CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck 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 $α = (0 1)$ and $β = (1 2)$ 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, $α β (0) = α (0) = 1$ , whereas $β α (0) = β (1) = 2$ .

givenL1

2.1

Therefore $α β \neq β α$ . 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 · two levels

6 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.