature-sourcedProof: AI-generatedSession-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.

Finite groups are amenable

Statement

Every finite group is amenable.

Facts & Assumptions

Given: A finite group G.

[L1]

A group is amenable exactly when it admits a left-invariant mean (Left-invariant means and amenable groups).

Proof

technique · direct
1.1

Define m(f)=∣G∣−1∑x∈Gf(x) on ℓ∞(G). This is linear, positive, and satisfies m(1G)=1, so it is a mean.

givenconstruct
2.1

For every g∈G, the map x↦g−1x is a permutation of G, so ∑x∈Gf(g−1x)=∑x∈Gf(x). Therefore m(g⋅f)=m(f), and [L1] makes G amenable.

L1step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

3 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