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: the fundamental group of a graph of groups is a topological fundamental group by definition
Statement
The fundamental group of a graph of groups is, by definition, a topological fundamental group of a space.
Facts & Assumptions
Given: The algebraic definition of the graph-of-groups fundamental group.
The graph-of-groups fundamental group is defined as a quotient of the path group by killing tree edges. (The fundamental group of a graph of groups relative to a maximal tree)
Refutation
By [L1], the definition is algebraic: it starts from generators, edge relations, and a quotient by tree-edge relations.
A topological realization may exist later, but it is not part of the definition recorded in step 1.1. Therefore the statement is false.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
3 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)