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 kernel row of a morphism of short exact sequences is short exact
Statement
For every morphism of short exact sequences in an abelian category, the induced kernel row is short exact.
Facts & Assumptions
Given: The universal short-exactness claim of the statement.
The general positive theorem gives exactness only at the first two nodes (The kernel row and cokernel row of a morphism of short exact sequences are exact at two nodes each).
The multiplication-by-two diagram gives a failure of short exactness (The kernel row of a morphism of short exact sequences need not be short exact).
Refutation
The witness in [L2] is a morphism of short exact sequences whose kernel row is not short exact. Therefore the universal statement fails.
Item [L1] records the correct surviving assertion: two exact nodes, not a short exact row.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
7 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)