False statementConstruction: AI-adaptedVerification: AI-adaptedSession-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.

FALSE: every word metric is invariant under right translation

Statement refuted

every word metric is invariant under right translation.

Facts & Assumptions

Given: The proposed claim together with the witness named in the Statement refuted.

[F1]

The word metric of $G$ with respect to $S$ is $d_{S} (g, h) = ∣ g^{- 1} h ∣_{S}$ (The word metric of a group with respect to a generating set).

[L1]

The word metric is a left-invariant metric and coincides with the path metric of the Cayley graph (The word metric is a left-invariant metric and coincides with the path metric of the Cayley graph).

[L2]

Right translation by a fixed element displaces every point of a word metric space by exactly the word length of that element (Right translation by a fixed element displaces every point of a word metric space by exactly the word length of that element).

Refutation

technique · contradiction

1.1

The claim asserts $d_{S} (g k, h k) = d_{S} (g, h)$ for all $g, h, k$ .

F1L1assume-contra

2.1

In the free group on two generators with its free basis, take $g = e$ , $h = a$ and $k = b$ : the left sides differ, because $d_{S} (b, a b) = ∣ b^{- 1} a b ∣_{S} = 3$ while $d_{S} (e, a) = 1$ .

F1L2step 1.1discharge-contradiction∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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. Loh, Geometric Group Theory: An Introduction (2015 course version), 264 pp. (standard reference, not scraped)
C. Drutu and M. Kapovich, Geometric Group Theory (with an appendix by B. Nica), 837 pp. (standard reference, not scraped)