Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26
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: the Cayley graph of a group is independent of the chosen generating set

Statement refuted

the Cayley graph of a group is independent of the chosen generating set.

Facts & Assumptions

Given: The proposed claim together with the witness named in the Statement refuted.

[F1]

The Cayley graph of a group G with respect to a subset S has vertex set G and edge set {{g,gs}:g∈G, s∈(S∪S−1)∖{e}} (The Cayley graph of a group with respect to a subset).

[L1]

Every vertex of a Cayley graph has the same degree, and the graph is locally finite exactly when the symmetrised generating set is finite (Cayley-graph neighbourhoods are equipotent, and local finiteness is equivalent to finiteness of the symmetrised subset).

Refutation

technique · contradiction
1.1

The claim asserts that the isomorphism type of the Cayley graph depends only on the group.

F1assume-contra
2.1

For the integers the generating sets {1} and {2,3} give graphs of degree two and four, so the claim fails.

F1L1step 1.1discharge-contradiction∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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