Alphabeta MathHow 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.
A quotient of a tree can have cycles
Statement refuted
The quotient of a simplicial tree by an action without inversions is always a tree.
Facts & Assumptions
Given: The quotient-graph definition and the translation action on the line.
A quotient graph identifies vertices and edges only up to orbit. (The quotient graph of an action without inversions)
Translation by one step on the bi-infinite line is a hyperbolic action on a simplicial tree. (The bi-infinite line and its translation action)
Counterexample
By [L2], the bi-infinite line is a simplicial tree. Let be translation by three steps, so acts without inversions on that tree.
In the quotient graph from [L1], the three vertex orbits are the residue classes modulo , and the edge orbits join them cyclically. The quotient is therefore a -cycle, not a tree. This refutes the statement.
Depends on
Used by
Nothing in the library uses this result yet.
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
- Yuriy Tumarkin, Groups Acting on Trees (standard reference, not scraped)