LemmaStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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.

Asymptoticity of Gromov sequences is an equivalence relation

Statement

In a proper geodesic hyperbolic space, asymptoticity of Gromov sequences is an equivalence relation.

Facts & Assumptions

Given: A proper geodesic hyperbolic space $X$ with basepoint $o$ .

[L1]

Hyperbolicity is equivalent to a Gromov-product inequality up to constants (Slim triangles, the Gromov product, and the four-point condition are equivalent up to constants).

[A1]

Reflexivity and symmetry are immediate from the definition of asymptoticity.

Proof

technique · direct

1.1

By [A1], every Gromov sequence is asymptotic to itself, and if $(x_{n})$ is asymptotic to $(y_{n})$ then $(y_{n})$ is asymptotic to $(x_{n})$ .

2.1

Suppose $(x_{n})$ is asymptotic to $(y_{n})$ and $(y_{n})$ is asymptotic to $(z_{n})$ . By [L1], there is $δ \geq 0$ with $(x_{m}, z_{n})_{o} \geq \min {(x_{m}, y_{k})_{o}, (y_{k}, z_{n})_{o}} - δ$ for all $m, n, k$ . Letting the indices go to infinity shows $(x_{m}, z_{n})_{o} \to \infty$ , so $(x_{n})$ is asymptotic to $(z_{n})$ .

L1step 1.1algebra∎

Depends on

Used by

FALSE: a proposed Gromov boundary quotient needs no equivalence check False statement
The boundary topology is well defined and quasi-isometry invariant Theorem

Cited to discharge well-definedness by The Gromov boundary via asymptotic sequences.

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

Brian H. Bowditch, A course on geometric group theory, Section 5.3 (standard reference, not scraped)