ExampleConstruction: AI-generatedVerification: Not suppliedSession-authored (Fable 5 assisted)judge 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.

A nonfree action can have 3 orbits on 4 points

Example

Let the group $C_{2} = {1, τ}$ act on

$X = {1, 2, 3, 4}$

by the transposition $τ = (1 2)$ and with $3, 4$ fixed.

The orbits are

${1, 2}, {3}, {4},$

so there are $3$ orbits. This is a nonfree action, and the naive quotient $∣ X ∣ / ∣ C_{2} ∣ = 4 / 2 = 2$ is the wrong count.

Depends on

Cauchy-Frobenius orbit counting: $|G|\,|X/G|=\sum_{g\in G}|X^g|$ for a finite group action

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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.