False statementConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 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.

FALSE: a proper isometric action has bounded orbits

Statement

A proper isometric action has bounded orbits.

Facts & Assumptions

Given: The translation action of $Z$ on $R$ from The integers act geometrically on the real line.

[L1]

That action is geometric, hence proper, and its orbit through $0$ is the unbounded subset $Z \subseteq R$ (The integers act geometrically on the real line).

Refutation

technique · direct

1.1

By [L1], the action is proper.

1.2

The orbit of $0$ is $Z$ , which is unbounded in $R$ . So the conclusion of the statement fails.

2.1

Steps 1.1 and 1.2 refute the statement.

step 1.1step 1.2∎

Depends on

The integers act geometrically on the real line

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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)