ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 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.

Z^n acts geometrically on Euclidean n-space

Example

For $n \geq 1$ , the group $Z^{n}$ acts geometrically on $R^{n}$ by integer translations $m \cdot x : = x + m .$ Hence $Z^{n}$ is quasi-isometric to Euclidean $n$ -space.

Facts & Assumptions

Given: The Euclidean metric $d_{2}$ on $R^{n}$ and the translation action of $Z^{n}$ on $R^{n}$ .

[L1]

The Euclidean distance $d_{2}$ is a metric on $R^{n}$ ( $R^{n}$ as the set of functions $n \to R$ , and $d_{1}$ , $d_{2}$ , $d_{\infty}$ are metrics on it).

[L2]

A geometric action is isometric, proper, and cobounded (Geometric actions on a metric space).

[L3]

Under a geometric action on a geodesic metric space, every orbit map is a quasi-isometry (The Svarc-Milnor lemma).

Verification

technique · direct

1.1

Translations preserve Euclidean distance, so the action is isometric. If bounded sets $B, C \subseteq R^{n}$ meet after translation by $m \in Z^{n}$ , then each coordinate of $m$ lies in a bounded interval, so only finitely many such integer vectors occur. Also every point of $R^{n}$ lies within Euclidean distance at most $\sqrt{n}$ of some integer lattice point, so the action is cobounded. Hence the action is geometric by [L2].

L1L2algebra

2.1

Euclidean space is geodesic, and step 1.1 gives a geometric action. Therefore [L3] makes the orbit map $m \mapsto m \cdot 0 = m$ a quasi-isometry from $Z^{n}$ to $R^{n}$ .

L1L3step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

26 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

C. Löh, Geometric Group Theory, Sections 4.4 and 5.1 (standard reference, not scraped)
C. Drutu and M. Kapovich, Lectures on Geometric Group Theory, Chapter 5 (standard reference, not scraped)