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: an inverse limit of groups can be empty
Statement
An inverse limit of groups can be empty.
Facts & Assumptions
Given: An inverse system of groups.
The inverse limit is the set of compatible tuples, and compatible tuples form a subgroup of the product group (The inverse limit is the set of compatible tuples in the Cartesian product, Compatible tuples form a subgroup of the product group).
Refutation
The identity tuple of the product is compatible, because every transition homomorphism preserves identity.
Therefore the inverse limit contains at least that identity tuple. So by [L1] it is never empty.
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
- Brian Osserman, Math 6112 notes on inverse limits and profinite groups (standard reference, not scraped)
- H. W. Lenstra, Profinite groups and Galois groups (standard reference, not scraped)