LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 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 nonempty fixed-vertex set of a tree automorphism is a subtree

Statement

Let $g$ be an automorphism of a simplicial tree $T$ . If $g$ fixes at least one vertex, then the set of fixed vertices of $g$ is a subtree of $T$ .

Facts & Assumptions

Given: An automorphism $g$ of a simplicial tree $T$ with a fixed vertex.

[L1]

The fixed subtree of a subgroup is the subtree spanned by its fixed vertices, when those fixed vertices are nonempty. (Fixed subtrees and minimal invariant subtrees)

[L2]

Between any two vertices of a simplicial tree there is a unique reduced path. (A simplicial graph is a tree exactly when every two vertices are joined by a unique reduced path)

Proof

technique · direct

1.1

Let $u$ and $v$ be fixed vertices of $g$ . By [L2] there is a unique reduced path $P$ from $u$ to $v$ . Applying $g$ to $P$ gives another reduced path from $u$ to $v$ , so uniqueness in [L2] forces $g (P) = P$ .

L2given

2.1

Every vertex on $P$ is therefore fixed by $g$ , because $g$ preserves the ordered path and fixes its endpoints. Hence the fixed vertices are closed under the unique geodesic between any two of them, which is exactly the subtree condition described in [L1].

L1L2step 1.1algebra∎

Depends on

Used by

Tree automorphisms without inversions are either elliptic or hyperbolic Theorem

Dependency tree · two levels

5 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)