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 members of an object form an abelian group
Statement
For every object of an abelian category, addition of representatives induces an abelian-group operation on its members modulo equivalence.
Facts & Assumptions
Given: The group-law claim of the statement.
The subtraction surrogate gives only an existence statement for a witness , not a binary operation on all member classes (The subtraction surrogate).
There is an explicit object whose members do not support such a group law (The members of an object do not form a group).
Refutation
The counterexample [L2] shows that the claimed group structure fails even in . So the statement is false.
This does not contradict [L1]: the subtraction surrogate is weaker than an additive law on member classes.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
8 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
- Saunders Mac Lane, Categories for the Working Mathematician, VIII.4 (standard reference, not scraped)