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.
FALSE: a paradoxical decomposition is just an abstract partition without prescribed translates
Statement
A paradoxical decomposition is nothing more than a set-theoretic partition of a group.
Facts & Assumptions
Given: The false claim above.
A paradoxical decomposition includes specified translating group elements and translated copies covering the whole group (Paradoxical decompositions of groups).
Refutation
Partition into the even integers and the odd integers. This is an ordinary set-theoretic partition.
By [L1], a paradoxical decomposition needs finitely many specified translates whose images each cover the whole group again. The even/odd partition by itself carries no such data, so it is not a paradoxical decomposition. Thus the statement is false.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
2 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. Löh, Geometric Group Theory: An Introduction (2015 course version) (standard reference, not scraped)