ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck 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.

The integers act geometrically on the real line

Example

The group $Z$ acts geometrically on the real line $R$ by integer translations $n \cdot x : = x + n .$ Consequently $Z$ is quasi-isometric to $R$ .

Facts & Assumptions

Given: The usual metric $d (x, y) = ∣ x - y ∣$ on $R$ and the translation action $n \cdot x : = x + n$ of $Z$ on $R$ .

[L1]

The absolute-value metric makes $R$ a metric space (The absolute value makes $R$ a metric space: $d (x, y) = ∣ x - y ∣$ is a metric, its open balls are the intervals $(x - r, x + r)$ , and it is unbounded).

[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

1.1L1L2algebra

Translations preserve absolute-value distance, so the action is isometric. If bounded sets $B, C \subseteq R$ are contained in intervals of lengths $M_{B}, M_{C}$ , then only finitely many integers $n$ can make $(B + n) \cap C \neq \emptyset$ . Also every real number lies within distance at most $1$ of some integer, so the action is cobounded. Thus the action is geometric by [L2].

2.1L1L3step 1.1∎

The real line is geodesic under its usual metric, and step 1.1 gives a geometric action. Hence [L3] makes the orbit map $n \mapsto n \cdot 0 = n$ a quasi-isometry from $Z$ to $R$ .

Depends on

Used by

FALSE: a proper isometric action has bounded orbits False statement

Dependency tree · two levels

21 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)