ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28

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 bi-infinite line and its translation action

Example

Let $L$ have vertex set $Z$ and one geometric edge between $n$ and $n + 1$ for each $n \in Z$ . Then $L$ is a simplicial tree, and translation $τ (n) = n + 1$ is a hyperbolic automorphism with axis $L$ itself and translation length $1$ .

Facts & Assumptions

Given: The bi-infinite line $L$ and the translation $τ$ .

[L1]

Translation length is the minimum displacement on the vertex set. (The translation length of a tree automorphism without inversions)

[L2]

A hyperbolic automorphism preserves a unique axis and translates along it by its translation length. (Tree automorphisms without inversions are either elliptic or hyperbolic)

Verification

technique · direct

1.1

Every two integers are joined in $L$ by the obvious consecutive edge path, and no reduced closed path exists, so $L$ is a simplicial tree. The map $τ (n) = n + 1$ preserves adjacency and has no fixed vertex.

L2given

2.1

Every vertex moves distance $1$ , so [L1] gives $ℓ (τ) = 1$ . Therefore [L2] places $τ$ in the hyperbolic case, with axis equal to the whole line $L$ .

L1L2step 1.1algebra∎

Depends on

Used by

A quotient of a tree can have cycles Counterexample

Dependency tree · two levels

6 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

Jean-Pierre Serre, Trees (standard reference, not scraped)