Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: 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.

Regular Cayley trees of free groups

Example

If Fr is a free group with free basis S of size r, then the undirected Cayley graph of (Fr,S) is a tree and every vertex has valence 2r. Orienting each geometric edge in both directions turns it into a simplicial tree.

Facts & Assumptions

Given: A free group Fr with free basis S.

[L1]

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)

[L2]

On finite pieces, the simplicial-tree notion matches the published tree notion. (For finite graphs, the simplicial-tree notion agrees with the published finite-tree notion)

Verification

technique · direct
1.1

By [L1], the underlying simple Cayley graph is a tree. Each vertex has one edge labelled by each basis element and by its inverse, so its valence is 2r.

L1given
2.1

Replacing every geometric edge by the two corresponding orientations does not create a cycle; it only records both directions explicitly. Thus the same graph becomes a simplicial tree, in agreement with the finite bridge principle [L2].

L1L2step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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