CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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.

Overlapping cycles need not commute

Statement refuted

False claim. Any two cycles commute, even when their supports overlap.

Facts & Assumptions

Given: The cycles $\alpha=(0\,1)$ and $\beta=(1\,2)$ in $S_3$ .

[L1]

Cycles with disjoint supports commute; disjointness is a hypothesis of that result (Cycles with disjoint supports commute).

Counterexample

technique · direct

1.1

With the right-hand factor acting first, $\alpha\beta(0)=\alpha(0)=1$ , whereas $\beta\alpha(0)=\beta(1)=2$ .

givenL1

2.1

Therefore $\alpha\beta\ne\beta\alpha$ . Their supports overlap at $1$ , so this explicit pair refutes the claim and shows why the disjoint-support hypothesis in [L1] is necessary.

step 1.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 10 results over 7 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.