DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Finite-group randomization setups

Definition

A finite-group randomization setup consists of a finite probability space $(Ω, w)$ as in Finite probability spaces, outcome weights, events, and event probabilities, a finite group $G$ acting on $Ω$ as in Left group actions, transitive actions, and faithful actions, a statistic $T : Ω \to R$ , and a family $P_{0}$ of null probability laws on $Ω$ that is invariant under the action. Invariance means that for every $P \in P_{0}$ , $g \in G$ , and $A \subseteq Ω$ , $P ({x : g \cdot x \in A}) = P (A)$ . The orbit of $x$ is $O (x) : = {g \cdot x : g \in G}$ , a finite subset of $Ω$ . The setup itself does not assert that the action is free or that distinct group elements give distinct orbit points.

Depends on

Used by

Without null invariance a group-rank p-value can be anti-conservative Counterexample
Measurable finite-group permutation setups Definition
Orbit upper-tail rank p-values Definition
A permutation orbit with tied ranks and nontrivial stabilizers Example
Exact-orbit and finite-B permutation p-value resolution Example
Exact finite-group permutation p-values and tests Theorem
Finite-group orbit-rank validity Theorem

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.

Sources

Peter B. Stark, Statistics 240 Lecture Notes, Part 8: Resampling Methods, §1.3, PDF pp. 10–11 (standard reference, not scraped)
Jesse Hemerik and Jelle Goeman, Exact testing with random permutations, Sections 2–3 (standard reference, not scraped)